# NGTS and HST insights into the long period modulation in GW Librae

P. Chote1, B. T. Gänsicke1, J. McCormac1, A. Aungwerojwit2, D. Bayliss1, M. R.
Burleigh3, S. L. Casewell3, Ph. Eigmüller4, S. Gill1, M. R. Goad3, J. J.
Hermes5, J. S. Jenkins6,7, A. S. Mukadam8, S. Poshyachinda9, L. Raynard3, D.
E. Reichart10, P. Szkody8, O. Toloza1, R. G. West1, P. J. Wheatley1
1Department of Physics, University of Warwick, Coventry CV4 7AL, United
Kingdom
2Department of Physics, Faculty of Science, Naresuan University, Phitsanulok
65000, Thailand
3Department of Physics and Astronomy, University of Leicester, Leicester, LE1
7RH, UK
4Institute of Planetary Research, German Aerospace Center, Rutherfordstrasse
2, 12489 Berlin, Germany
5Department of Astronomy, Boston University, Boston, MA 02215, USA
6Departamento de Astronomía, Universidad de Chile, Camino El Observatorio
1515, Las Condes, Santiago, Chile
7Centro de Astrofísica y Tecnologías Afines (CATA), Casilla 36-D, Santiago,
Chile
8Department of Astronomy, University of Washington, Box 351580, Seattle, WA
98195, USA
9National Astronomical Research Institute of Thailand (Public Organization),
Chiangmai, 50180, Thailand
10Department of Physics and Astronomy, University of North Carolina at Chapel
Hill, Chapel Hill, NC 27599, USA
E-mail<EMAIL_ADDRESS>
###### Abstract

Light curves of the accreting white dwarf pulsator GW Librae spanning a 7.5
month period in 2017 were obtained as part of the Next Generation Transit
Survey. This data set comprises 787 hours of photometry from 148 clear nights,
allowing the behaviour of the long (hours) and short period (20 min)
modulation signals to be tracked from night to night over a much longer
observing baseline than has been previously achieved. The long period
modulations intermittently detected in previous observations of GW Lib are
found to be a persistent feature, evolving between states with periods
$\simeq$ 83 min and 2 – 4 h on time-scales of several days. The 20 min signal
is found to have a broadly stable amplitude and frequency for the duration of
the campaign, but the previously noted phase instability is confirmed.
Ultraviolet observations obtained with the Cosmic Origin Spectrograph onboard
the Hubble Space Telescope constrain the ultraviolet-to-optical flux ratio to
$\simeq 5$ for the 4 h modulation, and $\lesssim 1$ for the 20 minute period,
with caveats introduced by non-simultaneous observations. These results add
further observational evidence that these enigmatic signals must originate
from the white dwarf, highlighting our continued gap in theoretical
understanding of the mechanisms that drive them.

###### keywords:

stars: individual: GW Librae, stars: variables: general, stars: dwarf novae,
white dwarfs

††pagerange: NGTS and HST insights into the long period modulation in GW
Librae–References

##  1 Introduction

It has now been over 20 years since coherent short-term variability was
discovered in the dwarf nova GW Librae (Warner & van Zyl, 1998), and
attributed to non-radial pulsations of the central white dwarf. This revealed
a new class of accreting white dwarf pulsators, of which more than a dozen are
now known: all residing in short-period, low accretion rate, cataclysmic
variables (CVs) (e.g. Warner & van Zyl, 1998; Woudt & Warner, 2004; Warner &
Woudt, 2004; Araujo-Betancor et al., 2005; Vanlandingham et al., 2005;
Patterson et al., 2005; Gänsicke et al., 2006; Nilsson et al., 2006; Mukadam
et al., 2007; Patterson et al., 2008; Pavlenko, 2009; Woudt & Warner, 2011;
Uthas et al., 2012; Mukadam et al., 2017). Their physical and
asteroseismological properties differ noticeably from those of the well-
studied single white dwarf pulsators with hydrogen-envelopes (DAV, ZZ Ceti
stars, Mukadam et al. 2004; Gianninas et al. 2006; Van Grootel et al. 2012)
and helium-envelopes (DBV, V777 Her stars, Beauchamp et al. 1999): the
envelopes of accreting white dwarf pulsators are hydrogen-dominated, but
enriched by non-negligible amounts of helium and trace metals from their
companion star. The H/He ratio affects the driving of pulsations, and has
hence to be considered as a third dimension, effectively establishing an
instability volume rather than the two-dimensional strips in
$T_{\mathrm{eff}}$ and $\log g$. Consequently, an additional HeII partial
ionisation zone may form that can drive pulsations at higher temperatures
compared to the ZZ Ceti stars (Townsley et al., 2004; Arras et al., 2006; Van
Grootel et al., 2015). In fact, the known accreting white dwarf pulsators span
a wide range of effective temperatures from 10500 K to above 15000 K (Szkody
et al., 2010). In addition, white dwarfs in CVs can be spun up to very short
periods (e.g. King et al., 1991; Cheng et al., 1997); $\sim$100 s for GW Lib
(van Spaandonk et al., 2010), compared with $\sim$ tens of hours for typical
isolated white dwarfs (e.g. Hermes et al., 2017), which further complicates
interpretation of the observed pulsation signals.

Observations of GW Lib obtained between 1997 and 2005 determined a binary
period of 76.78 minutes (Thorstensen et al., 2002) from H$\alpha$ radial
velocity measurements, and established the presence of three pulsation modes
near 650, 380, and 230 s visible in both optical and ultraviolet light curves
(van Zyl et al., 2000; Szkody et al., 2002; van Zyl et al., 2004; Copperwheat
et al., 2009). While these three pulsation signals were consistently detected,
they did not have the frequency stability seen in many ZZ Ceti or V777 Her
white dwarf pulsators, and the spread of periods observed between runs and the
apparent splitting of frequencies within individual long runs was interpreted
as signs of true underlying structure (van Zyl et al., 2004). In addition to
these short-period pulsation modes, Woudt & Warner (2002) discovered a quasi-
periodic 2.1 h modulation, whose origin remained unexplained.

The CVs hosting accreting white dwarf pulsators undergo dwarf novae
‘superoutbursts’ with recurrence times of tens of years, during which the
accretion disc reaches a critical density, becomes unstable, and rapidly dumps
its contents onto the surface of the white dwarf. This material compresses and
heats the envelope (Godon & Sion, 2003; Piro et al., 2005) so that its
temperature increases to a value outside the instability strip, halting
pulsations. Monitoring the white dwarf as it returns to its quiescent
temperature provides an opportunity to study the onset and evolution of
pulsations over human time-scales – a few years, as compared with $\sim
10^{8}\,$years for evolutionary cooling.

GW Lib was discovered on 1983 August 10, when it underwent a superoutburst
(Maza & Gonzalez, 1983), i.e. the white dwarf pulsations were identified after
the system had been 14 years back into quiescence. The next superoutburst was
detected on 2007 April 11 (Templeton et al., 2007), and observations during
the outburst only revealed superhumps from the accretion disc (Vican et al.
2011, hereafter VI11). The first short-period modulations were detected in
2008, though not at the periods seen prior to the outburst, but at $\simeq 19$
min and $\simeq 290$ s (Copperwheat et al. 2009, hereafter CO09), in line with
the expectations that heating of the envelope during the superoutburst affects
the driving of the pulsations. GW Lib was intensively monitored at optical and
ultraviolet wavelengths over the next decade, and presented overall a
consistent behaviour, exhibiting brightness modulations at $\simeq 20$ min and
$\simeq 300$ sec (Schwieterman et al. 2010, hereafter SC10; VI11; Bullock et
al. 2011, hereafter BU11; Szkody et al. 2012, hereafter SZ12; Chote & Sullivan
2016, hereafter CH16; Toloza et al. 2016, hereafter TO16; Szkody et al. 2016,
hereafter SZ16; Chote et al. 2017, hereafter CH17). Two puzzling facts emerged
from this vast array of data: (1) After an initial brief phase of cooling
(SZ12), the white dwarf has remained at a constant temperature since 2011, and
is still $\simeq 3000$ K hotter than in quiescence (TO16, SZ16) – contrasting
observations of similar large amplitude outburst dwarf novae which showed the
white dwarfs monotonously cooling back to their pre-outburst temperature on a
time-scale of $\simeq 5$ yr, e.g. WZ Sge (Slevinsky et al., 1999; Godon et
al., 2006) and AL Com (Szkody et al., 2003). (2) The long-period modulation
seen before the outburst was initially detected at 2.1 h in 2008 in both the
optical (CO09) and UV (BU11), but now often shows up at $\simeq 4$ h, which
led VI11 to suggest that this is the fundamental of the $\simeq 2.1$ h signal
and that this modulation originates in the accretion disc. However, TO16
demonstrated that the $\simeq 4$ h brightening event captured by ultraviolet
spectroscopy was clearly related to the heating and cooling of the white dwarf
in GW Lib. A more detailed assessment of the long-period modulations remains
particularly difficult as their time-scale is similar to that of a typical
observing run.

GW Lib was observed in the survey footprint of the Next Generation Transit
Survey (NGTS, Wheatley et al. 2018) between January and September 2017. Here
we report on the analysis of 787 h of time-series photometry obtained over 148
clear nights spanning 7.5 months. The observations and data reduction are
described in §2, before analysing the long- and short-timescale variability in
§3 and §4. Finally, §5 attempts to put these results into context against
earlier observations of GW Lib and other systems before closing in §6 with our
conclusions.

##  2 Observations and Data Reduction

###  2.1 NGTS

The NGTS facility (Wheatley et al., 2018) consists of 12 robotic 20 cm
telescopes that are housed in a common enclosure at the European Southern
Observatory (ESO) Cerro Paranal site in Chile. Each telescope observes a
$2.8\times 2.8$ degree field of view through a broad red (roughly $R+I$)
filter using a $2\text{k}\times 2\text{k}$ back-illuminated deep-depleted CCD
camera, producing a $5$ arcsec per pixel plate scale.

The NGTS survey strategy, as it operated in 2017, observed pre-selected fields
continuously with the same telescope every night while $\gtrsim 30^{\circ}$
elevation. Closed-loop auto-guiding (McCormac et al., 2013) ensured that stars
were observed using the same CCD pixels for the entire observing season. The
standard survey exposure was 10 s and it took 3 s to read out the CCD, giving
a survey cadence of 13 s.

The survey field NG1518-2518 containing GW Lib was scheduled for observations
between 2017 January 25 and 2017 September 21, but we chose to truncate the
data set on 2017 September 15 when the observability dropped below 1 h each
night. This provided a data set containing 218045 exposures from 148 nights,
corresponding to a total of 606 h integration time from 787 h time on target.

