# On Schrodinger operators
Modified by $\delta$ interactions

Kaya Güven Akbaş<EMAIL_ADDRESS>Fatih Erman<EMAIL_ADDRESS>O. Teoman Turgut<EMAIL_ADDRESS>
###### Abstract

We study the spectral properties of a Schrödinger operator $H_{0}$ modified by
delta interactions and show explicitly how the poles of the new Green’s
function are rearranged relative to the poles of original Green’s function of
$H_{0}$. We prove that the new bound state energies are interlaced between the
old ones and the ground state energy is always lowered if the delta
interaction is attractive. We also derive an alternative perturbative way of
finding the bound state energies and wave functions under the assumption of
small coupling constant in a somewhat heuristic way. We further show that
these results can be extended to the case where a renormalization process is
required. Finally, we consider the possible extensions of our results to the
multi center case, to delta interaction supported on curves, and to the case,
where the particle is moving in a compact two dimensional manifold under the
influence of a delta interaction.

Keywords: Dirac delta interactions, point interactions, Green’s function,
renormalization, Schrödinger operators, spectrum, resolvent.

##  1 Introduction

We consider a self-adjoint Schrödinger operator or Hamiltonian
$H_{0}=-\frac{\hbar^{2}}{2m}\Delta+V$ defined on some dense domain $D(H_{0})$
and assume that $H_{0}$ admits a discrete or point spectrum $\sigma_{d}(H)$
and a (purely absolutely) continuous real spectrum $\sigma_{c}(H)$, which does
not overlap with the discrete spectrum (no embedded eigenvalues). We further
assume that the discrete spectrum has no condensation point and the spectrum
is bounded below. These conditions put some restrictions on the regularity
properties of the potentials, but this is the typical situation in most of the
quantum mechanical problems with physically reasonable potentials. The
spectrum consists of eigenvalues $E_{n}$ of $H_{0}$ with finite
multiplicities,

$\displaystyle H_{0}\phi_{n}=E_{n}\phi_{n}\;,$ (1.1)

where $\phi_{n}\in L^{2}$ are the eigenvectors (or eigenfunctions)
corresponding to the eigenvalues $E_{n}$, and the values $\lambda$ in the
continuous spectrum are the “generalized eigenvalues” corresponding to the
“generalized eigenvectors” $\chi_{\lambda}$ (or generalized eigenfunctions),
which is outside of $L^{2}$. Here we omit the degeneracy/multiplicity labels
for simplicity. Although we are not aiming for a completely rigorous
presentation in this work, we point out that the proper meaning of the
generalized eigenfunctions has been understood in the context of so-called
Rigged Hilbert spaces [1, 2, 3] and the equation for the generalized
eigenvalues are given by $H_{0}\chi_{\lambda}=\lambda\chi_{\lambda}$ in the
sense of distributions. One may think of a non-zero tempered distribution
$\chi_{\lambda}$ as a generalized eigenfunction associated with the eigenvalue
$\lambda$ for $H_{0}$ if and only if $\langle
H_{0}\chi,\psi\rangle=\langle\chi_{\lambda},H_{0}\psi\rangle=\lambda\langle\chi_{\lambda},\psi\rangle$
for any infinitely differentiable rapidly decaying functions $\psi$ (Schwartz
functions), see e.g., [4] for a more elementary discussion. The function
$\chi_{\lambda}(x)=\frac{1}{\sqrt{2\pi}}e^{i\lambda x/\hbar}$ is the well-
known generalized eigenfunction of the momentum operator
$(P\psi)(x)=-i\hbar\frac{d}{dx}\psi(x)$ in $L^{2}(\mathbb{R})$. Under
relatively mild assumptions on the potential $V$ for dimensions $d\leq 3$
case, it is known that the generalized eigenfunctions actually can be selected
as continuous functions [5, 6]. This is extremely valuable for our
computations as we either evaluate them at a point or integrate them over a
curve in this work.

According to one of the fundamental assumptions of quantum mechanics, we can
formally expand any function $\psi\in L^{2}(\mathbb{R})$ as (see e.g., page 48
in [7]):

$\displaystyle\psi(x)=\sum_{n}a_{n}\phi_{n}(x)+\int_{\sigma_{c}(H_{0})}a(\lambda)\chi_{\lambda}(x)\;d\mu(\lambda)\;,$
(1.2)

where

$\displaystyle a_{n}=\int_{\mathbb{R}}\overline{\phi_{n}(x)}\psi(x)dx\;,\hskip
28.45274pta(\lambda)=\int_{\mathbb{R}}\overline{\chi_{\lambda}(x)}\psi(x)dx\;.$
(1.3)

The expression $d\mu(\lambda)$ in the above expansion is indeed a kind of
spectral measure associated with the self-adjoint operator $H_{0}$. If there
is a degeneracy, we need to sum over those indices as well. Fourier transform
is actually a formal eigenfunction expansion of a function $\psi$ in terms of
the generalized eigenfunctions $\chi_{\lambda}$ of the momentum operator $P$
and the action of $P$ on the function $\psi$ is given by:

$\displaystyle(P\psi)(x)=\int_{-\infty}^{\infty}\lambda\left(\int_{-\infty}^{\infty}\psi(\xi)\overline{\chi_{\lambda}(\xi)}d\xi\right)\chi_{\lambda}(x)d\lambda\;.$
(1.4)

The theory of such eigenfunction expansions was first described by Weyl [8]
for ordinary differential equations, and then developed partly by Titchmarsh
[9] and by Kodaira [10]. This is extended to the multidimensional case for
elliptic self-adjoint differential operators in [5]. In particular, such
expansions in terms of generalized eigenfunctions are rigorously constructed,
if we impose the uniformly locally square integrability condition on the
potential energy in the Schrödinger operator
$H_{0}=-\frac{\hbar^{2}}{2m}\Delta+V$, which is stated as Theorem 3 in [11].
In this paper, we tacitly assume that these mild conditions for the
eigenfunction expansions hold.

Our main interest here is to consider such Schrödinger operators $H_{0}$
modified by a $\delta$ interaction and show explicitly how the eigenvalues and
eigenfunctions change using the eigenfunction expansions of the Green’s
function for such generic $H_{0}$. The modification of the bound state
spectrum in one dimension and $d$ dimensional radial case, have been studied
in the framework of path integrals in an influential work of Grosche [12] for
some exactly solvable potentials $V$. One of the simplest choices for $H_{0}$
is the well-known harmonic oscillator problem in one dimension (as well as the
radially symmetric extension of it) and the effect of adding a delta
interaction to harmonic oscillator Hamiltonians has been studied by many
authors [13, 14] and the statistical properties of this system have been
worked out in [15] and the application of the problem to Bose-Einstein
condensation has been investigated in [16]. It is well-known that the
description of point like delta interactions in two and three dimensions
requires renormalization and they have been studied in [17, 18] in a
mathematically rigorous way and in [19, 20, 21, 22, 23] as toy models in
understanding of some quantum field theoretical concepts. The higher
dimensional version of the harmonic oscillator Hamiltionian with delta
potential requires renormalization as well and has been discussed in [24] and
more recently in [25]. A more interesting exactly solvable example, having
both discrete and continuous spectrum, is the so-called reflectionless
potential [12, 26], and we wish to show how its spectral properties change
under the influence of delta interaction as a case study.

It turns out that eigenvalues of $H_{0}$ modified by $\delta$ interactions
change according to some algebraic or transcendental equation and all the
eigenvalues of $H_{0}$ disappear unless some set of wave functions vanish at
the support of the $\delta$ interaction or if the support of Dirac delta
function is chosen to be at the nodes of the wave function of the initial
Hamiltonian $H_{0}$. This result was not completely illustrated for a generic
Hamiltonian $H_{0}$, but only shown for particular exactly solvable cases [12,
24]. In all these cases, the full Green’s function $G$ of the modified system
contains the Green’s function $G_{0}$ of $H_{0}$ added to another term
constructed again from $G_{0}$. It is not at all obvious that the poles of
$G_{0}$ cancel with the poles of this additional term (which has $G_{0}$
appearing in its expression in a nontrivial combination). This cancellation
has only been pointed out for $H_{0}$ being the one dimensional harmonic
oscillator Hamiltonian in [13] and for higher dimensional harmonic and linear
potentials in [25]. In this paper, we prove this explicitly by using the
eigenfunction expansion of the full Green’s function by taking the generalized
eigenfunction expansion into account and studying its pole structure for a
general class of potentials. Moreover, we prove that if the support of the
delta interaction is not at the node of the bound state eigenfunction of
$H_{0}$, then the new eigenvalues $E_{k}^{*}$ are interlaced between $E_{k-1}$
and $E_{k}$ for an attractive delta interaction. We then develop a
perturbative method from a different perspective to compute order by order the
new eigenvalues and wave functions (in a somewhat heuristic way) under the
assumption of small coupling, then compare the results with the standard
approach. All the results that we have found can easily be extended to the
multi center case, and to the case where we need to apply renormalization. Our
perturbation method applied to the problems which require renormalization
yields different results from the standard approach obtained by replacing the
bare coupling constant $\alpha$ with the renormalized one $\alpha_{R}$. We
further show that similar conclusions can be drawn for the deformations by
delta interactions supported on curves in the plane and for the problem, for
which the particle is intrinsically moving in a compact manifold under the
influence of a $\delta$ interaction.

This paper is organized as follows. In Section 2, using the eigenfunction
expansion of the Green’s function we have explicitly shown that the poles of
the Green’s function of $H_{0}$ modified by delta interaction in one dimension
cancels out the poles of the Green’s function of $H_{0}$ under some mild
conditions and prove that the new bound state energies are interlaced between
the old ones and discuss the results with an explicit exactly solvable
reflectionless potential. Then, we develop a new perturbative way of finding
the bound state energies and the bound state wavefunctions up to the second
order and compare them with the classical known results. In the last part of
this section, we show how the generalized eigenfunctions change after adding a
delta interaction to the initial Schrödinger operator $H_{0}$. Section 3 deals
with the extension of the results to the singular case, where the
renormalization of the problem is required. Finally, Section 4 is devoted to
the possible extensions of the results, e.g., $N$ center case, delta
interaction supported on a curve in the plane, and the case where a particle
is moving on a compact manifold interacting with a point center.

##  2 Modifications by $\delta$ Interactions in Regular Cases

Throughout the paper, we use some well-known properties of Green’s function.
For convenience of the reader, we state them here with our choice of notation.
For instance, the integral kernel of the resolvent $R_{H_{0}}(z)$ for the
Schrödinger operator $H_{0}=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+V(x)$ or
simply Green’s function defined by

$\displaystyle\left(R_{H_{0}}(E)\psi\right)(x)=\left(R_{0}(E)\psi\right)(x)=\left((H_{0}-E)^{-1}\psi\right)(x)=\int_{\mathbb{R}}G_{0}(x,y|E)\psi(y)dy\;,$
(2.1)

can be expressed as the following bilinear expansion

$\displaystyle
G_{0}(x,y|E)=\sum_{n=0}^{\infty}\frac{\phi_{n}(x)\overline{\phi_{n}(y)}}{E_{n}-E}+\int_{\sigma_{c}(H_{0})}\frac{\chi_{\lambda}(x)\overline{\chi_{\lambda}(y)}}{\lambda-E}\;d\mu(\lambda)\;.$
(2.2)

The Green’s function $G_{0}(x,y|E)$ is a square integrable function of $x$ for
almost all $y$ and vice versa [27]. It has poles at the eigenvalues $E_{n}$ of
$H_{0}$ and a branch cut along the continuous spectrum of $H_{0}$. Using the
Sokhotski–Plemelj formula $\lim_{\epsilon\to 0}\frac{1}{(\lambda-E)\pm
i\epsilon}=\mp i\pi\delta(\lambda-E)+P.V.\left(\frac{1}{\lambda-E}\right)$ in
the sense of distributions [28], the discontinuity of the Green’s function
$G_{0}$ along the cut $\sigma_{c}(H_{0})$ is given by

$\displaystyle G_{0}(x,y|E+i0)-G_{0}(x,y|E-i0)=2\pi
i\chi_{E}(x)\overline{\chi_{E}(y)}\;,$ (2.3)

where we have used the notation $G_{0}(x,y|E\pm i0):=\lim_{\epsilon\to
0}G_{0}(x,y|E\pm i\epsilon)$ in the sense of distribution. For a free
particle, $E$ is real and positive in the above formula. The above formulas
still hold in higher dimensions as well.

Suppose that we first consider one dimensional problems in which $H_{0}$ is
modified by a $\delta$ function interaction supported at the origin

$\displaystyle H=H_{0}-\alpha\delta\;,$ (2.4)

where $\alpha\in\mathbb{R}$. There are several ways to make sense of the above
formal expression $H$. One way is to consider the $\delta$ interaction as a
self-adjoint extension of $H_{0}$. A modern introduction to this subject is
the recent book by Gallone and Michelangeli [29] and the classic reference
elaborating this point of view is the monograph by Albeverio et all [17].
Delta interactions can also be defined through the strong limit of the
resolvent of Hamiltonians with delta interaction replaced by some scaled
function, see [18, 30] for the details. The formulation of the problem in this
section can be extended to higher dimensions as long as the co-dimension
(dimension of the space - dimension of the support of the $\delta$
interaction) is one. We will come back to this issue for possible
generalizations of the problem later on.

It is well-known that Green’s function for the Hamiltonian (2.4) is given by
Krein’s type of formula

$\displaystyle
G(x,y|E)=G_{0}(x,y|E)+\frac{G_{0}(x,0|E)G_{0}(0,y|E)}{\frac{1}{\alpha}-G_{0}(0,0|E)}\;,$
(2.5)

where $G_{0}(x,y|E)$ is the Green’s function for $H_{0}$. It is useful to
define

$\displaystyle\Phi(E):=\frac{1}{\alpha}-G_{0}(0,0|E)\;.$ (2.6)

Some of the books in the context of point interactions use the notation
$\Gamma$ for the function $\Phi$ here. Here and subsequently, as emphasized in
the introduction, we assume that
$H_{0}=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+V(x)$ satisfies some
conditions:

  * •

$H_{0}$ is self-adjoint on some dense domain $D(H_{0})\subset
L^{2}(\mathbb{R})$,

  * •

Spectrum of $H_{0}$ is a disjoint union of discrete $\sigma_{d}(H_{0})$ (set
of eigenvalues) and (absolutely) continuous spectrum $\sigma_{c}(H_{0})$,

  * •

The discrete spectrum has no accumulation point,

  * •

For stability, we assume $H_{0}$ has spectrum bounded below,

and these assumptions are assumed to hold in higher dimensions as well. These
conditions on the spectrum put some mild restrictions on the potential $V$.
Some of the possible choices are stated in the classical work of Reed and
Simon [31].

Then, we claim our first result about the spectral properties of the
Hamiltonian (2.4):

######  Proposition 2.1.

Let $\phi_{k}(x)$ be the bound state wave function of $H_{0}$ associated with
the bound state energy $E_{k}$. Then, the bound state energy $E_{k}^{*}$, when
$H_{0}$ is modified (perturbed) with a delta function interaction
($-\alpha\delta(x)$), satisfies the equation

$\displaystyle\Phi(E)=\frac{1}{\alpha}-G_{0}(0,0|E)=0\;,$ (2.7)

if $\phi_{k}(0)\neq 0$ for this particular $k$. If for this choice of $k$ we
have $\phi_{k}(0)=0$, the bound state energy does not change,
$E_{k}^{*}=E_{k}$. Moreover, the continuous spectrum of the Hamiltonian
modified with delta interaction is the same as that of $H_{0}$.

###### Proof.

We first explicitly show how the pole structure of the full Green’s function
$G(x,y|E)$ is rearranged and the poles of $G_{0}(x,y|E)$, which explicitly
appears as an additive factor in $G(x,y|E)$ actually are cancelled and new
poles appear. Note that from the explicit expression of the full Green’s
function (2.5), one may expect that the poles of the full Green’s function may
contain the poles of $G_{0}$ as well as the zeroes of the function $\Phi(E)$.
Using the eigenfunction expansion of the Green’s function (2.2) and splitting
the term in the summation associated with the isolated simple eigenvalue
$E_{k}$ of $H_{0}$, we obtain

$\displaystyle\hskip-56.9055ptG(x,y|E)=\frac{\phi_{k}(x)\overline{\phi_{k}(y)}}{E_{k}-E}\left(1-\left(1-\frac{(E_{k}-E)}{|\phi_{k}(0)|^{2}}D(\alpha,E)\right)^{-1}\right)+g(x,y|E)+h(x,y|E)$
$\displaystyle+\,\frac{(E_{k}-E)}{(E_{k}-E)D(\alpha,E)-|\phi_{k}(0)|^{2}}\bigg{(}g(x,0|E)g(0,y|E)+g(x,0|E)h(x,0|E)$
$\displaystyle\hskip 56.9055pt+\,g(x,0|E)h(0,y|E)+h(x,0|E)h(0,y|E)\bigg{)}$
$\displaystyle+\,\frac{1}{(E_{k}-E)D(\alpha,E)-|\phi_{k}(0)|^{2}}\bigg{(}g(x,0|E)\phi_{k}(0)\overline{\phi_{k}(y)}+g(0,y|E)\phi_{k}(x)\overline{\phi_{k}(0)}$
$\displaystyle\hskip
56.9055pt+\,h(x,0|E)\phi_{k}(0)\overline{\phi_{k}(y)}+h(0,y|E)\phi_{k}(x)\overline{\phi_{k}(0)}\bigg{)}\;,$
(2.8)

where we have defined the following functions, which are regular near $E_{k}$,
for simplicity

$\displaystyle g(x,y|E)$ $\displaystyle:=$ $\displaystyle\sum_{n\neq
k}\frac{\phi_{n}(x)\overline{\phi_{n}(y)}}{E_{n}-E}\;,$ (2.9) $\displaystyle
h(x,y|E)$ $\displaystyle:=$
$\displaystyle\int_{\sigma_{c}(H_{0})}\frac{\chi_{\lambda}(x)\overline{\chi_{\lambda}(y)}}{\lambda-E}\;d\mu(\lambda)\;,$
(2.10) $\displaystyle D(\alpha,E)$ $\displaystyle:=$
$\displaystyle\frac{1}{\alpha}-\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}}{E_{n}-E}-\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}}{\lambda-E}\;d\mu(\lambda)\;.$
(2.11)

The functions $g,h$ at $x=0$ and $y=0$, and the function $D$ are well-defined
thanks to the fact that the generalized eigenfunctions $\chi_{\lambda}$ in
here have continuous representatives [5, 6]. Except for the first term in
equation (2.8), it is easy to see that all terms are analytic in a
sufficiently small disk around $E=E_{k}$. If we choose $E$ sufficiently close
to $E_{k}$, i.e., if
$\frac{|E_{k}-E|}{|\phi_{k}(0)|^{2}}\left|D(\alpha,E)\right|<1$, the first
term in the above equation becomes

$\displaystyle-\frac{\phi_{k}(x)\overline{\phi_{k}(y)}}{|\phi_{k}(0)|^{2}}\left(\frac{1}{\alpha}-\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}}{E_{n}-E}-\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}}{\lambda-E}\;d\mu(\lambda)\right)+O(1)\;,$
(2.12)

so that $G(x,y|E)$ is regular near $E=E_{k}$ as long as $\phi_{k}(0)\neq 0$.
Hence, the pole $E=E_{k}$ of $G_{0}(x,y|E)$ is not a pole of $G(x,y|E)$ if
$\phi_{k}(0)\neq 0$. Then, the only poles of $G(x,y|E)$ must come from the
zeroes of the function $\Phi(E)$.

Finally, it follows from Weyl’s theorem [31] that the continuous spectrum of
the problem coincides with the initial Hamiltonian since the difference
between the resolvent of the Hamiltonian and the resolvent of the initial
Hamiltonian is of finite rank thanks to the explicit formula

$\displaystyle
R(E)=R_{0}(E)+(\Phi(E))^{-1}\langle\overline{G(\cdot,0|E},\cdot\rangle
G(\cdot,0|E)\;,$ (2.13)

defined on its resolvent set, or as can be seen more naturally in Dirac’s bra-
ket notation,

$\displaystyle R(E)=R_{0}(E)++(\Phi(E))^{-1}R_{0}(E)|0\rangle\langle
0|R_{0}(E)\;.$ (2.14)

∎

######  Lemma 2.1.

If $\alpha>0$ (attractive case) and $\phi_{k}(0)\neq 0$ for some $k\geq 1$,
then the new bound state energies $E_{k}^{*}$ are interlaced between the
eigenvalues of $H_{0}$:

$\displaystyle E_{k-1}<E_{k}^{*}<E_{k}\;.$ (2.15)

For the ground state ($k=0$), we always have $E_{0}^{*}<E_{0}$.

###### Proof.

Since the Green’s function $G_{0}$ has poles at the eigenvalues $E=E_{n}$ of
$H_{0}$ in the complex $E$ plane and has a branch cut along the generalized
eigenvalues of $H_{0}$, it is differentiable everywhere except at its poles
and along the branch cut. Then, by taking the derivative of $G_{0}(0,0|E)$
with respect to $E$ under the summation and integral sign, we formally obtain

$\displaystyle\frac{dG_{0}(0,0|E)}{dE}=\sum_{n=0}^{\infty}\frac{|\phi_{n}(0)|^{2}}{(E_{n}-E)^{2}}+\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}}{(\lambda-E)^{2}}\;d\mu(\lambda)>0\;.$
(2.16)

This implies that $G_{0}(0,0|E)$ is a monotonically increasing function of $E$
between its poles as well as between the largest eigenvalue below the
continuum branch cut and the infimum of the branch cut (which is typically
zero). Moreover, by isolating $n=k$ th term in the sum, it is easy to see that

$\displaystyle\lim_{E\to
E_{k}^{-}}\sum_{n=0}^{\infty}\frac{|\phi_{n}(0)|^{2}}{E_{n}-E}+\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}}{\lambda-E}\;d\mu(\lambda)$
$\displaystyle=$ $\displaystyle\infty\;,$ (2.17) $\displaystyle\lim_{E\to
E_{k}^{+}}\sum_{n=0}^{\infty}\frac{|\phi_{n}(0)|^{2}}{E_{n}-E}+\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}}{\lambda-E}\;d\mu(\lambda)$
$\displaystyle=$ $\displaystyle-\infty\;,$ (2.18)

for all $k=0,1,2,\cdots$, and by taking the limit under the summation and
integral we have

$\displaystyle\lim_{E\to-\infty}\sum_{n=0}^{\infty}\frac{|\phi_{n}(0)|^{2}}{E_{n}-E}+\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}}{\lambda-E}\;d\mu(\lambda)$
$\displaystyle=$ $\displaystyle 0^{+}\;.$ (2.20)

Then, if $\alpha>0$ it follows from the above results that the roots, say
$E_{k}^{*}$, of the equation (2.7) must be located at the points of
intersection of $1/\alpha$ and $G_{0}(0,0|E)$, as shown in Figure 1.

Figure 1: $1/\alpha$ and
$\sum_{n=0}^{\infty}\frac{|\phi_{n}(0)|^{2}}{E_{n}-E}+\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}}{\lambda-E}\;d\mu(\lambda)$
versus $E$ for $\alpha>0$.

This shows that the new bound state energies $E_{k}^{*}$ are shifted downwards
and interlaced between $E_{k-1}$ and $E_{k}$. Let $E_{0}$ be the ground state
energy of $H_{0}$. In contrast to excited states, the new ground state energy
can be as small as possible by choosing $\alpha$ sufficiently large. By the
positivity of the ground state wave functions of Schrödinger operators (thanks
to Kato’s inequality [31]), having no nodes $\phi_{0}(0)\neq 0$, it follows
from Proposition 2.1 that new ground state energy $E_{0}^{*}$ is always less
than the older one.