Due to issues with the shutter, the camera was unmounted twice during the
season. When the camera is remounted there is a small shift in the field
rotation and hence a new auto-guiding reference image is acquired for each
subset of nights.

GW Lib is too faint to be included in the standard NGTS analysis pipeline, so
we developed a stand-alone reduction procedure that operated on the raw
images. This allowed us to automatically extract light curves at the end of
each night and follow the behaviour of GW Lib in near-real time. The final
reduction procedure operates as follows:

Figure 1: The full 7.5 month NGTS light curve of GW Lib, smoothed with a 40
min boxcar filter to emphasize the long-period behaviour. Each fortnight is
offset vertically by 0.5 units, and the observation duty cycle for each
fortnight is listed in parentheses below the start date. The regular gaps in
the light curve are due to the day/night cycle; the gaps on February 1 – 2 and
August 11 – 15 were due to technical issues; the remaining gaps are nights
when NGTS was not operating due to cloud. Several nights are marked with
indicative periods to highlight the detection of the previously observed long
period modulations. The time windows of the HST observations are indicated
with shaded rectangles.

  1. (1)

A deep-stack reference frame is manually created for each auto-guider
reference by averaging 100 frames from the same night’s data.

  2. (2)

Accurate pixel positions are determined in each reference frame for GW Lib and
five nearby comparison stars. Comparison stars were selected based on their
brightness and lack of blending with nearby stars and confirmed to be non-
variable by inspection of their light curves.

  3. (3)

A SExtractor (Bertin & Arnouts, 1996) segmentation map is generated using the
SEP Python library (Barbary, 2016). The segmentation map is converted to a
pixel mask that identifies pixels that are suitable for calculating the
background sky level.

  4. (4)

Master bias and sky flat frames are created for each night using the afternoon
and morning calibration frames. On nights where calibration frames were not
taken or when the sky flats were affected by cloud (verified by manual
inspection) they are substituted with frames from the closest good night.

  5. (5)

Science frames are prepared by (optionally) applying the calibration frames
and then subtracting a background sky map that is calculated using the
standard SEP/SExtractor routine over the pixels defined by the reference
background mask.

  6. (6)

Aperture positions for the comparison stars are calculated by adding the auto-
guider offset (specified in the frame metadata) to the reference positions.
The apertures are then recentred to correct for the small drifts that occur
each night as the field distortion changes with airmass. The aperture for GW
Lib is placed using a blind offset from the mean comparison star position.

  7. (7)

Aperture photometry is done using one of two ways:

    1. (a)

Photometry for the long period analysis in Section 3 is extracted using the
SEP circular aperture routines with a 2.5 pixel aperture radius. This method
minimises systematic effects at the expense of increased noise in a given
measurement. This essentially white noise bins down as we combine many
measurements using a running mean.

    2. (b)

A second set of photometry is extracted omitting the bias and flat field
corrections and using a smaller 1.5 pixel aperture radius for GW Lib while
keeping the 2.5 pixel radius for the comparison stars. This extraction is used
in Section 4 where it improves the sensitivity to the short time-scale
variability at a trade-off of introducing long ($\simeq$ hours) time-scale
systematic trends.

  8. (8)

The extracted flux measurements and UTC timestamps are converted to tabular
files containing UTC and BJD${}_{\text{TDB}}$ time, relative flux, and
relative flux error using the tsreduce (Chote et al., 2014) software. These
files, along with the other data described below, are made available as
supplementary data in the online journal.

As GW Lib is bluer than most of the nearby stars, differential extinction
leads to reduced flux ratios at larger airmasses. This effect is reduced in
the NGTS data due to the redder bandpass and limited airmass range of
observations, but it can still be measured and removed following the technique
from CH16. The target / comparison flux ratio is fitted with a linear
coefficient as a function of airmass simultaneously for the entire data set.
The phase and amplitude of the long period modulations average out over many
nights, allowing the airmass effect to be extracted much more robustly than
the standard technique of fitting a low-order polynomial to each run.

###  2.2 APO & Prompt

Additional observations were obtained on 2017 March 4 using the Agile
photometer on the 3.5 m telescope at Apache Point Observatory, and on 2017 May
5 and 2017 May 6 using the 60 cm robotic Prompt8 telescope at CTIO. The APO
observations consisted of 515 30 s integrations obtained through a BG39
filter, and the Prompt observations consisted of 253 and 300 30 s integrations
obtained through a Baader Clear (C) filter.

The APO and Prompt data were reduced following our standard procedures (see
e.g. Chote et al., 2014, SZ16) to extract aperture photometry, with one
exception: the polynomial fit to remove long-period trends was omitted as
there was insufficient data to separate the effect of differential extinction
from the long-period variability intrinsic to GW Lib.

###  2.3 Hubble Space Telescope

Quasi-simultaneous Hubble Space Telescope (HST) far-ultraviolet (UV)
spectroscopy of GW Lib was obtained using the Cosmic Origins Spectrograph
(COS, Green et al. 2012) on the nights of 2017 August 31, September 6, and
September 13, near the end of the NGTS observation window. The first two
observations occurred during bad weather at Paranal, so simultaneous
observations were not possible. The weather was clear during the final run,
but GW Lib set past the NGTS elevation limit less than 2 min before the HST
observations started.

The time-tag COS data were reduced following the procedure from TO16 to create
ultraviolet light curves. Data were binned into 5 s sub-spectra, which were
then background-subtracted and the Ly$\alpha$ and OI  airglow lines at 1216
and 1302 Å were masked. Each spectrum was then binned in the spectral
direction to create individual photometric points with a 5 s cadence. We
present limited results from this data set where they provide important
context to the NGTS observations, but defer the full analysis to a future
publication (Toloza et al, in prep).

##  3 Long Period Variability

Photometry demonstrating the long-period behaviour was extracted using a 2.5
pixel aperture and all calibrations. Points more than 5 $\sigma$ from the
nightly mean (42 in total) were rejected, and then the data were smoothed with
a running 40 min mean (roughly 190 points) to reduce the scatter introduced by
noise and the 20 min signal. The resulting 34-week long light curve is
presented in Fig. 1.

The familiar signals near 83 minutes and between 2 – 4 h from earlier
observations (illustrated in CH17) are clearly visible, and we can see for the
first time these signals switching on and off, with each state lasting for 1 –
2 weeks. The larger amplitude modulations appear to grow rather haphazardly
out of the quieter states.

Figure 2: The 83 min periodicity has been seen frequently over the last
several years with a ragged multi-peaked appearance. The top panels show
examples observed with larger telescopes (from SZ12, CH16, CH17), and the
bottom panels show examples from the NGTS data set. Vertical black lines
denote the common 83 min period, and the blue curve shows the running 40 min
mean.

Fig. 2 compares the 83 min signals seen on several nights of NGTS observations
with earlier observations from larger aperture telescopes. The high-cadence
archival observations demonstrate the ragged nature of this signal, which is
smoothed out by the running mean. Similar average profiles are visible in the
NGTS light curves, and so we infer that this behaviour was present but not
resolved due to the low signal-to-noise ratio in the raw NGTS photometry. A
few nights in the smoothed NGTS data show amplitudes in excess of 10%, which
implies potentially much larger changes in the unresolved variability, more
than has been seen in earlier observations.

Figure 3: Near-simultaneous observations from NGTS and HST on the night of
2017 September 13 captured a cycle of the $\simeq 4$ h modulation, confirming
that these modulations are visible in the UV. The disparity in UV/Optical mode
amplitudes for the 20 min and 300 s periods is also visible. Note that
different scales have been used for the NGTS vs HST flux.

Fig. 3 presents the near-simultaneous optical and UV light curves on 2017
September 13. The NGTS light curve captures the rise on one of the long period
modulations, and the HST light curve captures the subsequent fall.

The full NGTS light curve shows that the brightness in successive minima of
the long period modulation is relatively constant (at the 10–20% level), so we
can estimate that the UV to optical amplitude ratio,
$A_{\text{UV}}/A_{\text{Optical}}\simeq 1.15/0.23\simeq 5$ assuming that the
same holds true in the UV, and that the modulation in the UV and optical are
in phase. This is comparable to the UV/Optical ratios measured for the
pulsation periods (e.g. Szkody et al., 2002, SZ16), reinforcing the suggestion
from TO16 that this long-period variability arises from the white dwarf rather
than the accretion disc.

##  4 Short-period signals

To improve the sensitivity to the short-period pulsations we extracted
photometry omitting the bias and flat field corrections, and using a smaller
(1.5 pixel) extraction radius for GW Lib. Reducing the aperture size removes
the sky and read noise contributed by the pixels in the wings of the PSF where
the signal is negligible. Omitting the calibration frames avoids their noise
contribution, taking advantage of the precise sub-pixel telescope guiding to
know that any per-pixel effects will remain consistent between exposures. The
stars do move on the CCD due to field distortion changes with airmass, but the
systematic trends this introduces are on a $\sim$hours time-scale, much longer
than the periods of interest (200 – 2000 seconds).

The time series data is converted to the frequency domain by normalising light
curves relative to their mean level and calculating the Discrete Fourier
Transform (DFT). Results are presented as amplitude spectra where the vertical
scale corresponds to the amplitude of an equivalent sinusoid in the time
domain data. The window function for each night is computed by taking the DFT
of a noise-free sinusoid sampled at the same times as the observations, and
shows the pattern of aliases that are associated with each ‘real’ sinusoidal
signal in the time domain data.

Fig. 4 presents an illustrative NGTS light curve alongside the APO and Prompt
light curves obtained during the NGTS campaign. The 20 min period is clearly
visible as the dominant feature in the light curves, and the NGTS DFTs compare
surprisingly well to the larger aperture telescopes in a large part due to the
much longer run lengths (up to 8.8 h per night).

Figure 4: Light curves and DFTs of the APO and Prompt8 runs, with a segment of
one NGTS run for comparison. The left column presents raw photometric
measurements as grey dots, with a blue line showing a running 5 min mean. The
$\simeq 4$ h modulation is clearly visible in the 2017 May 6 run. The right
column presents the DFTs and window functions for each night, with the 20 min
period and its harmonics marked with vertical red dashes and the 0.1% FAP
significance thresholds shown as horizontal grey lines. The vertical grey dash
on 2017 May 5 indicates a potential but non-significant detection of a 310 s
pulsation period. Figure 5: Trailed amplitude spectra computed from the NGTS
light curves. Each row represents the colour-coded amplitude of the DFT. The
Window column shows the associated window function for each night using the
same frequency scale. The FAP column indicates the amplitude at which a
feature in the DFT is deemed to be statistically significant. The vertical red
dashes indicate the location of the 20 min period and its harmonics. Periods
longer than 2000 s are omitted as they are contaminated by systematic effects
introduced by the reduction procedure.