∎

######  Remark 2.2.

If $\alpha<0$ (repulsive case), then the bound state energies of $H_{0}$ are
shifted upward and interlaced as $E_{k}<E_{k}^{*}<E_{k+1}$ if $\phi_{k}(0)\neq
0$. Otherwise, bound state energies of $H_{0}$ do not change.

######  Remark 2.3.

The above statements are still true if we consider the support of delta
potential at $x=a$. In this case, we have

$\displaystyle
G(x,y|E)=G_{0}(x,y|E)+\frac{G_{0}(x,a|E)G_{0}(a,y|E)}{\frac{1}{\alpha}-G_{0}(a,a|E)}\;,$
(2.21)

and the bound state energies under the addition of delta interaction do not
change if $a$ is at one of the nodes of the bound state wave function of
$H_{0}$, that is, $\phi_{k}(a)=0$. Otherwise, the bound state energies are
obtained from $\Phi(E)=\frac{1}{\alpha}-G_{0}(a,a|E)=0$.

######  Remark 2.4.

It is easy to see from Figure 1 that the number of bound states increases by
one if we add $\delta$ interaction to $H_{0}$. An extra pole is created below
the ground state energy of $H_{0}$ and all the other eigenvalues are
interlaced. Moreover, the bound state energy of $H_{0}$ is invariant under a
particular configuration of delta interaction added to $H_{0}$, that is, if
the support of the delta function is chosen to be at one of the nodes of the
bound state wave function associated with the bound state energy $E_{k}$ of
$H_{0}$, then this bound state energy $E_{k}$ does not change under the
addition (perturbation) of delta interaction to $H_{0}$.

######  Example 2.5.

We can illustrate what we have stated above by working out an exactly solvable
case for $H_{0}$, whose spectrum includes both discrete and continuous parts.
Consider the Schrödinger operator associated with the reflectionless
interaction $V$ given by

$\displaystyle(H_{0}\psi)(x)=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}\psi(x)-\frac{\hbar^{2}}{2m}\kappa^{2}\frac{N(N+1)}{\cosh^{2}(\kappa
x)}\;,$ (2.22)

where $N\in\mathbb{N}$. In this case, the discrete spectrum is given by the
set of eigenvalues
$E_{n}=-\frac{\hbar^{2}\kappa^{2}}{2m}\left(N-n\right)^{2}$, where
$n=0,1,2,\ldots N-1$. The number of bound states is finite and equal to $N$,
corresponding eigenfunctions are given by

$\displaystyle\phi_{n}^{(N)}(x)=\sqrt{\kappa}\left((N-n)\frac{(2N+1-n)!}{n!}\right)^{1/2}P_{N}^{n-N}(\tanh(\kappa
x))\;,$ (2.23)

where $P_{\mu}^{\nu}(x)$ is Legendre function, defined by
$P_{\nu}^{\mu}(x)=\frac{1}{\Gamma(1-\mu)}\left(\frac{1+x}{1-x}\right)^{1/2}{{}_{2}}F_{1}(-\nu,\nu+1;1-\mu;\frac{1-x}{2})$
in terms of hypergeometric functions ${{}_{2}}F_{1}$. In this example, the
continuous spectrum is the positive real axis, i.e,
$\sigma_{c}(H_{0})=[0,\infty)$ and the generalized eigenfunctions are given by
$\chi_{k}^{(N)}(x)=\left(\frac{k}{2\sinh(\pi
k/\kappa)}\right)^{1/2}P_{N}^{ik/\kappa}(\tanh(\kappa x))$ with the
generalized eigenvalues $E_{k}=\frac{\hbar^{2}k^{2}}{2m}$, with
$k\in\mathbb{R}$.

For simplicity, we choose $N=1$. In this case, there is a single bound state
and the associated normalized eigenfunction, given by

$\displaystyle E_{0}$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}\kappa^{2}}{2m}\;,$ (2.24)
$\displaystyle\phi_{0}(x)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\kappa}{2}}\frac{1}{\cosh(\kappa x)}\;.$ (2.25)

Its generalized eigenfunction for this case ($N=1$) is given by

$\displaystyle\chi_{E}(x)=\sqrt{\frac{\kappa}{2\pi}}e^{ikx}\left(\frac{ik-\kappa\tanh(\kappa
x)}{\kappa+ik}\right)\;,$ (2.26)

with the generalized eigenvalues $E=\frac{\hbar^{2}k^{2}}{2m}$. The Green’s
function of this potential problem in appropriate has been discussed in [12]
and it is given as

$\displaystyle\hskip-28.45274ptG_{0}(x,y|E)=\frac{1}{\hbar}\left(\frac{m}{-2E}\right)^{1/2}\exp\left(-\frac{|x-y|\sqrt{-2mE}}{\hbar}\right)-\frac{\kappa}{2\cosh(\kappa
x)\cosh(\kappa y)(E+\frac{\hbar^{2}\kappa^{2}}{2m})}$ $\displaystyle\hskip
56.9055pt\times\Bigg{[}1-\left(1-\frac{\hbar\kappa}{\sqrt{-2mE}}\right)\cosh\left(\kappa|x-y|\left(1+\frac{\sqrt{-2mE}}{\kappa\hbar}\right)\right)\Bigg{]}\;,$
(2.27)

in accordance with the conventions and units that we use here. Since
$\phi_{0}$ has no node, it follows from Proposition 2.1 that the new
eigenvalue due to the addition of $\delta$ interaction to $H_{0}$ must satisfy
the following equation

$\displaystyle\frac{1}{\alpha}-G(0,0|E)=\frac{1}{\alpha}-\frac{1}{\hbar}\left(\frac{m}{-2E}\right)^{1/2}+\frac{\kappa}{2\cosh^{2}(\kappa
a)(E+\frac{\hbar^{2}\kappa^{2}}{2m})}\left(\frac{\hbar\kappa}{\sqrt{-2mE}}\right)=0\;.$
(2.28)

It is easy to see from Figure 2 that

Figure 2: Graphical solution of the equation
$\frac{1}{\alpha}=\frac{1}{\hbar}\left(\frac{m}{-2E}\right)^{1/2}+\frac{\kappa}{2\cosh^{2}(\kappa
a)(E+\frac{\hbar^{2}\kappa^{2}}{2m})}\left(\frac{\hbar\kappa}{\sqrt{-2mE}}\right)$.
Here we have chosen $\alpha=1/2$, $\hbar=2m=1$, $\kappa=1$, and $a=1$.

as long as $a\neq 0$, we have always two bound state energies, one above
$-\frac{\hbar^{2}\kappa^{2}}{2m}$ and one below
$-\frac{\hbar^{2}\kappa^{2}}{2m}$, as expected by our analysis. It is
important to notice that the Green function $G_{0}(0,0|E)$ blows up near its
branch point at $E=0$, which ensures the existence of two new bound states
under the addition of $\delta$ interaction to the original problem. However,
if $a=0$, then $G_{0}(0,0|E)$ vanishes at $E=0$. This implies that we have
only one root and we have a single bound state energy below
$-\frac{\hbar^{2}\kappa^{2}}{2m}$. This is not a generic case and can be
explained by the following symmetry argument. Initially, we have a
reflectionless potential having $\mathbb{Z}_{2}$ symmetry, and it has a single
bound state with even parity. If we include delta interaction to this
potential, we still keep $\mathbb{Z}_{2}$ symmetry if $a=0$. Then, the excited
state must have an odd parity since the systems having such symmetries must
have a definite symmetry and the excited state must be orthogonal to the
ground state having even parity by the positivity [31]. This implies that the
excited state wave function must vanish at $x=0$, which removes the $\delta$
interaction term in the formal Hamiltonian. However, the initial system
$H_{0}$ has only a single bound state, namely the ground state, so the new
system modified by $\delta$ interaction must have a single bound state energy
level if $a=0$.

When $a\neq 0$, the bound state energies can be found explicitly and given by

$\displaystyle E_{0}^{*}$ $\displaystyle=$
$\displaystyle-\left(\sqrt{\frac{\hbar^{2}\kappa^{2}}{2m}+\frac{m\alpha^{2}}{8\hbar^{2}}}+\sqrt{\frac{m\alpha^{2}}{8\hbar^{2}}}\right)^{2}\;,$
(2.29) $\displaystyle E_{1}^{*}$ $\displaystyle=$
$\displaystyle-\left(\sqrt{\frac{\hbar^{2}\kappa^{2}}{2m}+\frac{m\alpha^{2}}{8\hbar^{2}}}-\sqrt{\frac{m\alpha^{2}}{8\hbar^{2}}}\right)^{2}\;.$
(2.30)

###  2.1 A Different Perspective for the Perturbative Estimates on the Bound
State Energies

We now develop a perturbative approximation in finding the eigenvalues of
$H_{0}-\alpha\delta$ with a different perspective. In the standard
perturbation theory [26, 32], we basically start with the so-called Rayleigh-
Schrödinger series expansion for the bound state energies and wave functions
and then substitute these into Schrödinger equation and solve each expansion
term recursively (which has been discussed more rigorously in [31, 33]).

Here, we will follow a somewhat more heuristic approach and assume that
$\alpha<<1$ and $\phi_{k}(0)\neq 0$. According to the previous analysis, new
bound state energies will shift due to the addition of $\delta$ interaction to
$H_{0}$. Let $E=E_{k}^{*}=E_{k}+\delta E_{k}$, where $E_{k}$ are the bound
state energies of $H_{0}$ and $\delta E_{k}$ is the change in the bound state
energy $E_{k}$. Then, according to Proposition 2.1 the new bound state
energies $E_{k}^{*}$ are given by the solutions of

$\displaystyle\frac{1}{\alpha}-\sum_{n=0}^{\infty}\frac{|\phi_{n}(0)|^{2}}{E_{n}-(E_{k}+\delta
E_{k})}-\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}}{\lambda-(E_{k}+\delta
E_{k})}\;d\mu(\lambda)$ $\displaystyle\hskip
28.45274pt=\frac{1}{\alpha}-\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}}{E_{n}-(E_{k}+\delta
E_{k})}+\frac{|\phi_{k}(0)|^{2}}{\delta
E_{k}}-\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}}{\lambda-(E_{k}+\delta
E_{k})}\;d\mu(\lambda)=0\;.$ (2.31)

This is an exact equation determining the bound state energies but it involves
the eigenfunctions at $x=0$. If we expand the terms in the summation and
integral in the powers of $\delta E_{k}$ and multiply the equation by
$\alpha\,\delta E_{k}$, we get

$\displaystyle\delta E_{k}-\alpha\,\delta E_{k}\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}}{E_{n}-E_{k}}\left(1+\frac{\delta
E_{k}}{E_{n}-E_{k}}+O(\delta E_{k}^{2})\right)$ $\displaystyle-\alpha\,\delta
E_{k}\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}}{\lambda-
E_{k}}\;\left(1+\frac{\delta E_{k}}{\lambda-E_{k}}+O(\delta
E_{k}^{2})\right)d\mu(\lambda)+\alpha|\phi_{k}(0)|^{2}+O(\delta
E_{k}^{3})=0\;.$ (2.32)

Let us assume that $\delta E_{k}$ has a power series in $\alpha$, that is,
$\delta E_{k}=E_{k}^{(1)}+E_{k}^{(2)}+\cdots$, where $E_{k}^{(n)}$ corresponds
to the change in the bound state energy of order $\alpha^{n}$. Then, solving
$E_{k}^{(1)}$ and $E_{k}^{(2)}$ term by term, we obtain

$\displaystyle E_{k}^{(1)}$ $\displaystyle=$
$\displaystyle-\alpha|\phi_{k}(0)|^{2}\;,$ (2.33) $\displaystyle E_{k}^{(2)}$
$\displaystyle=$ $\displaystyle-\alpha^{2}|\phi_{k}(0)|^{2}\left(\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}}{E_{n}-E_{k}}+\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}}{\lambda-
E_{k}}d\mu(\lambda)\right)\;,$ (2.34)

which are consistent with classical first and second order formulas in non-
degenerate perturbation theory [7, 26], given by

$\displaystyle E_{k}^{(1)}$ $\displaystyle=$
$\displaystyle\langle\phi_{k},-\alpha\delta\phi_{k}\rangle$ (2.35)
$\displaystyle E_{k}^{(2)}$ $\displaystyle=$ $\displaystyle-\sum_{n\neq
k}\frac{|\langle\phi_{k},-\alpha\delta\phi_{n}\rangle|^{2}}{E_{n}-E_{k}}-\int_{\sigma_{c}(H_{0})}\frac{|\langle\chi_{\lambda},-\alpha\delta\phi_{k}\rangle|^{2}}{\lambda-
E_{k}}d\mu(\lambda)\;.$ (2.36)

The important point to note here is that we are looking for the solution
$\delta E_{k}$ of (2.32) as the formal power series in $\alpha$ by assuming
both $\alpha$ and $\delta E_{k}$ are small so that it is sufficient to use
regular perturbation theory [34]. Let us summarize what we have found as the
following Proposition:

######  Proposition 2.6.

Let $\phi_{k}$ be the eigenfunctions associated with the eigenvalues $E_{k}$
of $H_{0}$ and $\chi_{\lambda}$ be the generalized eigenfunctions of $H_{0}$.
Then, under the addition of $\delta$ interaction with the coupling constant
$-\alpha$, the change in the bound state energies up to the first and second
order in $\alpha$ are formally given by the equations (2.33) and (2.34).