We define an amplitude threshold for statistical significance by applying a
False Alarm Probability (FAP; e.g. Sullivan et al., 2008) test. The 20 min
signal and its first harmonic are subtracted from the run, and any residual
coherent signals are destroyed by randomly shuffling the times associated with
each measurement. The DFT is calculated covering 200 – 2000 s, and the
amplitude of the largest peak is recorded. This is repeated 1000 times to
simulate different realizations with the same white noise characteristics. The
largest peak then defines the amplitude where there is a 1 in 1000 (0.1%)
probability that a peak of the same amplitude can be produced by white noise.

Fig. 5 presents the nightly runs as a single trailed DFT. Each horizontal
slice shows the DFT, window function, and FAP threshold of the corresponding
night with amplitude represented using colour. Several partial nights that
were interrupted by bad weather (ten in total) and have a very poor window
function were excluded to aid readability.

The 20 min period is clearly visible at $\simeq$1200 s with a $\simeq 3\,\%$
amplitude throughout the entire 7.5 month observing campaign. Traces of the
first and second harmonics can be seen by eye near 590 and 390 s. The $\simeq
300\,$s periods that have been repeatedly seen in the far-ultraviolet (SZ12,
SZ16) and occasionally in the optical (CO09, SZ12, SZ16, TO16, CH16) are not
detected in the NGTS data. The historical amplitude of this signal was
typically $\simeq 0.5-2$% in the optical, and so they would only be apparent
above the $\simeq 0.7$% mean noise floor if they remained coherent over
several nights – previous observations (e.g. SZ16, CH17) have shown this is
usually not the case.

Figure 6: Amplitude and frequency/period fits to the 20 min signal in the NGTS
run (blue) and APO, Prompt, and HST (see Section 3) runs (red). The histograms
on the right show that the majority of measurements are within $3\sigma$ of
the mean 1182 s period and 2.9% amplitude.

The period and amplitude of the 20 min signal on each night were fitted with a
sinusoid in the time domain. $1\sigma$ uncertainties were bootstrapped by
resampling the light curves with replacement (Efron, 1979) to generate 5000
alternative realizations of each light curve, which were each fitted to
produce a distribution of frequency and amplitude values. The majority of
these showed the expected normal distribution, which were fitted with a
Gaussian to measure $\sigma$. Nights with $\sigma>20\,\mu$Hz (five in total)
or non-Gaussian profiles (a further nine) were excluded as they do not
robustly measure a unique fit.

Fig. 6 presents the fitted frequency/period and amplitudes, which show that
the 20 min signal remains broadly stable in both frequency and amplitude
across the full run. The phase and frequency instability shown in CH16
remains, however, as DFTs calculated over multiple nights were found to smear
power in the frequency domain in a similar way.

Fourier transforms of the HST/COS observations obtained on 2017 August 31 and
2017 September 6 are presented in Fig. 7. Both observations were dominated by
the 300 s period, which had an amplitude in the range of 2 – 4%. Previous
observations (SZ12, SZ16) have shown $A_{\text{UV}}/A_{\text{Optical}}\approx
5$, which implies an optical amplitude below 1% – consistent with the NGTS
non-detection. A more detailed analysis on the behaviour of this pulsation
mode will be presented in a future publication.

The key result from the perspective of the NGTS observations is that the 20
min signal is only weakly detected in the UV, with an amplitude < 2%
constraining $A_{\text{UV}}/A_{\text{Optical}}\lesssim 1$. These observations
were obtained during a gap in NGTS observations, so we cannot rule out that
the modulation was intrinsically weaker at the time of the HST observations,
but this seems unlikely considering that the six months of optical
measurements before the observation and the few nights after show no evidence
of significant amplitude variability.

Figure 7: DFTs of the 2017-08-31 and 2017-09-06 HST COS observations show that
the $\sim$20 min signal is only weakly detected at periods consistent with the
optical observations. The $\sim 300\,$s pulsation period dominates the
observed variability, but demonstrates poor coherancy in the second visit with
power being smeared over a range of frequencies in the DFT.

##  5 Discussion

The 83 min signal was originally thought to be associated with a late
superhump period (BU11). As it has now been regularly observed to come and go
for nearly 10 years, it is likely related to the ‘quiescent superhump’ that
has been seen to come and go in EQ Lyn (Mukadam et al., 2013) and V455 And
(Araujo-Betancor et al., 2005). In EQ Lyn, the absence of any period in the UV
while near-simultaneous optical data showed superhumps suggested that the
periodicity may come from the outer accretion disc.

The UV flux from GW Lib is dominated by the white dwarf, with the cooler
accretion disc contributing just a few percent (see e.g. SZ12) to the total.
The accretion disc is more significant at optical wavelengths (VI11 suggest up
to 40% of the total flux), diluting the amplitude of any signals from the
white dwarf. Our near-simultaneous measurements showing the amplitude of the
$\simeq 4\,$h signal as $\simeq 5$ times larger in the UV than the optical
therefore further supports the suggestion from TO16 that this signal
originates from the white dwarf. The GALEX data from BU11 showed that this
signal had a larger amplitude in FUV than NUV, pointing to the inner disc or
white dwarf, but the optical amplitude was also comparable. Unfortunate timing
of the inter-orbit gaps in the 2015 observations (SZ16) mean that we can
neither definitively detect nor rule out the presence of the 2 h signal in the
UV at that time.

The NGTS observations appear to show that the behaviour switches betweenthese
2 – 4 h and 83 min states on a time-scale of roughly a week, which suggests
that they may share a common source.

The 20 min signal is similarly enigmatic. It was not recognized as a
persistent feature in the pre-2007 outburst photometry, despite van Zyl et al.
(2004) noting a detection in their Table 2. There have been at least three
extended periods in 2008, 2012, and 2015 – 2017 where this signal has been the
dominant source of variability. It was initially suggested (BU11, VI11) that
this signal was a disc-related feature due to its poor phase coherency between
nights and its prior non-detections. In the years since, it has become clear
that it must originate on the WD, as there are no known mechanisms that could
generate signals that maintain the level of observed stability over such a
long time-scale.

Hermes et al. (2017) showed that the stable pulsations in isolated DAVs also
lose phase coherency at periods greater than 800 s, so the lack of coherency
of the 20 min signal does not exclude the pulsation mode hypothesis.

The UV/optical amplitude ratio $A_{\text{UV}}/A_{\text{Optical}}<1$ is more
problematic, as the current understanding of g-mode pulsations predict that
the pulsation amplitude should increase significantly in the UV (Robinson et
al., 1995) due to increased limb darkening reducing the geometric cancellation
effect. It is not unprecedented, however: Szkody et al. (2010) report similar
ratios for two other accreting pulsators (the 1285 s period in PQ And and the
582 and 655 s periods in REJ 1255+266); Kotak et al. (2004) shows
$A_{\text{UV}}/A_{\text{Optical}}\simeq 1$ for the 272 and 304 s modes in the
prototypical ‘stable’ DAV G117-B15A; and Kepler et al. (2000) likewise for the
141 s mode in G185-32. So, while we do not yet understand the origin of these
signals, the amplitude ratios alone are not sufficient to rule out an origin
on the white dwarf.

Saio (2019) presents a case for the variability in these accreting systems not
being g-mode pulsations at all, but rather r-mode oscillations trapped in the
H-rich layers of the WD. While they can model the shorter period modulations
in GW Lib, they are not able to explain the $\sim$4 hour modulations and do
not attempt to model the 20 minute signal.

##  6 Conclusions

Observations of GW Librae between 2017 January 25 and 2017 September 21
provide the longest continuous monitoring campaign yet obtained on this
accreting white dwarf pulsator.

These observations demonstrate that the long-period modulations in the system
appear to change between states on a time-scale of days/weeks. While the
individual modes are not strictly periodic, they do appear to have distinctive
and repeating characteristics that can be identified in archival observations
going back nearly ten years. Near-simultaneous NGTS and HST observations
confirm that the 4 h modulation seen by (TO16) in the UV has the same origin
as in the optical, and provide an estimate of the UV/Optical flux ratio of
$\simeq 5$.

The steady presence of the 20 min signal over the full observation baseline
strongly suggests that this signal originates on the WD, and is not a
transient disc phenomenon. The weak detection of this signal in UV is
difficult to explain, but has been seen before in both accreting and non-
accreting WD pulsators.

These NGTS observations add significantly to the body of observational data on
GW Lib, which document several phenomenological behaviours that so far remain
unexplained. Future advancements in explaining these observations will require
theoretical developments towards the pulsation (or other potential) mechanisms
that can drive near-coherent variability in these systems.

## Acknowledgments

Based on data collected under the NGTS project at the ESO La Silla Paranal
Observatory. The NGTS facility is operated by the consortium institutes with
support from the UK Science and Technology Facilities Council (STFC) under
projects ST/M001962/1 and ST/S002642/1. The research leading to these results
has received funding from the European Research Council under the European
Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n.
320964 (WDTracer). BG, OT and PJW received support from the UK STFC
consolidated grants ST/L000733/1 and ST/P000495/1. PS and AM acknowledge
support from HST GO-13807 and GO-14912.002-A, as well as NSF AST-1514737. AA
received support from the Program Management Unit for Human Resources &
Institutional Development, Research and Innovation grant B05F630110. JSJ
acknowledges support by FONDECYT grant 1201371 and partial support from
CONICYT project Basal AFB-170002. This work is based on observations made with
the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science
Institute, which is operated by the Association of Universities for Research
in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are
associated with program #14912. This work has made use of observations
obtained with the Apache Point Observatory (APO) 3.5-meter telescope, which is
owned and operated by the Astrophysical Research Consortium (ARC). This work
has made use of data obtained by the PROMPT-8 telescope, owned by National
Astronomical Research Institute of Thailand, and operated by the Skynet
Robotic Telescope Network.

## Data availability

The data underlying this article are available in the article and in its
online supplementary material.

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#  Jost function based on the Hartree-Fock-Bogoliubov formalism

K. Mizuyama1, N. Nhu Le2,3, T. Dieu Thuy2, T. V. Nhan Hao2,3 1 Institute of
Research and Development, Duy Tan University, Da Nang City, Vietnam
2 Faculty of Physics, University of Education, Hue University, 34 Le Loi
Street, Hue City, Vietnam
3 Center for Theoretical and Computational Physics, College of Education, Hue
University, 34 Le Loi Street, Hue City, Vietnam
corresponding author<EMAIL_ADDRESS>
###### Abstract

We formulate the Jost function formalism based on the Hartree-Fock-Bogoliubov
(HFB) theory which has been used to represent the nature of the superfluidity
of nucleus. The Jost function based on the HFB can give the analytic
representation of the S-matrix for the nucleon elastic scattering targeting on
the open-shell nucleus taking into account the pairing effect. By adopting the
Woods-Saxon potential, we show the numerical results of S-matrix poles and
their trajectories with varying the pairing strength in two cases of Fermi
energies: $\lambda=-8.0$ and $-1.0$ MeV. The total cross sections in each
cases for neutron elastic scattering are also analyzed, and we confirmed some
staggering shapes or sharp resonances originated from the effect of pairing
can be seen in the cross section.

###### pacs:

21.60.Jx, 21.10.Tg, 24.10.-i, 24.10.Ht, 25.40.Dn, 33.25.+k

##  I Introduction

The coefficient functions of the regular solution of the Schrödinger equation
for scattering introduced by Jost jost  has been known as the Jost function.
They have been used to identify the S-matrix poles for the bound states and
resonances of the system on the complex energy/momentum plane.

The S-matrix poles represent the bound states and the resonance energy and
width. The analysis of S-matrix poles has been performed on the scattering
data to investigate the properties of the potential of the target. An
interesting study is to investigate the trajectories of the S-matrix poles on
the complex energy/momentum plane shown as a function of the potential depth
Hussein . It is also possible to analyze the channel coupling effect in terms
of the S-matrix poles on the Riemann sheets defined for each channels Rod ;
Hans .

The optical potential is indispensable to a reliable extraction of nuclear
structure information from experimental data of various direct nuclear
reactions such as inelastic scattering and transfer reactions. The global
optical potentials(GOP) have been investigated for long time watson ; Bec ;
Nada ; Var ; Hama ; Cooper ; Koning ; Kunieda ; Han1 ; Perey ; Dae ; Boj ; An
; Han2 , and succeeded to reproduce the experimental cross section with high
accuracy. The GOP is given by the complex function, the imaginary part gives
the absorption of the target nucleus for the current of the incident beam. The
existence of the imaginary part of the GOP has been justified qualitatively by
the Feshbach projection theoryfeshbach .

Recently, the self-consistent particle-vibration coupling (PVC) method has
been applied to the nucleon-nucleus ($NA$) scattering mizuyama ; mizuyama2 ;
branchon ; hao . The experimental data for neutron elastic scattering on 16O
mizuyama , proton inelastic scattering on 24O mizuyama2 , neutron elastic
scattering on 40Ca, its analyzing power branchon , and neutron elastic
scattering on 16O and 208Pb mizuyama  were successfully reproduced. In the PVC
framework, the microscopic optical potential is given as the non-local complex
potential which gives the proper absorption for the reproduction of the
experimental cross section. These results are consistent with the Feshbach
projection theory, and give the quantitative justification for the existence
of the imaginary part of the GOP.

It is well known that there are many sharp peaks are observed at low-energy in
the total and elastic cross section of the $NA$-scattering. Such resonances
have been analyzed by using the R-matrix theory rmatrix , The R-matrix theory
is a phenomenological framework in which reactions of neutrons, charged
particles + nuclei are described quantum mechanically based on boundary
conditions for various channels and the scattering matrix (S-matrix) is
acquired from measurement data of cross section. Obtained resonance parameters
have been published as the nuclear data jendl ; jeff ; endf  and used for many
applications (Nuclear engineering, radiotherapy, etc.) because nuclear
resonances are very useful characteristic for the applications.

It was also shown that the sharp resonances appeared in the neutron elastic
scattering can be partially reproduced due to the coupling between the induced
neutron and the giant resonance of the target nucleus mizuyama . This is also
consistent with the Feshbach theory which justified the R-matrix theory and
its resonance formula. Very recently, the multichannel algebraic scattering
method (MCAS) which is based on the phenomenological vibrational model has
been applied to study the bound and resonance properties of the 17O and 17F
nuclei below and above the nucleon -core threshold. The narrow resonances of
the experimental data could be reproduced successfully amos .

The results of these recent progress and attempts at low energy $NA$
scattering may indicate the possibility of a new phenomenological model based
on the microscopic theory of nuclear structure and reaction. However, the
pairing correlation has not been discussed in the direction of these recent
progress.

The importance of the pairing correlation has been discussed mainly in the
nuclear structure. The pairing correlation in nuclei is very important to
explain the fundamental properties of nuclei, such as the magic number, the
separation energy and the quadrupole 1st excited state of nucleus, and so on.
In the last decades, the investigation of the pairing effect on nuclear
structure has been held based on the microscopic theoretical framework, such
as the Hartree-Fock-Bardeen-Cooper-Schrieffer (HF+BCS) hfbcs2017 , the
Hartree-Fock-Bogoliubov (HFB) hfb2010 , the Highly Truncated Diagonalization
Approach (HTDA) htda2002 ; htda2012  and the Quasiparticle Random Phase
Approximation (QRPA) hfb1 ; qrpa1 ; qrpaC ; qrpaT . Di-neutron correlation
resulting from the coherent overlap of the continuum states due to the pair
correlation at both ground state and the low-lying excitation of the neutron
drip line nuclei has been discussed qrpa1 . Also the quasiparticle resonances
of the neutron drip-line nuclei have been investigated within the framework of
HFBhamamoto ; hamamoto2 ; zhang ; kobayashi ; grasso ; fayans2 ; michel ; Pei
; zhang2 ; oba ; zhang3 ; Sand ; betan ; sand2 . The experimental cross
section of d(9Li,10Li)p was also analyzed from the point of the pair resonance
based on the HFB formalism orrigo2 .

Nevertheless, the role of the pairing correlation for the reaction is not
clear yet. The pairing correlation may have the important role for the
reaction and the channel coupling in which two neutron transfer channel is
relevant ptrf . In order clarify the role of the pairing in the reaction, it
may be necessary to investigate the pairing effect in terms of the S-matrix
poles. The aim of this work is to derive the Jost function formalism based on
the HFB formalism in order to discuss the pairing effect on $NA$-scattering in
terms of the S-matrix poles.

This paper is organized as follows. We introduce the definition of the complex
quasiparticle energy and momentum basing on the HFB equation in Sec.II. We
show the derivation of the Jost function based on the HFB equation, the
definition of the S-matrix and its unitarity in Sec.III. In Sec.IV, we
demonstrate the numerical results adopting the Woods-Saxon potential. The
poles of the S-matrix and their trajectories with varying the paring strength
are shown. The appearance of the quasiparticle resonances on the total cross
section of the neutron elastic scattering cross section is also discussed.

##  II Definitions of complex energy & momentum planes

The Hartree-Fock-Bogoliubov(HFB) equation can be represented as

$\displaystyle\left[\left(\frac{\partial^{2}}{\partial
r^{2}}-\frac{l(l+1)}{r^{2}}\right)\mbox{\boldmath$1$}-\widehat{S}_{z}\mathcal{U}_{lj}(r)\right]\phi_{lj}(r;E)$
$\displaystyle\hskip 56.9055pt=-\begin{pmatrix}k_{1}^{2}(E)&0\\\
0&k_{2}^{2}(E)\end{pmatrix}\phi_{lj}(r;E),$ (1)
$\displaystyle\mathcal{U}_{lj}(r)=\frac{2m}{\hbar^{2}}\begin{pmatrix}U_{lj}(r)&\Delta(r)\\\
\Delta(r)&-U_{lj}(r)\end{pmatrix},$ (2)
$\displaystyle\phi_{lj}(r;E)=\begin{pmatrix}\varphi_{1,lj}(r;E)\\\
\varphi_{2,lj}(r;E)\end{pmatrix},$ (3)

where $E$ is the quasi-particle energy defined as positive value, $U_{lj}(r)$
and $\Delta(r)$ are the (Hartree-Fock) mean-field potential and the pairing
potential, respectively. Where $1$ is the unit matrix, and $\widehat{S}_{z}$
is the third component of the Pauli matrix given by

$\displaystyle\widehat{S}_{z}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},$ (4)

$k_{1}(E)$ and $k_{2}(E)$ are momentum defined by

$\displaystyle k_{1}(E)$ $\displaystyle\equiv$
$\displaystyle\sqrt{\frac{2m}{\hbar^{2}}(\lambda+E)},$ (5) $\displaystyle
k_{2}(E)$ $\displaystyle\equiv$
$\displaystyle\sqrt{\frac{2m}{\hbar^{2}}(\lambda-E)},$ (6)

for upper and lower components of quasiparticle wave functions of HFB
equation, respectively. $\lambda$ is the Fermi energy defined as negative
value in the case of the bound target. On the complex-$E$ plane, $\lambda\pm
E$ can be represented as

$\displaystyle\lambda+E$ $\displaystyle=$
$\displaystyle|\lambda+E|e^{i\theta_{1}(E)},$ (7) $\displaystyle\lambda-E$
$\displaystyle=$ $\displaystyle|E-\lambda|e^{i(\theta_{2}(E)+\pi)},$ (8)

where $\theta_{1}(E)$ and $\theta_{2}(E)$ are angles defined on the
complex-$E$ plane as shown in Figs.1 and 2. The definition range of angle is
$0\leq\theta_{1,2}(E)\leq 2\pi$.

Inserting Eqs.(7) and (8) into Eqs.(5) and (6), respectively, we can obtain

$\displaystyle k_{1}(E)$ $\displaystyle=$
$\displaystyle|k_{1}(E)|e^{i\theta_{1}(E)/2}e^{in\pi}\hskip
14.22636pt(n=0,1),$ (9) $\displaystyle|k_{1}(E)|$ $\displaystyle\equiv$
$\displaystyle\sqrt{\frac{2m}{\hbar^{2}}|\lambda+E|},$ (10) $\displaystyle
k_{2}(E)$ $\displaystyle=$ $\displaystyle
i|k_{2}(E)|e^{i\theta_{2}(E)/2}e^{in\pi}\hskip 14.22636pt(n=0,1),$ (11)
$\displaystyle|k_{2}(E)|$ $\displaystyle\equiv$
$\displaystyle\sqrt{\frac{2m}{\hbar^{2}}|E-\lambda|},,$ (12)

the factor $e^{in\pi}(n=0,1)$ is due to the phase ambiguity which shows that
the complex energy plane consists of two Riemann sheets. By using $n(=0,1)$,
let us define the $(n+1)$th Riemann sheet, $E_{n+1}$. We chose the energy
dependent angles $\theta_{1,2}(E)$ have the relation as

$\displaystyle\theta_{1,2}(E)+2n\pi$
$\displaystyle\to\theta_{1,2}(E_{2})=\theta_{1,2}(E_{1})+2\pi,$ (13)

so that the first Riemann sheet has the scattering and bound states, and the
second Riemann sheet has the resonance states. Inserting this condition into
Eqs.(9) and (11), it is easy to find that $k_{1,2}(E)$ have the relation

$\displaystyle k_{1,2}(E_{2})=-k_{1,2}(E_{1}).$ (14)

and the upper/lower half of the complex momentum-$k_{1,2}$ correspond to the
first/second Riemann sheets respectively. The branch cut is set for each
$k_{1}(E)$ and $k_{2}(E)$ as shown in Fig.1, Fig.2, Fig.3 and Fig.4.