###  2.2 A Different Perspective for the Perturbative Estimates on the Bound
State Wave Function

The contour integral of the resolvent $R(E)=(H-E)^{-1}$ around each simple
eigenvalue $E_{k}^{*}$, we can find the projection operator onto the
eigenspace associated with the eigenvalue $E_{k}^{*}$,

$\displaystyle\mathbb{P}_{k}=-\frac{1}{2\pi i}\oint_{\Gamma_{k}}R(E)\;dE\;,$
(2.37)

where $\Gamma_{k}$ is the closed contour around each simple pole $E_{k}^{*}$,
or equivalently

$\displaystyle\psi_{k}(x)\overline{\psi_{k}(y)}=-\frac{1}{2\pi
i}\oint_{\Gamma_{k}}G(x,y|E)\;dE\;.$ (2.38)

From the explicit expression of the Green’s function (2.5) and residue
theorem, we obtain

$\displaystyle\psi_{k}(x)=\frac{G_{0}(x,0|E_{k}^{*})}{\left(\frac{dG_{0}(0,0|E)}{dE}\bigg{|}_{E=E_{k}^{*}}\right)^{1/2}}\;.$
(2.39)

Using the eigenfunction expansion of $G_{0}$, and $E_{k}^{*}=E_{k}+\delta
E_{k}$, we get

$\displaystyle\psi_{k}(x)=\frac{\left(\sum_{n=0}^{\infty}\frac{\phi_{n}(x)\overline{\phi_{n}(0)}}{E_{n}-(E_{k}+\delta
E_{k})}+\int_{\sigma_{c}(H_{0})}\frac{\chi_{\lambda}(x)\overline{\chi_{\lambda}(0)}}{\lambda-(E_{k}+\delta
E_{k})}d\mu(\lambda)\right)}{\left(\sum_{n=0}^{\infty}\frac{|\phi_{n}(0)|^{2}}{\left(E_{n}-(E_{k}+\delta
E_{k})\right)^{2}}+\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}}{(\lambda-(E_{k}+\delta
E_{k}))^{2}}d\mu(\lambda)\right)^{1/2}}\;.$ (2.40)

If we split the $n=k$ th term in the sums and integrals, and then expand the
above terms in powers of $\delta E_{k}$, then the wave function becomes

$\displaystyle\hskip-28.45274pt\psi_{k}(x)=-\frac{\phi_{k}(x)\overline{\phi_{k}(0)}}{|\phi_{k}(0)|}\left[1+\frac{\delta
E_{k}^{2}}{|\phi_{k}(0)|^{2}}\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}}{(E_{n}-E_{k})^{2}}+\frac{\delta
E_{k}^{2}}{|\phi_{k}(0)|^{2}}\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}}{(\lambda-
E_{k})^{2}}d\mu(\lambda)+O(\delta E_{k}^{3})\right]^{-1/2}$
$\displaystyle\hskip 56.9055pt+\left[1+\frac{\delta
E_{k}^{2}}{|\phi_{k}(0)|^{2}}\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}}{(E_{n}-E_{k})^{2}}+\frac{\delta
E_{k}^{2}}{|\phi_{k}(0)|^{2}}\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}}{(\lambda-
E_{k})^{2}}d\mu(\lambda)+O(\delta E_{k}^{3})\right]^{-1/2}$
$\displaystyle\times\left(\sum_{n\neq
k}\frac{\phi_{n}(x)\overline{\phi_{n}(0)}}{(E_{n}-E_{k})|\phi_{k}(0)|}\left(\delta
E_{k}+\frac{\delta E_{k}^{2}}{(E_{n}-E_{k})}+O(\delta
E_{k}^{3})\right)\right)$ $\displaystyle\hskip 56.9055pt+\left[1+\frac{\delta
E_{k}^{2}}{|\phi_{k}(0)|^{2}}\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}}{(E_{n}-E_{k})^{2}}+\frac{\delta
E_{k}^{2}}{|\phi_{k}(0)|^{2}}\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}}{(\lambda-
E_{k})^{2}}d\mu(\lambda)+O(\delta E_{k}^{3})\right]^{-1/2}$
$\displaystyle\times\left(\int_{\sigma_{c}(H_{0})}\frac{\chi_{\lambda}(x)\overline{\chi_{\lambda}(0)}}{(\lambda-
E_{k})|\phi_{k}(0)|}\left(\delta E_{k}+\frac{\delta E_{k}^{2}}{(\lambda-
E_{k})}+O(\delta E_{k}^{3})\right)d\mu(\lambda)\right)+O(\delta E_{k}^{3})\;,$
(2.41)

where we have assumed $|\frac{\delta E_{k}}{(E_{n}-E_{k})}|<1$ and
$|\frac{\delta E_{k}}{(\lambda-E_{k})}|<1$. Keeping the terms in $\delta
E_{k}$ up to the second order, we obtain

$\displaystyle\hskip-28.45274pt\psi_{k}(x)=\phi_{k}(x)e^{-i\theta_{k}+i\pi}+\sum_{n\neq
k}\frac{\phi_{n}(x)\overline{\phi_{n}(0)}}{|\phi_{k}(0)|}\frac{E_{k}^{(1)}}{E_{n}-E_{k}}+\int_{\sigma_{c}(H_{0})}\frac{\chi_{\lambda}(x)\overline{\chi_{\lambda}(0)}}{|\phi_{k}(0)|}\frac{E_{k}^{(1)}}{\lambda-
E_{k}}d\mu(\lambda)$ $\displaystyle\hskip
56.9055pt-\frac{1}{2}\phi_{k}(x)e^{-i\theta_{k}+i\pi}\left(\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}}{|\phi_{k}(0)|)^{2}}\frac{(E_{k}^{(1)})^{2}}{(E_{n}-E_{k})^{2}}+\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}}{|\phi_{k}(0)|^{2}}\frac{(E_{k}^{(1)})^{2}}{(\lambda-
E_{k})^{2}}d\mu(\lambda)\right)$ $\displaystyle+\sum_{n\neq
k}\frac{\phi_{n}(x)\overline{\phi_{n}(0)}}{(E_{n}-E_{k})|\phi_{k}(0)|}\left(E_{k}^{(2)}+\frac{(E_{k}^{(1)})^{2}}{(E_{n}-E_{k})}\right)$
$\displaystyle\hskip
56.9055pt+\left(\int_{\sigma_{c}(H_{0})}\frac{\chi_{\lambda}(x)\overline{\chi_{\lambda}(0)}}{(\lambda-
E_{k})|\phi_{k}(0)|}\left(E_{k}^{(2)}+\frac{(E_{k}^{(1)})^{2}}{(\lambda-
E_{k})}\right)d\mu(\lambda)\right)+O(\delta E_{k}^{3})\;,$ (2.42)

where $\phi_{k}(0)=|\phi_{k}(0)|e^{i\theta_{k}}$. Using the first order and
second order results (2.35) and (2.36) for the bound state energies, we
finally obtain the bound state wave function for each order:

$\displaystyle\hskip-28.45274pt\psi_{k}^{(0)}(x)$ $\displaystyle=$
$\displaystyle e^{-i\theta_{k}+i\pi}\phi_{k}(x)$ (2.43)
$\displaystyle\psi_{k}^{(1)}(x)$ $\displaystyle=$
$\displaystyle\alpha\;\phi_{k}(0)e^{-i\theta_{k}+i\pi}\sum_{n\neq
k}\frac{\phi_{n}(x)\overline{\phi_{n}(0)}}{E_{n}-E_{k}}+\alpha\;\phi_{k}(0)e^{-i\theta_{k}+i\pi}\int_{\sigma_{c}(H_{0})}\frac{\chi_{\lambda}(x)\overline{\chi_{\lambda}(0)}}{\lambda-
E_{k}}d\mu(\lambda)$ (2.44) $\displaystyle\psi_{k}^{(2)}(x)$ $\displaystyle=$
$\displaystyle-\frac{\alpha^{2}}{2}\phi_{k}(x)e^{-i\theta_{k}+i\pi}|\phi_{k}(0)|^{2}\left(\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}}{(E_{n}-E_{k})^{2}}+\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}}{(\lambda-
E_{k})^{2}}d\mu(\lambda)\right)$ (2.45)
$\displaystyle\hskip-28.45274pt+\alpha^{2}\phi_{k}(0)e^{-i\theta_{k}+i\pi}\sum_{n\neq
k}\frac{\phi_{n}(x)\overline{\phi_{n}(0)}}{(E_{n}-E_{k})}\Bigg{(}\sum_{m\neq
k}\frac{|\phi_{m}(0)|^{2}}{E_{m}-E_{k}}+\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}}{\lambda-
E_{k}}d\mu(\lambda)-\frac{|\phi_{k}(0)|^{2}}{(E_{n}-E_{k})}\Bigg{)}$
$\displaystyle\hskip-71.13188pt+\alpha^{2}\phi_{k}(0)e^{-i\theta_{k}+i\pi}\int_{\sigma_{c}(H_{0})}\frac{\chi_{\lambda}(x)\overline{\chi_{\lambda}(0)}}{(\lambda-
E_{k})}\Bigg{(}\sum_{m\neq
k}\frac{|\phi_{m}(0)|^{2}}{E_{m}-E_{k}}+\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda^{\prime}}(0)|^{2}}{\lambda^{\prime}-E_{k}}d\mu(\lambda^{\prime})-\frac{|\phi_{k}(0)|^{2}}{\lambda-
E_{k}}\Bigg{)}d\mu(\lambda)\;.$

######  Remark 2.7.

According to standard regular perturbation theory results, the normalized wave
functions for the bound states up to second order in $\alpha$ are given by