Figure 1: (Color online) Complex quasiparticle energy planes with S-matrix
poles corresponding bound states, anti-bound states, resonances and anti-
resonances. The angle $\theta_{1}$ which is used in Eq.(9) is also shown. The
branch cut corresponding Re $k_{1}$ line of scattering states is shown by the
black solid arrow on the real axis above $-\lambda$. Four quadrants of the
first Riemann sheet (a)-(d) correspond to four areas of the upper half plane
of $k_{1}$ shown as (a)-(d) in Fig.2. Four quadrants of the second Riemann
sheet (a’)-(d’) correspond to four areas of the lower half plane of $k_{1}$
shown as (a’)-(d’) in Fig.2. Figure 2: (Color online) Complex $k_{1}$-momentum
plane corresponding the complex energy plane shown in Fig.1. Corresponding
states are shown by using the same colors and symbols. The black solid arrow
on Re $k_{1}$ indicates the scattering states corresponding the branch cut on
the complex energy plane. The upper/lower half plane correspond to the
first/second Riemann sheets of the complex energy. The hyperbola shown by
dotted red curve represent the $k_{1}$-momentum with Re $E=0$. $k_{F}$ is
defined by $k_{F}=\sqrt{\frac{2m|\lambda|}{\hbar^{2}}}$. Figure 3: (Color
online) Same figure with Fig.1 but with branch cut defined by
$k_{2}$-momentum. Figure 4: (Color online) Same figure with Fig.2 but relevant
to $k_{2}$-momentum.

##  III Jost function based on the HFB formalism

In general, the double differential equation such as the Schrödinger equation
has the regular and irregular solutions. The regular solution can be
represented by the linear combination of the linear independent irregular
solutions. If the irregular solutions satisfy the out-going (or in-coming)
boundary condition, the Jost function is defined as the coefficients for the
linear combination jost .

In this section, we derive the Jost function basing on the HFB formalism. The
HFB equation has two independent solutions for regular solutions and irregular
solutions, respectively.

Applying the Green’s theorem, it is possible to describe the regular and
irregular solution in the integral equation form. The Green’s theorem for the
HFB equation is given by

$\displaystyle\left[\chi_{l}^{\mathrm{T}}(r;E)\widehat{S}_{z}\frac{\partial\phi_{lj}(r;E)}{\partial
r}-\frac{\partial\chi_{l}^{\mathrm{T}}(r;E)}{\partial
r}\widehat{S}_{z}\phi_{lj}(r;E)\right]_{r=a}^{r=b}$
$\displaystyle=\frac{2m}{\hbar^{2}}\int_{a}^{b}dr^{\prime}\chi_{l}^{\mathrm{T}}(r^{\prime};E)\mathcal{U}_{lj}(r^{\prime})\phi_{lj}(r^{\prime};E),$
(15)

where $\chi_{l}^{\mathrm{T}}(r;E)$ is the free particle wave function which
satisfies

$\displaystyle\left(\frac{\partial^{2}}{\partial
r^{2}}-\frac{l(l+1)}{r^{2}}\right)\chi_{l}(r;E)=-\begin{pmatrix}k_{1}^{2}(E)&0\\\
0&k_{2}^{2}(E)\end{pmatrix}\chi_{l}(r;E).$ (16)

In general, the second derivative equation has two independent solutions;
$\phi_{lj}^{(rs)}(r;E)$ which is regular at the origin $r=0$ and
$\phi_{lj}^{(\pm s)}(r;E)$ satisfies the out-going boundary condition at the
asymptotic limit $r\to\infty$. The HFB equation has two independent solutions
for each $\phi_{lj}^{(rs)}(r;E)$ and $\phi_{lj}^{(\pm s)}(r;E)$ which are
described with the subscription $s=1,2$ as introduced in matsuo .

By supposing

$\displaystyle\lim_{r\to 0}\left[\chi_{l}^{(\pm
s)\mathrm{T}}(r;E)\widehat{S}_{z}\frac{\partial\phi_{l}^{(rs^{\prime})}(r;E)}{\partial
r}\right.$ $\displaystyle\hskip 85.35826pt\left.-\frac{\partial\chi_{l}^{(\pm
s)\mathrm{T}}(r;E)}{\partial
r}\widehat{S}_{z}\phi_{l}^{(rs^{\prime})}(r;E)\right]$
$\displaystyle=\mp\frac{i}{k_{s}(E)}\langle
s|\widehat{S}_{z}|s^{\prime}\rangle,$ (17)

with $\chi_{l}^{(\pm s)\mathrm{T}}(r;E)$ defined by the spherical Hankel
function as

$\displaystyle\chi_{l}^{(\pm 1)}(r;E)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}rh^{(\pm)}_{l}(k_{1}(E)r)\\\ 0\end{pmatrix},$
(18) $\displaystyle\chi_{l}^{(\pm 2)}(r;E)$ $\displaystyle=$
$\displaystyle\begin{pmatrix},0\\\ rh^{(\pm)}_{l}(k_{2}(E)r)\end{pmatrix}$
(19)

and taking $a\to 0$ and $b\to r$ for Eq.(15), we can derive the integral form
of $\phi_{lj}^{(rs)}(r;E)$ as

$\displaystyle\phi_{lj}^{(rs)}(r;E)$ $\displaystyle=$
$\displaystyle\chi_{l}^{(rs)}(r;E)$
$\displaystyle+\int_{0}^{\infty}dr^{\prime}\mathcal{G}_{FR,l}(rr^{\prime};E)\mathcal{U}_{lj}(r^{\prime})\phi_{lj}^{(rs)}(r^{\prime};E),$
(20) $\displaystyle\chi_{l}^{(rs)}(r;E)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\chi_{l}^{(+s)}(r;E)+\chi_{l}^{(-s)}(r;E)\right],$
(21)

with the Green’s function defined by

$\displaystyle\mathcal{G}_{FR,l}(rr^{\prime};E)$
$\displaystyle=-\frac{2m}{\hbar^{2}}\sum_{s=1,2}\frac{k_{s}(E)}{2i}\theta(r-r^{\prime})\widehat{S}_{z}$
$\displaystyle\times\left[\chi_{l}^{(-s)}(r;E)\chi_{l}^{(+s)\mathrm{T}}(r^{\prime};E)-\chi_{l}^{(+s)}(r;E)\chi_{l}^{(-s)\mathrm{T}}(r^{\prime};E)\right]$
(22) $\displaystyle=\begin{pmatrix}g_{FR,l}(rr^{\prime};k_{1}(E))&0\\\
0&-g_{FR,l}(rr^{\prime};k_{2}(E))\end{pmatrix},$ (23)

where

$\displaystyle g_{FR,l}(rr^{\prime};k)$
$\displaystyle\equiv-\frac{2m}{\hbar^{2}}\frac{k}{2i}\theta(r-r^{\prime})rr^{\prime}$
$\displaystyle\times\left[h^{(-)}_{l}(kr)h^{(+)}_{l}(kr^{\prime})-h^{(-)}_{l}(kr^{\prime})h^{(+)}_{l}(kr)\right],$
(24)

Note that Eq.(17) is the so-called Robin boundary condition robin  at the
origin. The Robin boundary condition is the combination of the Dirichlet and
Neumann boundary conditions.

The asymptotic boundary condition of $\phi_{lj}^{(\pm s)}(r;E)$ at the limit
$r\to\infty$ is given by

$\displaystyle\lim_{r\to\infty}\phi_{lj}^{(\pm 1)}(r;E)$ $\displaystyle\to$
$\displaystyle\begin{pmatrix}rh^{(\pm)}_{l}(k_{1}(E)r)\\\ 0\end{pmatrix},$
(25) $\displaystyle\lim_{r\to\infty}\phi_{lj}^{(\pm 2)}(r;E)$
$\displaystyle\to$ $\displaystyle\begin{pmatrix}0\\\
rh^{(\pm)}_{l}(k_{2}(E)r)\end{pmatrix}.$ (26)

Taking $a\to r$ and $b\to\infty$ for Eq.(15) with these boundary conditions,
we can obtain the integral form of $\phi_{lj}^{(\pm s)}(r;E)$ as

$\displaystyle\phi_{lj}^{(\pm s)}(r;E)$ $\displaystyle=\chi_{l}^{(\pm
s)}(r;E)+\int_{0}^{\infty}dr^{\prime}\mathcal{G}_{FL,l}(rr^{\prime};E)\mathcal{U}_{lj}(r^{\prime})\phi_{lj}^{(\pm
s)}(r^{\prime};E)$ , (27)

with the Green’s function defined by

$\displaystyle\mathcal{G}_{FL,l}(rr^{\prime};E)$
$\displaystyle=\frac{2m}{\hbar^{2}}\sum_{s=1,2}\frac{k_{s}(E)}{2i}\theta(r^{\prime}-r)\widehat{S}_{z}$
$\displaystyle\times\left[\chi_{l}^{(-s)}(r;E)\chi_{l}^{(+s)\mathrm{T}}(r^{\prime};E)-\chi_{l}^{(+s)}(r;E)\chi_{l}^{(-s)\mathrm{T}}(r^{\prime};E)\right]$
(28) $\displaystyle=\begin{pmatrix}g_{FL,l}(rr^{\prime};k_{1}(E))&0\\\
0&-g_{FL,l}(rr^{\prime};k_{2}(E))\end{pmatrix},$ (29)

where

$\displaystyle g_{FL,l}(rr^{\prime};k)$
$\displaystyle\equiv\frac{2m}{\hbar^{2}}\frac{k}{2i}rr^{\prime}\theta(r^{\prime}-r)$
$\displaystyle\times\left[h^{(-)}_{l}(kr)h^{(+)}_{l}(kr^{\prime})-h^{(-)}_{l}(kr^{\prime})h^{(+)}_{l}(kr)\right]$
(30)

. It should be noted that we used the value of Wronskian for the spherical
Hankel function given by

$\displaystyle W(rh^{(+)}_{l}(kr),rh^{(-)}_{l}(kr))=-\frac{2i}{k},$ (31)

in order to derive Eq.(27).

By notifying the relation between $\mathcal{G}_{FR,l}$ and
$\mathcal{G}_{FL,l}$ given by

$\displaystyle\mathcal{G}_{FR,l}(rr^{\prime};E)=\mathcal{G}_{FL,l}(rr^{\prime};E)$
$\displaystyle-\frac{2m}{\hbar^{2}}\sum_{s=1,2}\frac{k_{s}(E)}{2i}\widehat{S}_{z}$
$\displaystyle\times\left[\chi_{l}^{(-s)}(r;E)\chi_{l}^{(+s)\mathrm{T}}(r^{\prime};E)-\chi_{l}^{(+s)}(r;E)\chi_{l}^{(-s)\mathrm{T}}(r^{\prime};E)\right],$

we can easily find the relation between $\phi_{lj}^{(rs)}$ and
$\phi_{lj}^{(\pm s)}$ given by

$\displaystyle\phi_{lj}^{(rs)}(r;E)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{s^{\prime}=1,2}\left[\left(\mathcal{J}_{lj}^{(+)}(E)\right)_{ss^{\prime}}\phi_{lj}^{(-s^{\prime})}(r;E)\right.$
(33)
$\displaystyle\left.+\left(\mathcal{J}_{lj}^{(-)}(E)\right)_{ss^{\prime}}\phi_{lj}^{(+s^{\prime})}(r;E)\right],$

where $\left(\mathcal{J}_{lj}^{(\pm)}(E)\right)_{ss^{\prime}}$ is the Jost
function defined by

$\displaystyle\left(\mathcal{J}_{lj}^{(\pm)}(E)\right)_{ss^{\prime}}\equiv\delta_{ss^{\prime}}$
$\displaystyle\mp\langle
s^{\prime}|\widehat{S}_{z}|s^{\prime}\rangle\frac{2m}{\hbar^{2}}\frac{k_{s^{\prime}}(E)}{i}\int_{0}^{\infty}dr\chi_{l}^{(\pm
s^{\prime})\mathrm{T}}(r;E)\mathcal{U}_{lj}(r)\phi_{lj}^{(rs)}(r;E).$ (34)

This Jost function can be also expressed as

$\displaystyle\left(\mathcal{J}_{lj}^{(\pm)}(E)\right)_{ss^{\prime}}$
$\displaystyle=\delta_{ss^{\prime}}$ $\displaystyle\mp\langle
s^{\prime}|\widehat{S}_{z}|s^{\prime}\rangle\frac{k_{s^{\prime}}(E)}{i}\frac{2m}{\hbar^{2}}\int_{0}^{\infty}dr\chi_{l}^{(rs)\mathrm{T}}(r;E)\mathcal{U}_{lj}(r)\phi_{lj}^{(\pm
s^{\prime})}(r;E)$ (35) $\displaystyle=\pm\langle
s^{\prime}|\widehat{S}_{z}|s^{\prime}\rangle\frac{k_{s^{\prime}}(E)}{i}\frac{2m}{\hbar^{2}}w_{lj}^{s,\pm
s^{\prime}}(E),$ (36)

where $w_{lj}^{s,\pm s^{\prime}}(E)$ is the Wronskian for the HFB solutions
defined by

$\displaystyle w_{lj}^{s,\pm
s^{\prime}}(E)\equiv\frac{\hbar^{2}}{2m}\left[\frac{\partial\phi_{l}^{(\pm
s^{\prime})\mathrm{T}}(r;E)}{\partial
r}\widehat{S}_{z}\phi_{l}^{(rs)}(r;E)\right.$ $\displaystyle\hskip
56.9055pt-\left.\phi_{l}^{(\pm
s^{\prime})\mathrm{T}}(r;E)\widehat{S}_{z}\frac{\partial\phi_{l}^{(rs)}(r;E)}{\partial
r}\right],$ (37)

as introduced in matsuo ; fayans ; belyaev .

Figure 5: (Color online) The absolute value surface of the denominator ((a)
red colored surface) and the denominator ((b) blue colored surface) of the
S-matrix Eq.(44) for $s_{1/2}$. The black solid curve is a curve where the red
and blue surfaces ((a) and (b)) are touched each other.

Note that the off-diagonal components of the Jost function becomes
$\left(\mathcal{J}_{lj}^{(\pm)}(E)\right)_{12}=\left(\mathcal{J}_{lj}^{(\pm)}(E)\right)_{21}=0$
at the no pairing limit $\langle\Delta\rangle\to 0$.

The asymptotic behaviour at $r\to\infty$ of Eq.(33) is given by

$\displaystyle\lim_{r\to\infty}\phi_{lj}^{(rs)}(r;E)$
$\displaystyle\sim\frac{i^{l+1}}{2k_{1}(E)}\left[\left(\mathcal{J}_{lj}^{(+)}(E)\right)_{s1}\begin{pmatrix}e^{-ik_{1}(E)r}\\\
0\end{pmatrix}\right.$ $\displaystyle\hskip
42.67912pt\left.+(-)^{l+1}\left(\mathcal{J}_{lj}^{(-)}(E)\right)_{s1}\begin{pmatrix}e^{+ik_{1}(E)r}\\\
0\end{pmatrix}\right]$
$\displaystyle+\frac{i^{l+1}}{2k_{2}(E)}\left[\left(\mathcal{J}_{lj}^{(+)}(E)\right)_{s2}\begin{pmatrix}0\\\
e^{-ik_{2}(E)r}\end{pmatrix}\right.$ $\displaystyle\hskip
42.67912pt\left.+(-)^{l+1}\left(\mathcal{J}_{lj}^{(-)}(E)\right)_{s2}\begin{pmatrix}0\\\
e^{+ik_{2}(E)r}\end{pmatrix}\right].$ (38)

If the the complex quasiparticle energy $E$ belongs to the 1st Riemann sheet
($E=E_{1}$), the complex momentum $k_{1}$ and $k_{2}$ belong to the upper-half
plane of each momentum planes as explained in Sec.II. The condition for the
bound states are, therefore, given by

$\displaystyle\left(\mathcal{J}_{lj}^{(+)}(E_{1})\right)_{s1}=\left(\mathcal{J}_{lj}^{(+)}(E_{1})\right)_{s2}=0,$
(39)

or

$\displaystyle\det\left(\mathcal{J}_{lj}^{(+)}(E_{1})\right)=0.$ (40)

When $\det\left(\mathcal{J}_{lj}^{(+)}(E)\right)\neq 0$, the scattering wave
function $\psi_{lj}^{(+)}(r;E)$ can be defined using the inverse of the Jost
function $\left(\mathcal{J}_{lj}^{(+)}(E)\right)^{-1}$ as

$\displaystyle\psi_{lj}^{(+)}(r;E)\equiv\sum_{s=1,2}\left(\mathcal{J}_{lj}^{(+)}(E)\right)^{-1}_{1s}\phi_{lj}^{(rs)}(r;E).$
(41)

since the upper component of the HFB solution corresponding to particle states
at no pairing limit ($\Delta(r)\to 0$) in the positive energy region (Re
$E>0$).

The asymptotic behaviour of $\psi_{lj}^{(+)}(r;E)$ is given by

$\displaystyle\lim_{r\to\infty}\psi_{lj}^{(+)}(r;E)$
$\displaystyle\sim\frac{i^{l+1}}{2k_{1}(E)}\left[\begin{pmatrix}e^{-ik_{1}(E)r}\\\
0\end{pmatrix}\right.$ $\displaystyle\hskip
71.13188pt\left.+(-)^{l+1}S^{11}_{lj}(E)\begin{pmatrix}e^{+ik_{1}(E)r}\\\
0\end{pmatrix}\right]$ $\displaystyle\hskip
14.22636pt+\frac{(-i)^{l+1}}{2k_{2}(E)}S^{12}_{lj}(E)\begin{pmatrix}0\\\
e^{+ik_{2}(E)r}\end{pmatrix},$ (42)

where

$\displaystyle
S^{ss^{\prime}}_{lj}(E)\equiv\sum_{s^{\prime\prime}=1,2}\left(\mathcal{J}_{lj}^{(+)}(E)\right)^{-1}_{ss^{\prime\prime}}\left(\mathcal{J}_{lj}^{(-)}(E)\right)_{s^{\prime\prime}s^{\prime}}.$
(43)

Since $k_{1}$ becomes real, and $k_{2}$ becomes pure imaginary on the upper-
half plane of each momentum as seen in Fig.1-4 on the scattering states (the
real axis of $E$, Re $E>-\lambda(>0)$), the second term is vanished at
$r\to\infty$ in the r.h.s of Eq.(42). And $S_{lj}^{11}(E)$ can be interpreted
as the S-matrix for the nucleon scattering on open-shell nucleus within the
HFB formalism.

According to the definition of $S^{ss^{\prime}}_{lj}(E)$ (Eq.(43)),
$S_{lj}^{11}(E)$ can be expressed as

$\displaystyle
S_{lj}^{11}(E)=\frac{\left(\mathcal{J}_{lj}^{(+)}(E)\right)_{22}\left(\mathcal{J}_{lj}^{(-)}(E)\right)_{11}-\left(\mathcal{J}_{lj}^{(+)}(E)\right)_{12}\left(\mathcal{J}_{lj}^{(-)}(E)\right)_{21}}{\left(\mathcal{J}_{lj}^{(+)}(E)\right)_{22}\left(\mathcal{J}_{lj}^{(+)}(E)\right)_{11}-\left(\mathcal{J}_{lj}^{(+)}(E)\right)_{12}\left(\mathcal{J}_{lj}^{(+)}(E)\right)_{21}}.$
(44)

If the quasiparticle energy $E$ moves from the 1st quadrant to the 4th
quadrant by passing through branch cut corresponding to $k_{1}$ ($E\to E^{*}$
from (a) to (d’) in Fig.1), $k_{1}(E^{*})$ is given by

$\displaystyle k_{1}(E^{*})=k_{1}^{*}(E),$ (45)

which is corresponding to the motion from (a) to (d’) in Fig.2.

Since there is no branch cut for $k_{2}$ between the 1st and 4th quadrants,

$\displaystyle k_{2}(E^{*})=-k_{2}^{*}(E).$ (46)

(corresponding the motion from (a) to (d) in Fig.4).