$\displaystyle\psi_{k}^{(0)}(x)$ $\displaystyle=$ $\displaystyle\phi_{k}(x)$
(2.46) $\displaystyle\psi_{k}^{(1)}(x)$ $\displaystyle=$
$\displaystyle\sum_{n\neq
k}\frac{\langle\phi_{n},-\alpha\delta\phi_{k}\rangle}{E_{k}-E_{n}}\phi_{n}(x)+\int_{\sigma_{c}(H_{0})}\frac{\langle\chi_{\lambda},-\alpha\delta\phi_{k}\rangle}{\lambda-
E_{k}}\chi_{\lambda}(x)d\mu(\lambda)$ (2.47) $\displaystyle\psi_{k}^{(2)}(x)$
$\displaystyle=$ $\displaystyle\sum_{n\neq k}\sum_{m\neq
k}\frac{\langle\phi_{n},-\alpha\delta\phi_{m}\rangle\langle\phi_{m},-\alpha\delta\phi_{k}\rangle}{(E_{k}-E_{n})(E_{k}-E_{m})}\phi_{n}(x)-\sum_{n\neq
k}\frac{\langle\phi_{n},-\alpha\delta\phi_{k}\rangle\langle\phi_{k},-\alpha\delta\phi_{k}\rangle}{(E_{k}-E_{n})^{2}}\phi_{n}(x)$
(2.48) $\displaystyle-$ $\displaystyle\frac{1}{2}\sum_{n\neq
k}\frac{\langle\phi_{k},-\alpha\delta\phi_{n}\rangle\langle\phi_{n},-\alpha\delta\phi_{k}\rangle}{(E_{k}-E_{n})^{2}}\phi_{k}(x)$
$\displaystyle+$ $\displaystyle\int_{\sigma_{c}(H_{0})}\left(\sum_{n\neq
k}\frac{\langle\phi_{n},-\alpha\delta\chi_{\lambda}\rangle\langle\chi_{\lambda},-\alpha\delta\phi_{k}\rangle}{(E_{k}-E_{n})(E_{k}-\lambda)}\phi_{n}(x)\right)d\mu(\lambda)$
$\displaystyle+$ $\displaystyle\int_{\sigma_{c}(H_{0})}\left(\sum_{m\neq
k}\frac{\langle\chi_{\lambda},-\alpha\delta\phi_{m}\rangle\langle\phi_{m},-\alpha\delta\phi_{k}\rangle}{(E_{k}-E_{m})(E_{k}-\lambda)}\chi_{\lambda}(x)\right)d\mu(\lambda)$
$\displaystyle+$
$\displaystyle\int_{\sigma_{c}(H_{0})}\left(\int_{\sigma_{c}(H_{0})}\frac{\langle\chi_{\lambda},-\alpha\delta\chi_{\lambda^{\prime}}\rangle\langle\chi_{\lambda^{\prime}},-\alpha\delta\phi_{k}\rangle}{(E_{k}-\lambda)(E_{k}-\lambda^{\prime})}\chi_{\lambda}(x)d\mu(\lambda^{\prime})\right)d\mu(\lambda)$
$\displaystyle\hskip-65.44142pt-\int_{\sigma_{c}(H_{0})}\frac{\langle\chi_{\lambda},-\alpha\delta\phi_{k}\rangle\langle\phi_{k},-\alpha\delta\phi_{k}\rangle}{(E_{k}-\lambda)^{2}}d\mu(\lambda)\;\chi_{\lambda}(x)-\frac{1}{2}\int_{\sigma_{c}(H_{0})}\frac{\langle\phi_{k},-\alpha\delta\chi_{\lambda}\rangle\langle\chi_{\lambda},-\alpha\delta\phi_{k}\rangle}{(E_{k}-\lambda)^{2}}d\mu(\lambda)\;\phi_{k}(x)\;.$

These are completely consistent with our results up to a phase factor
$e^{-i\theta_{k}+i\pi}$. (Second order perturbation result for the normalized
wave function only in the discrete case has been given in [35], but clearly it
can be generalized as we have stated here).

Let us summarize our findings as

######  Proposition 2.8.

The bound state wave functions $\psi_{k}(x)$ for the Hamiltonian $H_{0}$
modified by the $\delta$ interaction are given by (2.39) in terms of the
associated bound state energies $E_{k}^{*}$, which are the solutions of
$\frac{1}{\alpha}-G_{0}(0,0|E)=0$. The expansion of the bound state wave
function $\psi_{k}(x)$ in terms of the coupling constants $\alpha$ up to
second order are given by (2.43), (2.44), and (2.45).

###  2.3 Generalized Eigenfunctions for $H_{0}$ Modified by $\delta$
Interactions

Using the Green’s function (2.5), we get

$\displaystyle G(x,y|E+i0)-G(x,y|E-i0)=G_{0}(x,y|E+i0)-G_{0}(x,y|E-i0)$
$\displaystyle\hskip
56.9055pt+\frac{G_{0}(x,0|E+i0)G_{0}(0,y|E+i0)}{\Phi(E+i0)}-\frac{G_{0}(x,0|E-i0)G_{0}(0,y|E-i0)}{\Phi(E-i0)}\;,$
(2.49)

where we have defined $\Phi(E)=\frac{1}{\alpha}-G_{0}(0,0|E)$. Using the fact
that

$\displaystyle G_{0}(x,y|E+i0)-G_{0}(x,y|E-i0)=2\pi
i\chi_{E}(x)\overline{\chi_{E}(y)}\;,$ (2.50)

and adding and subtracting the terms
$\frac{G_{0}(x,0|E+i0)G_{0}(0,y|E-i0)}{\Phi(E+i0)}$ and
$\frac{G_{0}(x,0|E+i0)G_{0}(0,y|E-i0)}{\Phi(E-i0)}$, we obtain

$\displaystyle G(x,y|E+i0)-G(x,y|E-i0)=2\pi
i\chi_{E}(x)\overline{\chi_{E}(y)}$ $\displaystyle\hskip
56.9055pt+\frac{G_{0}(x,0|E+i0)G_{0}(0,y|E+i0)}{\Phi(E+i0)}-\frac{G_{0}(x,0|E+i0)G_{0}(0,y|E-i0)}{\Phi(E+i0)}$
$\displaystyle\hskip
56.9055pt+\frac{G_{0}(x,0|E+i0)G_{0}(0,y|E-i0)}{\Phi(E+i0)}-\frac{G_{0}(x,0|E-i0)G_{0}(0,y|E-i0)}{\Phi(E-i0)}$
$\displaystyle\hskip
56.9055pt+\frac{G_{0}(x,0|E+i0)G_{0}(0,y|E-i0)}{\Phi(E-i0)}-\frac{G_{0}(x,0|E+i0)G_{0}(0,y|E-i0)}{\Phi(E-i0)}\;.$
(2.51)

The Second and the third terms in the above equation can be combined as

$\displaystyle\hskip-28.45274pt\frac{G_{0}(x,0|E+i0)G_{0}(0,y|E+i0)}{\Phi(E+i0)}-\frac{G_{0}(x,0|E+i0)G_{0}(0,y|E-i0)}{\Phi(E+i0)}$
$\displaystyle\hskip 85.35826pt=2\pi
i\frac{G_{0}(x,0|E+i0)}{\Phi(E+i0)}\overline{\chi_{E}(y)}\chi_{E}(0)\;.$
(2.52)

It follows from the explicit form of the function $\Phi$ that

$\displaystyle\left(\frac{1}{\Phi(E+i0)}-\frac{1}{\Phi(E-i0)}\right)$
$\displaystyle=$
$\displaystyle\frac{\Phi(E-i0)-\Phi(E+i0)}{\Phi(E+i0)\Phi(E-i0)}=\frac{G_{0}(0,0|E+i0)-G_{0}(0,0|E-i0)}{\Phi(E+i0)\Phi(E-i0)}$
(2.53) $\displaystyle=$ $\displaystyle\frac{2\pi
i\chi_{E}(0)\overline{\chi_{E}(0)}}{\Phi(E+i0)\Phi(E-i0)}\;.$

Hence, we obtain

$\displaystyle\frac{G_{0}(x,0|E+i0)G_{0}(0,y|E-i0)}{\Phi(E+i0)}-\frac{G_{0}(x,0|E+i0)G_{0}(0,y|E-i0)}{\Phi(E-i0)}$
$\displaystyle\hskip 56.9055pt=2\pi
i\frac{G_{0}(x,0|E+i0)\chi_{E}(0)}{\Phi(E+i0)}\overline{\left(\frac{\chi_{E}(0)G_{0}(0,y|E-i0)}{\Phi(E-i0)}\right)}\;,$
(2.54)

where we have used the fact that $\overline{\Phi(E)}=\Phi(\overline{E})$ and
$\overline{G(x,y|E)}=G(x,y|\overline{E})$. For the rest of the terms, we have

$\displaystyle\frac{G_{0}(x,0|E+i0)G_{0}(0,y|E-i0)}{\Phi(E-i0)}-\frac{G_{0}(x,0|E-i0)G_{0}(0,y|E-i0)}{\Phi(E-i0)}$
$\displaystyle\hskip 56.9055pt=2\pi
i\chi_{E}(x)\overline{\left(\frac{\chi_{E}(0)G_{0}(0,y|E+i0)}{\Phi(E+i0)}\right)}\;.$
(2.55)

Putting all these results together, we finally obtain

$\displaystyle\hskip-28.45274ptG(x,y|E+i0)-G(x,y|E-i0)=2\pi
i\chi_{E}(x)\overline{\chi_{E}(y)}+2\pi
i\frac{G_{0}(x,0|E+i0)\chi_{E}(0)}{\Phi(E+i0)}\overline{\chi_{E}(y)}$
$\displaystyle\hskip-28.45274pt+2\pi
i\chi_{E}(x)\overline{\left(\frac{\chi_{E}(0)G_{0}(0,y|E+i0)}{\Phi(E+i0)}\right)}+2\pi
i\left(\frac{G_{0}(x,0|E+i0)\chi_{E}(0)}{\Phi(E+i0)}\right)\overline{\left(\frac{\chi_{E}(0)G_{0}(0,y|E-i0)}{\Phi(E-i0)}\right)}\;,$
(2.56)

or this can be further simplified to

$\displaystyle\hskip-28.45274ptG(x,y|E+i0)-G(x,y|E-i0)=2\pi
i\bigg{(}\chi_{E}(x)+\frac{G_{0}(x,0|E+i0)\chi_{E}(0)}{\Phi(E+i0)}\bigg{)}$
$\displaystyle\hskip
170.71652pt\times\overline{\bigg{(}\chi_{E}(y)+\frac{G_{0}(y,0|E+i0)\chi_{E}(0)}{\Phi(E+i0)}\bigg{)}}\;.$
(2.57)

This shows that the new generalized eigenfunction $\eta_{E}(x)$ for the
modified system is given by

$\displaystyle\eta_{E}(x)=\chi_{E}(x)+\frac{G_{0}(x,0|E+i0)\chi_{E}(0)}{\Phi(E+i0)}\;.$
(2.58)

This result can also be justified formally by directly computing the action of
the operator $H_{0}-\alpha\delta(x)-E$:

$\displaystyle\hskip-28.45274pt(H_{0}-E-\alpha\delta(x))\left(\chi_{E}(x)+\frac{G_{0}(x,0|E+i0)\chi_{E}(0)}{\Phi(E+i0)}\right)$
$\displaystyle\hskip
56.9055pt=-\alpha\chi_{E}(0)\delta(x)+\frac{\chi_{E}(0)}{\Phi(E+i0)}\delta(x)-\alpha\frac{G_{0}(0,0|E+i0)\chi_{E}(0)}{\Phi(E+i0)}\delta(x)=0\;,$
(2.59)

where we have used $\Phi(E+i0)=\frac{1}{\alpha}-G_{0}(0,0|E+i0)$. Let us
summarize our findings as

######  Proposition 2.9.

The generalized eigenfunctions of $H_{0}$ modified by $\delta$ interactions
are given by the equation (2.58).

######  Example 2.10.

Let $H_{0}$ be the free Hamiltonian in $L^{2}(\mathbb{R})$. It is well-known
that the generalized eigenfunctions are given by $\chi_{E}(x)=e^{ikx}$, with
generalized eigenvalues $E=\frac{\hbar^{2}k^{2}}{2m}$. Then, using the
explicit expression for the Green’s function
$G_{0}(x,y|E+i0)=\frac{im}{\hbar^{2}k}e^{ik|x-y|}$, the above formula (2.58)
yields

$\displaystyle\eta_{E}(x)=e^{ikx}+\frac{im}{\hbar^{2}k}\frac{e^{ik|x|}}{\left(\frac{1}{\alpha}-\frac{im}{\hbar^{2}k}\right)}\;.$
(2.60)

From this, one can easily find the well-known reflection and transmission
coefficients by expressing (2.60) for $x<0$ and $x>0$, respectively [36].

##  3 Singular Modifications by $\delta$ Interactions

When the co-dimension (dimension of the space - dimension of the support of
the $\delta$ interaction) is greater than one (e.g., point delta interaction
in two and three dimensions, delta interaction supported by a surface in three
dimensions), we need to define delta function interaction by a renormalization
procedure. The reason for this is essentially based on the singularity of the
Green’s function for free Hamiltonians $H_{0}$ in two and three dimensions.
The history of the subject is rather rich and there has been a vast amount of
material in the physics literature, see e.g., [19, 20, 21, 22, 23]. The
subject has been also discussed thoroughly from a more mathematical point of
view in classic monographs [17, 18] as well as in a more recent work [29].
Here we now consider the Schrödinger operators $H_{0}$ with the same
assumption discussed before and we study the spectrum of $H_{0}$ modified by
delta interactions.

It is useful to express the Green’s function $G_{0}$ in terms of the heat
kernel $K_{t}(x,y)$ associated with the operator
$H_{0}=-\frac{\hbar^{2}}{2m}\Delta+V$, given by

$\displaystyle G_{0}(x,y|E)=\int_{0}^{\infty}K_{t}(x,y)e^{tE}dt\;,$ (3.1)

where $\operatorname{Re}(E)<0$ and $H_{0}K_{t}(x,y)=i\frac{\partial}{\partial
t}K_{t}(x,y)$ and defined for other values of $E$ in the complex $E$ plane
through analytical continuation. We note that the first term in the short time
asymptotic expansion of the diagonal heat kernel for any self-adjoint elliptic
second order differential operator [37] in $d$ dimensions, is given by

$\displaystyle K_{t}(x,x)\sim t^{-d/2}\;.$ (3.2)

This gives rise to the divergence near $t=0$ in the integral representation of
the Green’s function (3.1). In order to make sense of such singular
interactions, one must first introduce a cut-off $\epsilon>0$ and regularize
the Hamiltonian. This could be done by replacing the delta interaction by the
heat kernel $K_{\epsilon/2}(x,0)$, which converges to $\delta(x)$ as
$\epsilon\to 0$ in the distributional sense. Then, we make the coupling
constant $\alpha$ dependent on the cut-off in such a way that the regularized
Hamiltonian has a non-trivial limit (in the norm resolvent sense) as we remove
the cut-off. A natural choice for the coupling constant is given by

$\displaystyle\frac{1}{\alpha(\epsilon)}=\frac{1}{\alpha_{R}(M)}+\int_{\epsilon}^{\infty}K_{t}(0,0)e^{tM}dt\;,$
(3.3)

where $M$ is the renormalization scale and could be eliminated in favor of a
physical parameter by imposing the renormalization condition. For instance,
$M$, and therefore $\alpha_{R}$, can be related to the bound state energy of
the particle moving under the addition of $\delta$ interaction to $H_{0}$, say
$-\mu^{2}$ (this requirement, in general, leads to a flow in the space of
coupling constants $\alpha_{R}$ [38]). A special choice is to set
${1\over\alpha_{R}}=0$ while demanding $M=-\mu^{2}$, which is particularly
convenient for bound state calculations.

Then, taking the formal limit as $\epsilon\to 0$, we obtain the integral
kernel of the resolvent or Green’s function, given by Krein’s type of formula

$\displaystyle
G(x,y|E)=G_{0}(x,y|E)+\frac{G_{0}(x,0|E)G_{0}(0,y|E)}{\frac{1}{\alpha_{R}}-\sum_{n=0}^{\infty}\frac{|\phi_{n}(0)|^{2}(E+\mu^{2})}{(E_{n}-E)(E_{n}+\mu^{2})}-\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}(E+\mu^{2})}{(\lambda+\mu^{2})(\lambda-E)}\;d\mu(\lambda)}\;,$
(3.4)

where we have used the eigenfunction expansion of the heat kernel. Such
specific examples are examined in [24] in the context of path integrals in two
and three dimensions. However, there is no explicit derivation in this work,
showing that the poles of the free resolvent are cancelled in the final
expression.

By following the same line of argument developed for the one-dimensional case,
we can now explicitly show how the poles of the Hamiltonian $H_{0}$ disappear
with the addition of $\delta$ interaction under the same assumptions about the
spectrum of $H_{0}$ as in the one dimensional case.

Tom simplify our arguments, we assume a purely discrete spectrum, the
generalization to include a continuum woth generalized eigenfunctions is
fairly straightforward. We again split the term in the eigenfunction expansion
of the Green’s function associated with the isolated simple eigenvalue $E_{k}$
of $H_{0}$:

$\displaystyle\hskip-28.45274ptG(x,y|E)=\sum_{n\neq
k}\frac{\phi_{n}(x)\overline{\phi_{n}(y)}}{E_{n}-E}+\frac{\phi_{k}(x)\overline{\phi_{k}(y)}}{E_{k}-E}+\frac{\left(\sum_{n\neq
k}\frac{\phi_{n}(x)\overline{\phi_{n}(0)}}{E_{n}-E}\right)\left(\sum_{n\neq
k}\frac{\phi_{n}(0)\overline{\phi_{n}(y)}}{E_{n}-E}\right)}{\frac{1}{\alpha_{R}}-\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}(E+\mu^{2})}{(E_{n}-E)(E_{n}+\mu^{2})}-\frac{|\phi_{k}(0)|^{2}(E+\mu^{2})}{(E_{k}-E)(E_{k}+\mu^{2})}}$
$\displaystyle\hskip 28.45274pt+\frac{\left(\sum_{n\neq
k}\frac{\phi_{n}(x)\overline{\phi_{n}(0)}}{E_{n}-E}\right)\left(\frac{\phi_{k}(0)\overline{\phi_{k}(y)}}{E_{k}-E}\right)}{\frac{1}{\alpha_{R}}-\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}(E+\mu^{2})}{(E_{n}-E)(E_{n}+\mu^{2})}-\frac{|\phi_{k}(0)|^{2}(E+\mu^{2})}{(E_{k}-E)(E_{k}+\mu^{2})}}$
$\displaystyle\hskip
56.9055pt+\frac{\left(\frac{\phi_{k}(x)\overline{\phi_{k}(0)}}{E_{k}-E}\right)\left(\sum_{n\neq
k}\frac{\phi_{n}(0)\overline{\phi_{n}(y)}}{E_{n}-E}\right)}{\frac{1}{\alpha_{R}}-\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}(E+\mu^{2})}{(E_{n}-E)(E_{n}+\mu^{2})}-\frac{|\phi_{k}(0)|^{2}(E+\mu^{2})}{(E_{k}-E)(E_{k}+\mu^{2})}}$
$\displaystyle\hskip
85.35826pt+\frac{\left(\frac{\phi_{k}(x)\overline{\phi_{k}(0)}}{E_{k}-E}\right)\left(\frac{\phi_{k}(0)\overline{\phi_{k}(y)}}{E_{k}-E}\right)}{\frac{1}{\alpha_{R}}-\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}(E+\mu^{2})}{(E_{n}-E)(E_{n}+\mu^{2})}-\frac{|\phi_{k}(0)|^{2}(E+\mu^{2})}{(E_{k}-E)(E_{k}+\mu^{2})}}\;.$
(3.5)

If we combine the second and the last term in the above expression, we obtain

$\displaystyle\hskip-28.45274ptG(x,y|E)=\frac{\phi_{k}(x)\overline{\phi_{k}(y)}}{E_{k}-E}\left(1-\left(1-\frac{(E_{k}-E)}{|\phi_{k}(0)|^{2}}\left(\frac{1}{\alpha_{R}}-\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}}{E_{n}-E}+\frac{|\phi_{k}(0)|^{2}}{E_{k}+\mu^{2}}\right)\right)^{-1}\right)$
$\displaystyle\hskip 28.45274pt+\sum_{n\neq
k}\frac{\phi_{n}(x)\overline{\phi_{n}(y)}}{E_{n}-E}+(E_{k}-E)\frac{\left(\sum_{n\neq
k}\frac{\phi_{n}(x)\overline{\phi_{n}(0)}}{E_{n}-E}\right)\left(\sum_{n\neq
k}\frac{\phi_{n}(0)\overline{\phi_{n}(y)}}{E_{n}-E}\right)}{(E_{k}-E)\left(\frac{1}{\alpha_{R}}-\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}(E+\mu^{2})}{(E_{n}-E)(E_{n}+\mu^{2})}\right)-\frac{|\phi_{k}(0)|^{2}(E+\mu^{2})}{(E_{k}+\mu^{2})}}$
$\displaystyle\hskip 56.9055pt+\frac{\left(\sum_{n\neq
k}\frac{\phi_{n}(x)\overline{\phi_{n}(0)}}{E_{n}-E}\right)\left(\phi_{k}(0)\overline{\phi_{k}(y)}\right)}{(E_{k}-E)\left(\frac{1}{\alpha_{R}}-\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}(E+\mu^{2})}{(E_{n}-E)(E_{n}+\mu^{2})}\right)-\frac{|\phi_{k}(0)|^{2}(E+\mu^{2})}{(E_{k}-E)(E_{k}+\mu^{2})}}$
$\displaystyle\hskip
85.35826pt+\frac{\left(\phi_{k}(x)\overline{\phi_{k}(0)}\right)\left(\sum_{n\neq
k}\frac{\phi_{n}(0)\overline{\phi_{n}(y)}}{E_{n}-E}\right)}{(E_{k}-E)\left(\frac{1}{\alpha_{R}}-\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}(E+\mu^{2})}{(E_{n}-E)(E_{n}+\mu^{2})}\right)-\frac{|\phi_{k}(0)|^{2}(E+\mu^{2})}{(E_{k}+\mu^{2})}}\;.$
(3.6)

Except for the first term, it is easy to see that all terms are regular near
$E=E_{k}$. For the first term, if we choose $E$ sufficiently close to $E_{k}$,
i.e., if
$\frac{|E_{k}-E|}{|\phi_{k}(0)|^{2}}\left|\frac{1}{\alpha_{R}}-\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}}{E_{n}-E}+\frac{|\phi_{k}(0)|^{2}}{E_{k}+\mu^{2}}\right|<1$,
the first term in the above equation becomes

$\displaystyle-\frac{\phi_{k}(x)\overline{\phi_{k}(y)}}{|\phi_{k}(0)|^{2}}\left(\frac{1}{\alpha_{R}}-\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}}{E_{n}-E}+\frac{|\phi_{k}(0)|^{2}}{E_{k}+\mu^{2}}\right)+O(|E_{k}-E|^{2})$
(3.7)

so that $G(x,y|E)$ is regular near $E=E_{k}$ as long as $\phi_{k}(0)\neq 0$.
If we assume the presence of a continuous spectrum, this part of the spectrum
does not change by the same reasoning as given before. Hence we have

######  Proposition 3.1.

Let $\phi_{k}(x)$ be the bound state wave function of $H_{0}$ associated with
the bound state energy $E_{k}$. Then, the bound state energies $E_{k}^{*}$ for
$H_{0}$ modified (perturbed) with attractive delta interactions satisfy the
equation

$\displaystyle\frac{1}{\alpha_{R}}-\sum_{n=0}^{\infty}\frac{|\phi_{n}(0)|^{2}(E+\mu^{2})}{(E_{n}-E)(E_{n}+\mu^{2})}-\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}(E+\mu^{2})}{(\lambda-E)(\lambda+\mu^{2})}\;d\mu(\lambda)=0\;,$
(3.8)

if $\phi_{k}(0)\neq 0$ for some this particular $k$. If for this choice of $k$
we have $\phi_{k}(0)=0$, the bound state energies do not change,
$E_{k}^{*}=E_{k}$. Moreover, the continuous spectrum of the Hamiltonian
modified with delta interaction is the same as that of $H_{0}$.