By applying Eqs.(45) and (46) to Eqs.(20) and (27), we find

$\displaystyle\phi_{lj}^{(r1)}(r;E^{*})$ $\displaystyle=$
$\displaystyle\phi_{lj}^{(r1)*}(r;E),$ (47)
$\displaystyle\phi_{lj}^{(r2)}(r;E^{*})$ $\displaystyle=$
$\displaystyle(-)^{l}\phi_{lj}^{(r2)*}(r;E),$ (48)
$\displaystyle\phi_{lj}^{(\pm 1)}(r;E^{*})$ $\displaystyle=$
$\displaystyle\phi_{lj}^{(\mp 1)*}(r;E),$ (49) $\displaystyle\phi_{lj}^{(\pm
2)}(r;E^{*})$ $\displaystyle=$ $\displaystyle(-)^{l}\phi_{lj}^{(\pm
2)*}(r;E),$ (50)

and the Jost function as a function of $E^{*}$ have the following properties
as

$\displaystyle\left(\mathcal{J}_{lj}^{(\pm)}(E^{*})\right)_{11}$
$\displaystyle=$
$\displaystyle\left(\mathcal{J}_{lj}^{(\mp)*}(E)\right)_{11},$ (51)
$\displaystyle\left(\mathcal{J}_{lj}^{(\pm)}(E^{*})\right)_{12}$
$\displaystyle=$
$\displaystyle(-)^{l}\left(\mathcal{J}_{lj}^{(\pm)*}(E)\right)_{12},$ (52)
$\displaystyle\left(\mathcal{J}_{lj}^{(\pm)}(E^{*})\right)_{21}$
$\displaystyle=$
$\displaystyle(-)^{l}\left(\mathcal{J}_{lj}^{(\mp)*}(E)\right)_{21},$ (53)
$\displaystyle\left(\mathcal{J}_{lj}^{(\pm)}(E^{*})\right)_{22}$
$\displaystyle=$
$\displaystyle\left(\mathcal{J}_{lj}^{(\pm)*}(E)\right)_{22},$ (54)

Thus, we finally can find that $S_{lj}^{11}$ can be represented by

$\displaystyle S_{lj}^{11}(E^{*})$
$\displaystyle=\frac{\left(\mathcal{J}_{lj}^{(+)*}(E)\right)_{22}\left(\mathcal{J}_{lj}^{(+)*}(E)\right)_{11}-\left(\mathcal{J}_{lj}^{(+)*}(E)\right)_{12}\left(\mathcal{J}_{lj}^{(+)*}(E)\right)_{21}}{\left(\mathcal{J}_{lj}^{(+)*}(E)\right)_{22}\left(\mathcal{J}_{lj}^{(-)*}(E)\right)_{11}-\left(\mathcal{J}_{lj}^{(+)*}(E)\right)_{12}\left(\mathcal{J}_{lj}^{(-)*}(E)\right)_{21}}=1/S_{lj}^{11*}(E),$
(55)

when $E^{*}$ belongs to the 2nd Riemann sheet defined by $k_{1}$ in the
positive energy region (Re $E>0$). From this result, we can see that zero
points of the denominator and numerator of Eq.(44) represent the S-matrix
poles on the 1st and 2nd Riemann sheet of the complex energy plane,
respectively.

Also it is obvious that

$\displaystyle|S_{lj}^{11}(E)|^{2}=1,$ (56)

when $E=E^{*}$ in Eq.(55). This shows that $S_{lj}^{11}(E)$ satisfies the
Unitarity of the S-matrix if $E$ stands on the branch cut for the positive
energy region.

The denominator of the S-matrix Eq.(44) is equal to the determinant of the
Jost function. Since the following properties of the Jost function

$\displaystyle\left(\mathcal{J}_{lj}^{(\pm)}(E)\right)_{22}$ $\displaystyle=$
$\displaystyle\left(\mathcal{J}_{lj}^{(\pm)}(-E)\right)_{11},$ (57)
$\displaystyle\left(\mathcal{J}_{lj}^{(\pm)}(E)\right)_{12}$ $\displaystyle=$
$\displaystyle-\left(\mathcal{J}_{lj}^{(\pm)}(-E)\right)_{21},$ (58)

can be found by applying $k_{2}(-E)=k_{1}(E)$ to the definition of the Jost
function (Eqs.(34)-(36)), we can find the symmetric property of the
determinant of the Jost function as

$\displaystyle\det\left(\mathcal{J}_{lj}^{(\pm)}(-E)\right)=\det\left(\mathcal{J}_{lj}^{(\pm)}(E)\right).$
(59)

Therefore, the zeros of the denominator are given symmetrically to the
imaginary axis of the complex energy (Im $E$), and are corresponding the eigen
solutions of the HFB Hamiltonian. The S-matrix poles given by zeros of the
numerator represent the resonances.

In Fig.5, we show the absolute value surfaces for the denominator and the
numerator of the S-matrix (Eq.(44) for $s_{1/2}$ which is calculated
numerically using the parameter sets and Woods-Saxon potential introduced in
Sec.IV adopting $\lambda=-8$ MeV and $\langle\Delta\rangle=2.17$ MeV for the
Fermi energy and the mean pairing gap, respectively. The denominator and
numerator of the S-matrix are touched along the branch cut defined in the
positive energy region.

Figure 6: (Color online) The square of the denominator (panel (a)) and the
numerator (panel (b)) of the S-matrix (Eq.(44)) for $d_{5/2}$ with
$\langle\Delta\rangle=3.0$ MeV and $\lambda=-8.0$ MeV are shown on the complex
quasiparticle energy plane. The yellow X-symbols show the position of zeros
which represent the poles of the S-matrix. The yellow curves is the
trajectories of zeros obtained by varying the pairing gap
$\langle\Delta\rangle$ from 0.0 to 10.0 MeV. The arrows represent the
direction of X.

The T-matrix for the neutron elastic scattering on open-shell nucleus can be
derived as

$\displaystyle T_{lj}(E)\equiv\frac{i}{2}\left(S_{lj}^{11}(E)-1\right)$
$\displaystyle=\frac{i}{2}\sum_{s=1,2}\left(\mathcal{J}_{lj}^{(+)}(E)\right)^{-1}_{1s}\left[\left(\mathcal{J}_{lj}^{(-)}(E)\right)_{s1}-\left(\mathcal{J}_{lj}^{(+)}(E)\right)_{s1}\right]$
$\displaystyle=\frac{2mk_{1}(E)}{\hbar^{2}}\int_{0}^{\infty}dr\chi_{l}^{(r1)\mathrm{T}}(r;E)\mathcal{U}_{lj}(r)\psi_{lj}^{(+)}(r;E).$
(60)

Note that Eqs.(34) and (41) was used to derive Eq.(60).

When the S-matrix satisfies the Unitarity (Eq.(56), the T-matrix satisfies the
optical theorem given by

$\displaystyle-\mbox{ Im }T_{lj}(E)=|T_{lj}(E)|^{2},$ (61)

where $E$ is supposed to be existing on the branch cut. The total cross
section for nucleon scattering $\sigma_{lj}(\hbar\omega)$ is given by

$\displaystyle\sigma(E_{i})$ $\displaystyle=$
$\displaystyle\sum_{lj}\sigma_{lj}(E_{i}),$ (62)
$\displaystyle\sigma_{lj}(E_{i})$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2m}\frac{\pi}{E_{i}}\frac{2j+1}{2}|T_{lj}(E_{i}-\lambda)|^{2},$
(63)

where $E_{i}$ is the incident nucleon energy defied by

$\displaystyle E_{i}=E+\lambda.$ (64)

Figure 7: (Color online) The Woods-Saxon potential and the single-particle
levels obtained in the no pairing limit. The lines of Fermi energy
$\lambda=-1$ and $-8$ MeV are displayed together.

##  IV Numerical results

In this study, we adopt the Woods-Saxon potential which is given by

$\displaystyle U_{lj}(r)$ $\displaystyle=$
$\displaystyle\left[V_{0}+V_{ls}\mbox{\boldmath$l$}\\!\cdot\\!\mbox{\boldmath$s$}\frac{r_{0}^{2}}{r}\frac{d}{dr}\right]f(r),$
(65) $\displaystyle f(r)$ $\displaystyle=$
$\displaystyle\frac{1}{1+\exp\left(\frac{r-R}{a}\right)},$ (66)

in the standard form for the Hartree-Fock mean field $U_{lj}(r)$ in Eq.(1)
with use of the standard parameters given by hamamoto ; bohr . The mass number
$A=24$ is used for the parameter given by $R=r_{0}A^{\frac{1}{3}}$.

The volume-type pairing field $\Delta(r)$ described by

$\displaystyle\Delta(r)=V_{pair}f(r)$ (67)

is adopted in the paper.

We define the average pairing gap $\langle\Delta\rangle$ by

$\displaystyle\langle\Delta\rangle\equiv\frac{\int r^{2}dr\Delta(r)f(r)}{\int
r^{2}drf(r)},$ (68)

as introduced in hamamoto . We solve the integral equation Eq.(27)
numerically, and obtain $\phi_{lj}^{(\pm s)}(r;E)(s=1,2)$ in order to
calculate the Jost function with Eq.(35) in this study. It should be noted
that we confirmed that the Jost function by Eqs.(34)-(36) give the same
numerical results.

The square of the denominator (panel (a)) and the numerator (panel (b)) of the
S-matrix (Eq.(44)) for $d_{5/2}$ with $\langle\Delta\rangle=3.0$ MeV and
$\lambda=-8.0$ MeV are shown on the complex quasiparticle energy plane in
Fig.6. The yellow colored X-symbols are zeros of the plotted function
determined by the Nelder-Mead method nelder . The yellow curves are the
trajectories of the S-matrix poles with varying $\langle\Delta\rangle$ from
0.0 to 10.0 MeV. The arrows represent the direction of motion of zeros
indicated by X-symbols.

The same results are shown on the complex $k_{1}$-plane in Fig.8. Panels (a)
and (b) in Fig.6 correspond to the (a)upper and (b)lower figures in Fig.8,
respectively. The dashed curves seen in the upper-half and lower-half plane
represent Re $E=0$ which pass through the points at $k_{1}=\pm ik_{F}$.
$k_{F}$ is the Fermi momentum defined by
$k_{F}\equiv\sqrt{\frac{2m|\lambda|}{\hbar^{2}}}$. The above/below regions on
upper/lower panels is corresponding to the negative energy regions (left-half
planes) of 1st/2nd Riemann sheets, respectively, as explained in Figs.1 and 2.