In the renormalized case, the interlacing of the energy eigenvalues is exactly
the same as the one in Lemma 2.1, where renormalization is not required, and
if $\alpha_{R}<0$, then all the bound state energies are shifted upward and we
have $E_{k}<E_{k}^{*}<E_{k+1}$.

######  Remark 3.2.

Note that these results can be interpreted as a generalization of the well-
known Sturm comparison theorems to the singular delta interactions, it is
remarkable that even the renormalized case has this property.

By following the same line of arguments as in the regular problem, we now
develop a perturbative approximation to the eigenvalues for $H_{0}$ by the
addition of the singular $\delta$ interaction. We assume that $\alpha_{R}<<1$.
Let $E=E_{k}^{*}=E_{k}+\delta E_{k}$, where $E_{k}$ are the bound state
energies of $H_{0}$. Then, the poles of the Green’s function $G(x,y|E)$ are
given by the solutions of

$\displaystyle\frac{1}{\alpha_{R}}-\sum_{n=0}^{\infty}\frac{|\phi_{n}(0)|^{2}(E_{k}+\delta
E_{k}+\mu^{2})}{(E_{n}+\mu^{2})(E_{n}-(E_{k}+\delta
E_{k}))}-\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}(E_{k}+\delta
E_{k}+\mu^{2})}{(\lambda+\mu^{2})(\lambda-(E_{k}+\delta
E_{k}))}\;d\mu(\lambda)$ $\displaystyle\hskip
56.9055pt=\frac{1}{\alpha_{R}}-\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}(E_{k}+\delta
E_{k}+\mu^{2})}{(E_{n}+\mu^{2})(E_{n}-(E_{k}+\delta
E_{k}))}+\frac{|\phi_{k}(0)|^{2}(E_{k}+\delta
E_{k}+\mu^{2})}{(E_{k}+\mu^{2})(\delta E_{k})}$ $\displaystyle\hskip
85.35826pt-\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}(E_{k}+\delta
E_{k}+\mu^{2})}{(\lambda+\mu^{2})(\lambda-(E_{k}+\delta
E_{k}))}\;d\mu(\lambda)=0\;.$ (3.9)

If we expand above expressions in the powers of $\delta E_{k}$ and multiply
the equation by $\alpha_{R}\delta E_{k}$, we get

$\displaystyle\delta
E_{k}+\alpha_{R}|\phi_{k}(0)|^{2}+\frac{\alpha_{R}|\phi_{k}(0)|^{2}}{E_{k}+\mu^{2}}\delta
E_{k}-\alpha_{R}\delta E_{k}\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}(E_{k}+\mu^{2})}{(E_{n}-E_{k})(E_{n}+\mu^{2})}\left(1+\frac{\delta
E_{k}}{E_{n}-E_{k}}+O(\delta E_{k}^{2})\right)$
$\displaystyle-\alpha_{R}\delta E_{k}\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}\delta
E_{k}}{(E_{n}-E_{k})(E_{n}+\mu^{2})}\left(1+\frac{\delta
E_{k}}{E_{n}-E_{k}}+O(\delta E_{k}^{2})\right)$
$\displaystyle-\alpha_{R}\delta
E_{k}\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}(E_{k}+\mu^{2})}{(\lambda-
E_{k})(\lambda+\mu^{2})}\left(1+\frac{\delta E_{k}}{\lambda-E_{k}}+O(\delta
E_{k}^{2})\right)\;d\mu(\lambda)$ $\displaystyle-\alpha_{R}\delta
E_{k}\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}\delta
E_{k}}{(\lambda-E_{k})(\lambda+\mu^{2})}\left(1+\frac{\delta E_{k}}{\lambda-
E_{k}}+O(\delta E_{k}^{2})\right)\;d\mu(\lambda)+O(\delta E_{k}^{3})=0\;.$
(3.10)

Let us assume that $\delta E_{k}$ can be expandable in the power series of
$\alpha_{R}$, that is, $\delta E_{k}=E_{k}^{(1)}+E_{k}^{(2)}+\cdots$, where
$E_{k}^{(n)}$ corresponds to the change in the bound state energy of order
$\alpha_{R}^{n}$. Then, solving $E_{k}^{(1)}$ and $E_{k}^{(2)}$ term by term,
we obtain

$\displaystyle\hskip-28.45274ptE_{k}^{(1)}$ $\displaystyle=$
$\displaystyle-\alpha_{R}|\phi_{k}(0)|^{2}\;,$ (3.11)
$\displaystyle\hskip-28.45274ptE_{k}^{(2)}$ $\displaystyle=$
$\displaystyle\alpha_{R}^{2}|\phi_{k}(0)|^{2}\left(\frac{|\phi_{k}(0)|^{2}}{E_{k}+\mu^{2}}-\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}(E_{k}+\mu^{2})}{(E_{n}-E_{k})(E_{n}+\mu^{2})}-\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}(E_{k}+\mu^{2})}{(\lambda-
E_{k})(\lambda+\mu^{2})}d\mu(\lambda)\right)\;.$ (3.12)

It is important to notice that the first order result in the bound state
energy is the same as the case where the renormalization is not required,
except that $\alpha$ is replaced by the renormalized coupling constant
$\alpha_{R}$. However, the above second order result in the bound state energy
is completely different from the case where the renormalization is not
required. For the wave function, we formally obtain the same formula (2.42)
for the wave function expansion in $\delta E_{k}$. Substituting the results
for first and second order eigenvalues given above, we find

$\displaystyle\psi_{k}^{(0)}(x)$
$\displaystyle=e^{-i\theta_{k}+i\pi}\phi_{k}(x)\;,$ (3.13)
$\displaystyle\psi_{k}^{(1)}(x)$
$\displaystyle=\alpha_{R}\phi_{k}(0)e^{-i\theta_{k}+i\pi}\left(\sum_{n\neq
k}\frac{\phi_{n}(x)\overline{\phi_{n}(0)}}{E_{n}-E_{k}}+\int_{\sigma_{c}(H_{0})}\frac{\chi_{\lambda}(x)\overline{\chi_{\lambda}(0)}}{\lambda-
E_{k}}d\mu(\lambda)\right)\;,$ (3.14)

and

$\displaystyle\psi_{k}^{(2)}(x)$
$\displaystyle=-\frac{\alpha_{R}^{2}}{2}\phi_{k}(x)e^{-i\theta_{k}+i\pi}|\phi_{k}(0)|^{2}\left(\sum_{n\neq
k}\frac{|\phi_{n}(0)|^{2}}{(E_{n}-E_{k})^{2}}+\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}}{(\lambda-
E_{k})^{2}}d\mu(\lambda)\right)$
$\displaystyle\hskip-28.45274pt+\alpha_{R}^{2}\sum_{n\neq
k}\frac{\phi_{n}(x)\overline{\phi_{n}(0)}}{(E_{n}-E_{k})}\phi_{k}(0)e^{-i\theta_{k}+i\pi}\Bigg{[}\sum_{m\neq
k}\frac{|\phi_{m}(0)|^{2}(E_{k}+\mu^{2})}{(E_{m}-E_{k})(E_{m}+\mu^{2})}+\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda}(0)|^{2}(E_{k}+\mu^{2})}{(\lambda-
E_{k})(\lambda+\mu^{2})}d\mu(\lambda)$ $\displaystyle\hskip
113.81102pt-\frac{|\phi_{k}(0)|^{2}(E_{n}+\mu^{2})}{(E_{n}-E_{k})(E_{k}+\mu^{2})}\Bigg{]}$
$\displaystyle\hskip-28.45274pt+\alpha_{R}^{2}\int_{\sigma_{c}(H_{0})}\frac{\chi_{\lambda}(x)\overline{\chi_{\lambda}(0)}}{(\lambda-
E_{k})}\phi_{k}(0)e^{-i\theta_{k}+i\pi}\Bigg{[}\sum_{m\neq
k}\frac{|\phi_{m}(0)|^{2}}{E_{m}-E_{k}}+\int_{\sigma_{c}(H_{0})}\frac{|\chi_{\lambda^{\prime}}(0)|^{2}}{\lambda^{\prime}-E_{k}}d\mu(\lambda^{\prime})$
$\displaystyle\hskip
113.81102pt-\frac{|\phi_{k}(0)|^{2}(\lambda+\mu^{2})}{(\lambda-
E_{k})(E_{k}+\mu^{2})}\Bigg{]}d\mu(\lambda)\;.$ (3.15)

######  Remark 3.3.

Note that the generalized eigenvectors, corresponding to the continuous
spectrum, can be found as the jump discontinuity across the part of the real
axis containing the continuous spectrum, this leads to the same formula as in
(2.58).

##  4 Possible Generalizations

###  4.1 $N$ Center Case

It is possible to generalize our approach to $N$-point delta interaction case.
When there is no need for renormalization, we will assume that all the
couplings are actually positive (hence corresponds to the attractive case).
Let us enumerate these points as $a_{1},a_{2},...,a_{N}$ with $a_{i}\neq
a_{j}$ whenever $i\neq j$ and the associated couplings with
$\alpha_{1},\alpha_{2},...\alpha_{N}$. We will proceed recursively, and
suppose that $H_{0}$ has the same spectral properties discussed in Section 2.
If some set of eigenfunctions satisfies $\phi_{k}(a_{1})\neq 0$ then they lead
to a shift of this eigenvalue to a new value $E_{k}^{*_{1}}$ in between
$E_{k-1}$ and $E_{k}$. For these, the eigenfunctions change to
${\mathcal{N}}_{0}G_{0}(x,a_{1}|E_{k}^{*_{1}})$ where ${\mathcal{N}_{0}}$ is
the normalization constant. So, we have a new Hamiltonian $H_{1}$ with a new
set of discrete states $\phi_{k}^{(1)}$ (and possibly a new set of continuum
states as found before by the formula (2.58), that we represent by
$\eta^{(1)}_{\lambda}$ here). We now are back to the initial case, if we add
the point interaction at $a_{2}$, the same construction is repeated, and new
eigenvalues $E_{k}^{*_{2}}$ for $\phi_{k}^{(1)}(a_{2})\neq 0$ fall in between
$E_{k-1}^{*_{1}}$ and $E_{k}^{*_{1}}$. The poles associated with the old
eigenvalues are removed, and there are new wave functions given by
${\mathcal{N}}_{1}G_{1}(x,a_{2}|E_{k}^{*_{2}})$ (similarly we can talk about
the continuum, yet it is more natural to write a direct formula for the
continuum case). We can now proceed recursively, and define
$H_{N}=H_{N-1}-\alpha_{N}\delta(x,a_{N})$ and thus find the Green’s function
[39, 40]

$G_{N}(x,y|E)=G_{N-1}(x,y|E)+G_{N-1}(x,a_{N}|E)\Phi^{-1}(E)G_{N-1}(a_{N},y|E),$
(4.1)

with $\Phi(E)={1\over\alpha_{N}}-G_{N-1}(a_{N},a_{N}|E)$. By this recursive
argument, the new eigenvalues $E_{k}^{*_{N}}$ are in between the eigenvalues
$E_{k-1}^{*_{N-1}}$ and $E_{k}^{*_{N-1}}$. The resulting eigenfunction
$\phi_{k}^{(N)}(x)$ is given by
${\mathcal{N}}_{N-1}G_{N-1}(x,a_{N}|E_{k}^{*_{N}})$, note that
$G_{N-1}(x,a_{N}|E_{k}^{*_{N}})=G_{0}(x,a_{N}|E_{k}^{*_{N}})+\sum_{i,j=1}^{N-1}G_{0}(x,a_{i}|E_{k}^{*_{N}})\Phi^{-1}_{ij}(E_{k}^{*_{N}})G_{0}(a_{j},a_{N}|E_{k}^{*_{N}})$,
where $\Phi_{ij}(E)=\frac{1}{\alpha_{i}}\delta_{ij}-G_{0}(a_{i},a_{j}|E)$ is
the matrix formed by the point centers $a_{1},...a_{N-1}$.