Figure 8: (Color online) The same figure with Fig.6 but on the complex
$k_{1}$-momentum plane. The upper- and lower-half plane ((a) and (b))
correspond to the panels (a) and (b) in Fig.6, respectively. The dashed curves
show curves where Re $E=0$ MeV.

The S-matrix poles on the 1st Riemann sheet are existing on the real axis, and
have a tendency to go away from $E=0$ along the real axis. Corresponding poles
on $k_{1}$-momentum plane are existing on the imaginary axis and go away from
the Fermi momentum along the imaginary axis as the pairing gap is increased.

Figure 9: (Color online) Trajectories of the S-matrix poles on the 2nd Riemann
sheet with varying the pairing gap $\langle\Delta\rangle$ from 0 to 10 MeV.
o-symbols represent the position of poles at the no pairing limit
($\langle\Delta\rangle=0$ MeV). The hole-like and particle-like poles are
classified in the upper and lower panels, respectively (see text).  Figure 10:
(Color online) The total cross section and the partial cross sections for the
neutron elastic scattering with $\lambda=-8.0$ MeV.

The S-matrix poles existing on the real axis of the 2nd Riemann sheet move
along the real axis by taking a circular trajectory as the pairing gap is
increased. It seems to avoid a certain point existing around
$E=-\frac{3}{4}\lambda=6$ MeV shown as (b’) in Fig.6. The certain point
corresponds to $k_{1}=-ik_{F}/2$ on the complex $k_{1}$-momentum plane. The
poles become resonances at $E>-\lambda$ with the large pairing gap limit. We
need further investigation in order to clarify this point.

The resonances which are found at $E=10.60\pm 4.60\hskip 2.0pti$ MeV with
$\Delta(r)=0$ move on the trajectories as shown in Fig.6 due to the pairing
effect. The trajectory of resonances show that the pairing effect increases
both resonance energy and width, since the imaginary value of the pole on the
complex energy plane corresponds to the width of resonance.

In Fig.9, we show the trajectories of S-matrix all poles existing on the 2nd
Riemann sheet below 20 MeV quasiparticle energy obtained by varying the
pairing gap $\langle\Delta\rangle$ from 0 to 10 MeV. It seems to be possible
to classify the trajectories from their behaviour into two types except the
poles of $d_{5/2}$, $f_{5/2}$ and $f_{7/2}$ denoted by the text in Fig.9. The
$p_{1/2}$ and $p_{3/2}$ poles shown in the upper panel become resonances by
the configuration mixing between hole states existing below the Fermi energy
as shown in Fig.7 and the continuum states due to the pairing effect. The
resonance energy and the width are monotonically increased as the pairing gap
is increased. This type of the resonances are called as “hole-like”
quasiparticle resonance kobayashi .

The resonances above $E=-\lambda(=8$MeV) are called as the “particle-like”
quasiparticle resonances in kobayashi . The resonances at the no pairing limit
are formed by the centrifugal barrier and the spin-orbit potential. Such
resonances are called as “the shape resonance”. The trajectories in the lower
panel above $E=-\lambda(=8$MeV) show the pairing effect on the “shape
resonance” except the $d_{5/2}$ resonance which is represented by text in the
figure with large width.

The poles of $d_{5/2}$ and $d_{3/2}$ existing below $E=-\lambda(=8$MeV) are
anti-bound quasiparticle states. These poles are moved to the positive
direction of the real axis, and become the “particle-like” quasiparticle
resonance above $E=-\lambda(=8$MeV) at the large pairing limit.

Table 1: S-matrix poles of hole-like quasiparticle resonances which give the
clear contribution to the total cross section shown in Fig.10.

$\langle\Delta\rangle$ | $p_{1/2}$ | $p_{3/2}$
---|---|---
$0.0$ | $11.71$ | $15.68$
$3.0$ | $12.65\pm i0.28$ | $16.39\pm i0.08$

(For $\lambda=-8.0$ MeV, Unit:MeV)

The trajectory’s behaviour of the poles of $d_{5/2}$, $f_{5/2}$ and $f_{7/2}$
denoted by the text in Fig.9 is distinct from both “particle-like” and “hole-
like” resonances. The $d_{5/2}$ resonance may contribute to the nucleon
scattering cross section as a background because it has very wide width. These
states may be categorized into “echo” state of the nucleon scattering as
suggested by Sasakawa sasakawa . However, more detailed analysis of these
poles would be necessary to determine the classification.

We show the total neutron elastic scattering cross section as a function of
the incident energy of neutron defined by Eq.(64) in Fig.10. The partial cross
sections for each angular momentum components are also shown in the same
figure. We use the formulas given by Eqs.(62) and (63) to calculate the cross
section. The incident energy $E_{i}$ is multiplied to the cross section in
order to see the contribution of the partial cross section clearly. The cross
sections with $\langle\Delta\rangle=0$ and $3.0$ MeV are shown in the upper
and lower panels, respectively. In the lower panel, the total cross section
with $\langle\Delta\rangle=0$ MeV is also plotted by the black dashed curve
for comparison.

A peak found around $E_{i}\sim 9.5$MeV in the upper panel is a typical shape
resonances formed by $g_{9/2}$ and $f_{5/2}$ resonances. It is very difficult
to see the pairing effect on those resonances in comparison between the upper
and lower panel because those resonances have wide width and the energy shift
is also very small. On the other hand, we can find staggering shapes of cross
section around $E_{i}\sim 4.5$ and $8$ MeV. Those staggering shapes of cross
section are due to the asymmetric shapes of the partial cross section caused
by the hole-like quasiparticle resonances of $p_{1/2}$ and $p_{3/2}$,
respectively. The quasiparticle energies found on the 2nd Riemann sheet of the
complex quasiparticle energy plane are listed in Table 1. The widths for each
resonances, $p_{1/2}$ and $p_{3/2}$, are determined as $\Gamma=0.56,0.16$ MeV
respectively, because Im $E=\frac{\Gamma}{2}$.

Figure 11: (Color online) The same figure with Fig.9 but with $\lambda=-1$
MeV. Table 2: S-matrix poles of particle-like and hole-like quasiparticle
resonances which give the clear contribution to the total cross section shown
in Fig.12.

Particle-like
---
$\langle\Delta\rangle$ | $p_{3/2}$ | $f_{7/2}$
$0.0$ | $0.89$ | $0.91\pm i0.11$
$3.0$ | $1.35\pm i0.17$ | $2.46\pm i0.01$

Hole-like
---
$\langle\Delta\rangle$ | $d_{3/2}$ | $s_{1/2}$ | $d_{5/2}$ | $p_{1/2}$
$0.0$ | $4.12$ | $7.69$ | $11.22$ | $18.70$
$3.0$ | $4.98\pm i0.26$ | $8.30\pm i0.48$ | $11.84\pm i0.33$ | $19.29\pm i0.03$

(For $\lambda=-1.0$ MeV, Unit:MeV)

In Fig.11, we show the same plot for the case of $\lambda=-1.0$ MeV to see the
trajectories of the S-matrix poles for the neutron-rich unstable nuclei. The
number of the hole-like quasiparticle resonances shown in the upper panel are
increased because of the shallow Fermi energy. The paring effect on the
particle-like quasiparticle resonances ($p_{1/2}$, $p_{3/2}$, $f_{5/2}$ and
$g_{9/2}$) are same qualitatively in shape or form of the trajectories, but
more sensitive to the pairing gap than those shown in Fig.9. The
$d_{5/2}$-resonance indicated by text in the Fig.11 is existing with the same
width in the no pairing limit as the one shown in Fig.9. However, the
sensitivity for the pairing is quite different. Almost no pairing effect can
be seen in the behaviour of $d_{5/2}$-resonance trajectory even at the large
pairing limit ($\langle\Delta\rangle=10.0$ MeV).

Figure 12: (Color online) The total cross section and the partial cross
sections for the neutron elastic scattering with $\lambda=-1.0$ MeV.

The total cross section for the neutron elastic scattering is shown in Fig.12.
The partial cross sections for each angular momentum are also shown together.
In the lower panel, we can find 6 peaks of resonances which are caused by the
pairing effect. The two peaks indicated as $p_{3/2}$ and $f_{7/2}$ are the
particle-like quasiparticle resonances. The 4 peaks indicated by text as
$s_{1/2}$, $p_{1/2}$, $d_{3/2}$ and $d_{5/2}$ in Fig.11 are the hole-like
quasiparticle resonances corresponding the S-matrix poles represented in the
upper panel of Fig.11. The corresponding quasiparticle energies on the complex
quasiparticle energy plane are listed in Table 2. The widths for each
resonances can be obtained by multiplying factor $2$ to the imaginary values
of the complex quasiparticle energies.

##  V summary

We have extended the Jost function formalism based on the Hartree-Fock-
Bogoliubov formalism in order to give the analytic expression of the S-matrix
for the nucleon elastic scattering targeting on the open-shell nucleus. We
showed that the S-matrix satisfies the unitarity on the branch-cut (continuum)
in the positive quasiparticle energy region. Adopting the Woods-Saxon
potential, we showed the S-matrix poles on the complex quasiparticle
energy/momentum plane by using the analytic expression of the S-matrix
represented by the Jost function. The trajectories of the S-matrix poles
obtained by varying the pairing strength were also shown for two cases of the
Fermi energies given by $\lambda=-8.0$ and $-1.0$ MeV. We pointed out that the
trajectories of resonances can be categorized into two types, the hole-like
and particle-like quasiparticle resonances aside from a couples of exceptions.
We confirmed that such quasiparticle resonances can be observed as the
staggering shape or sharp peaks of the total cross section of the neutron
elastic scattering. We did not compare our results with the experimental data
in this study. For the quantitative discussion, we need to adopt the proper
optical potential in order to take account the proper absorption given by the
imaginary part of the optical potential. One of the proper way is to adopt the
microscopic optical potential based on the PVC method. However, it may be
necessary to extend the existing the PVC method to the quasiparticle-vibration
coupling method for the self-consistency of the method although it is a matter
of concern that excessive calculation time is required. An alternative way is
to adopt the GOP, however, readjustment of the parameters may be necessary
because the current parameters have been adjusted without pairing correlation.

##  VI Acknowledgments

We thank Dr. Yoshihiko Kobayashi (Kyushu University) for useful discussions.
This work is funded by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under grant number “103.04-2018.303”. T. Dieu Thuy
partially thanks the financial support of the Nuclear Physics Research Group
(NP@HU) at Hue University through the Hue University Grant 43/HD-DHH.

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