Finally, we comment that in the case of $N$ point centers, working out the
jump discontinuity across the continuous spectrum, we can find the generalized
eigenfunctions;

$\displaystyle\eta_{E}(x)=\chi_{E}(x)+\sum_{i,j=1}^{N}G_{0}(x,a_{i}|E+i0)\chi_{E}(a_{j})\Phi^{-1}_{ij}(E+i0)\;.$
(4.2)

Incidentally, the above derivation does not make use of the finiteness of the
matrix $\Phi$, even in case the matrix $\Phi(E)$ requires a renormalization,
our derivation remains valid.

###  4.2 Delta Interaction Supported on Curves in Plane

Note that none of the derivations actually rely on the interaction being
concentrated at a point, we can generalize to the curve case easily. In fact,
by extending the above discussion, we can accommodate multiple non-
intersecting curves and points cases easily. To make the discussion simpler we
consider a single rectifiable curve $\Gamma$ in the plane first, and assume
that the spectrum of $H_{0}$ consists of only a discrete part. The Hamiltonian
is formally written as

$\displaystyle H=H_{0}-\alpha\delta(\cdot-\Gamma)\;.$ (4.3)

There are various ways to define the above Hamiltonian in a mathematically
rigorous way. One way is to interpret the above formal interaction by the
quadratic form
$\int_{\mathbb{R}^{2}}|\nabla\psi|^{2}d^{2}x-\lambda\int_{\Gamma}|\psi|^{2}ds\;,$
and then prove that there is an associated self-adjoint Hamiltonian [41, 42,
43]. Other ways are to impose the continuity and the jump discontinuity
conditions at $\Gamma$, (see Remark 4.1 in [41] and [44]) or use scaled
potentials [45], or direct construction of the resolvent operator [41, 46].
The physical motivation for studying such Hamiltonians is to give a realistic
model for trapped electrons due to interfaces between two different
semiconductor materials, which are known as leaky graphs, curves or surfaces
in the literature [47].

In this section, we consider rank one modification (perturbations) of $H_{0}$
in the sense described in [18] and the Hamiltonian is formally given by

$\displaystyle H_{0}-\alpha|\Gamma\rangle\langle\Gamma|\;,$ (4.4)

where $H_{0}=-\frac{\hbar^{2}}{2m}\Delta+V$ and we have used the Dirac
notation for the inner products, and the Dirac delta function
$\delta_{\Gamma}$ supported by the curve $\Gamma$ with length $L$ is defined
by their action on test functions $\psi$ [4]

$\displaystyle\langle\delta_{\Gamma}|\psi\rangle=\langle\Gamma|\psi\rangle$
$\displaystyle:=$ $\displaystyle\frac{1}{L(\Gamma)}\int_{\Gamma}\psi\;ds\;,$
(4.5)

where $ds$ is the integration element over the curve $\Gamma$ and
$|\Gamma\rangle\langle\Gamma|$ written in Dirac’s bra-ket notation is
$\langle\delta_{\Gamma},\cdot\rangle\delta_{\Gamma}$.

It is well known that the resolvent of free Hamiltonians modified by delta
interaction supported on a curve can be expressed by some explicit formulae
involving the resolvent of the free Hamiltonian, and they are known as Krein’s
formula in the literature [17, 18, 47]. Instead of free Hamiltonian we have
here a general Schrödinger operator but the formula for the resolvent would be
exactly the same as before. Hence, we obtain

$\displaystyle
R(E)=R_{0}(E)+\frac{1}{\Phi(E)}R_{0}(E)|\Gamma\rangle\langle\Gamma|R_{0}(E)\;,$
(4.6)

where $\Phi(E)=\frac{1}{\alpha}-G_{0}(\Gamma,\Gamma|E)$ and
$G_{0}(\Gamma,\Gamma|E)=\frac{1}{L^{2}}\iint_{\Gamma\times\Gamma}G_{0}(\gamma(s),\gamma(s^{\prime}))ds\;ds^{\prime}$.
This can be expressed in terms of the Green’s functions as

$\displaystyle
G(x,y|E)=G_{0}(x,y|E)+\frac{1}{\Phi(E)}G_{0}(x,\Gamma|E)G_{0}(\Gamma,y|E)\;.$
(4.7)

Here $G_{0}(x,\Gamma|E)=\frac{1}{L}\int_{\Gamma}G_{0}(x,\gamma(s))ds$. A
generalization of such delta interactions supported on curves embedded in
manifolds is studied in [48].

Let us define $g(x,y|E)=\sum_{n\neq k}\phi_{n}(x)\phi_{n}(y)(E-E_{n})^{-1}$ as
well as projections onto the curve as $\langle
x|g(E)|\Gamma\rangle=\sum_{n\neq
k}\phi_{n}(x)\langle\Gamma|\phi_{k}\rangle(E-E_{n})^{-1}$. For clarity we only
assume a discrete spectrum, since generalizing to continuous spectrum is not
difficult. These are holomorphic functions of $E$ in a sufficiently small
neighborhood of $E_{k}$. By following the same steps as we have done before,
and using the expansion of the Green’s function we find

$\displaystyle G(x,y|E)$ $\displaystyle=$ $\displaystyle
g(x,y|E)+\frac{\phi_{k}(x)\phi_{k}(y)}{E-E_{k}}+\frac{{\phi_{k}(x)\langle\phi_{k}|\Gamma\rangle\over
E-E_{k}}{\langle\Gamma|\phi_{k}\rangle\phi_{k}(y)\over
E-E_{k}}}{{1\over\alpha}-\langle\Gamma|g(E)|\Gamma\rangle-|\langle\Gamma|\phi_{k}\rangle|^{2}(E-E_{k})^{-1}}$
(4.8) $\displaystyle+\frac{\langle
x|g(E)|\Gamma\rangle{\langle\Gamma|\phi_{k}\rangle\phi_{k}(y)\over
E-E_{k}}}{{1\over\alpha}-\langle\Gamma|g(E)|\Gamma\rangle-|\langle\Gamma|\phi_{k}\rangle|^{2}(E-E_{k})^{-1}}+\frac{{\phi_{k}(x)\langle\phi_{k}|\Gamma\rangle\over
E-E_{k}}\langle\Gamma|g(E)|y\rangle}{{1\over\alpha}-\langle\Gamma|g(E)|\Gamma\rangle-|\langle\Gamma|\phi_{k}\rangle|^{2}(E-E_{k})^{-1}}$
$\displaystyle+\frac{\langle
x|g(E)|\Gamma\rangle\langle\Gamma|g(E)|y\rangle}{{1\over\alpha}-\langle\Gamma|g(E)|\Gamma\rangle-|\langle\Gamma|\phi_{k}\rangle|^{2}(E-E_{k})^{-1}}\;.$

Here we have assumed that there is a continuous representative for each
eigenvector, so that the expression
$\langle\Gamma|\phi_{k}\rangle=\frac{1}{L}\int_{\Gamma}\phi_{k}(\gamma(s))ds$
is well defined, this is true even for the generalized eigenvectors of a
Laplacian modified by some potential which satisfies some conditions [6]
(indeed for the free Hamiltonian, the domain of $H_{0}$ is the Sobolev space
and these expressions for arbitrary vectors in the domain are always well-
defined by the Sobolev embedding theorem [49]). Just as before, multiplying by
$(E-E_{k})$ the numerators and denominators, and expanding around $E_{k}$, we
see the cancellation of the pole at $E_{k}$, as long as
$\langle\Gamma|\phi_{k}\rangle\neq 0$.

In this case we have the new wave functions given by

$\psi_{k}(x)={\mathcal{N}}G_{0}(x,\Gamma|E_{k}^{*})={\mathcal{N}}\int_{\Gamma}G_{0}(x,\gamma(s)|E_{k}^{*})ds\;.$
(4.9)

By the same token, the generalized eigenvectors for the continuous spectrum
are given by

$\eta_{E}(x)=\chi_{E}(x)+G_{0}(x,\Gamma|E+i0)\langle\Gamma|\chi_{E}\rangle\Phi^{-1}(E+i0)\;.$
(4.10)

Eigenvalues and eigenvectors, for multiple curves and points are now
constructed by iterating the above set of arguments for each additional rank
one perturbation.

######  Remark 4.1.

In all our discussions, it is essential to have elliptic operators with some
special properties (typically a summability condition on the potentials) to
allow for generalized eigenvectors with certain regularity properties, as we
evaluate them at a point or integrate them over a curve.

###  4.3 A Particle in a Compact Manifold under the Influence of a $\delta$
Interaction

Another possibility for $H_{0}$ could correspond to a system where a particle
is intrinsically moving on a two or three dimensional compact and connected
manifold $M$ with the metric structure $g$:

$\displaystyle(H_{0}\psi)(x)=-\frac{\hbar^{2}}{2m}\left(\frac{1}{\sqrt{\det
g}}\sum_{i,j=1}^{d}\frac{\partial}{\partial x^{i}}\left(\sqrt{\det
g}g^{ij}\frac{\partial\psi(x)}{\partial x^{j}}\right)\right)\;,$ (4.11)

where $x=(x^{1},\ldots,x^{d})$ are the local coordinates, and $g^{ij}$ are the
components of inverse of the metric $g$. Then it is well known [50, 51]) that
there exists a complete orthonormal system of $C^{\infty}$ eigenfunctions
$\\{\phi_{n}\\}_{n=0}^{\infty}$ in $L^{2}(M)$ and the spectrum
$\sigma(H_{0})=\\{E_{n}\\}=\\{0=E_{0}\leq E_{1}\leq E_{2}\leq\dots\\}$, with
$E_{n}$ tending to infinity as $n\rightarrow\infty$ and each eigenvalue has
finite multiplicity: $H_{0}\phi_{n}=E_{n}\phi_{n}$. Some eigenvalues are
repeated according to their multiplicity. The multiplicity of the first
eigenvalue $E_{0}=0$ is one and the corresponding eigenfunction is constant.
This result is also valid for Neumann, Dirichlet and mixed eigenvalue problems
except for $E_{0}>0$, provided that the appropriate boundary condition is
imposed.

When we add a point like delta interaction on such Schrödinger operator
$H_{0}$, we have physically a particle moving intrinsically in a two
dimensional compact manifold $\mathcal{M}$ and interacting with delta
interaction located at some point $a$ in $\mathcal{M}$. This problem requires
renormalization due to the short distance singular behaviour of the free
Green’s function on a manifold, which can be similarly seen from the short
time expansion of the heat kernel $K_{t}(a,a)\sim 1/t$ as $t\to 0^{+}$ and the
formula (3.1) for Riemannian manifolds [52]. This has been discussed in our
previous works [53, 54, 55, 56] for finitely many delta centers. Then, the
Green’s function can be similarly expressed as

$\displaystyle
G(x,y|E)=G_{0}(x,y|E)+\frac{G_{0}(x,a|E)G_{0}(a,y|E)}{\frac{1}{\alpha_{R}}-\sum_{n=0}^{\infty}\frac{|\phi_{n}(a)|^{2}(E+\mu^{2})}{(E_{n}-E)(E_{n}+\mu^{2})}}\;,$
(4.12)

for any points $x,y$ and $a$ in $\mathcal{M}$. Then, all the results that we
have obtained for previous problems follow easily.

## Acknowledgments

F. Erman would like to thank H. Uncu for useful discussions on the spectrum of
harmonic oscillator modified by delta interaction, and A. Bohm and M. Gadella
for giving helpful insights on the Rigged Hilbert spaces and eigenfunction
expansions. O. T. Turgut would like to thank A. Michelangeli for various
discussions on singular interactions and J. Hoppe for discussions and
encouragement on this project.

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