Regularization of energy-dependent pointlike interactions in 1D quantum
mechanics

Etienne Granet1

1 Kadanoff Center for Theoretical Physics, University of Chicago, 5640 South
Ellis Ave, Chicago, IL 60637, USA

## Abstract

We construct a family of hermitian potentials in 1D quantum mechanics that
converges in the zero-range limit to a $\delta$ interaction with an energy-
dependent coupling. It falls out of the standard four-parameter family of
pointlike interactions in 1D. Such classification was made by requiring the
pointlike interaction to be hermitian. But we show that although our
Hamiltonian is hermitian for the standard inner product when the range of the
potential is finite, it becomes hermitian for a different inner product in the
zero-range limit. This inner product attributes a finite probability (and not
probability density) for the particle to be exactly located at the position of
the potential. Such pointlike interactions can then be used to construct
potentials with a finite support with an energy-dependent coupling.

## Introduction

Pointlike interactions have long been part of the theoretical physics
landscape, as idealized versions of short-range potentials that make explicit
computations possible, see e.g. [1, 2, 3, 4, 5]. Their most physical
definition is through regularization, namely as the zero-range limit of a
family of regular potentials. In 1D the archetypal pointlike interaction is
the $\delta$ potential [6, 7], to which any potential of the form
$\tfrac{1}{a}V(\tfrac{x}{a})$ converges in the zero-range limit $a\to 0$, with
coupling $\int V(x){\rm d}x$ [8]. In higher dimensions the definition and
regularization of the $\delta$ potential is more involved [9, 10, 11]. But in
1D there also exist other pointlike interactions that find applications and
that require a more involved functional form and scaling [12, 13, 14].

In this context, the question of classifying and regularizing pointlike
interactions in 1D naturally arose [15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25]. An appealing approach to this classification is to determine the
constraints on the resulting Hamiltonian once the zero-range limit is taken.
Given a particle in a potential $V_{a}(x)$ such that $V_{a}(x)\to 0$ when
$a\to 0$ at fixed $x\neq 0$, the following reasoning is tempting. In the zero-
range limit $a\to 0$, the particle does not see any potential for $x\neq 0$,
so its Hamiltonian should be merely $\partial_{x}^{2}$. The only possible
remaining effect of the interaction is thus to impose some boundary conditions
on the wave function at the position of the potential. But since the
Hamiltonian is hermitian for $a>0$ these boundary conditions are constrained
by hermiticity. Such constructions are called self-adjoint extensions of the
operator $\partial_{x}^{2}$ [26, 27], and they have been classified to be
given by a four-parameter family of connection conditions on the wave function
and its derivatives before and after the position of the potential [18, 19,
20]. Besides, the Hamiltonian for $a>0$ is time reversal-symmetric, which
restricts this further to a three-parameter family in the zero-range limit
[20]. Reciprocally, it has been shown that all this family of connection
conditions can indeed be obtained through appropriate regularizations [28,
23]. Hence, according to this reasoning, pointlike interactions in 1D quantum
mechanics obtained from a regular potential should exactly be this three-
parameter family of connection conditions.

But there is actually a subtlety in this reasoning. The weak point is that
hermiticity is inner product-dependent [29]. Although the Hamiltonian is
certainly hermitian for the standard inner product when $a>0$, it can become
hermitian for a different inner product in the zero-range limit, yielding more
general connection conditions than previously argued. In this paper, we
precisely construct a family of smooth potentials $V_{a}(x)$ that are
hermitian for the standard inner product, but whose zero-range limit is not
described by the previous connection conditions. We show that in the limit
$a\to 0$ it converges instead to a $\delta$ interaction with an energy-
dependent coupling. This means that the resulting eigenstates of the
Hamiltonian satisfy the connection conditions of a $\delta$ interaction, but
with an amplitude that depends on the energy of that eigenstate. Such a
dependence on the energy is allowed only because the Hamiltonian becomes
hermitian for a different inner product when $a\to 0$. This inner product
attributes a finite probability (and not just probability density) for a
particle to be located at the position of the potential. Similarly, we present
as well a potential $V_{a}(x)$ that in the zero-range limit converges to the
so-called ”$\delta^{\prime}$ interaction” between fermions, but with an
energy-dependent coupling. These pointlike interactions can then be used to
construct energy-dependent potentials that have a finite support.

We then discuss the dynamics of a particle interacting with this potential and
the influence of the non-standard inner product. Because of this inner
product, the particle has a tendency to stick to the potential, i.e. can be
captured by the potential that acts like a pointlike potential well. When two
particles interact with each other via this potential, this means a finite
probability of forming a pointlike pair of particles, that is however not
stable. We investigate such properties with the two pointlike interactions
constructed in this paper.

We note that quantum mechanics with non-hermitian Hamiltonians or with non-
standard inner products has been the object of many studies in the past, see
e.g. [29, 30, 31, 32], in particular in $\mathcal{P}\mathcal{T}$-symmetric
systems [33, 34, 35] and open systems [36, 37]. But in our case, this unusual
inner product results from the limit of Hamiltonians that are perfectly
hermitian for the usual inner product. The fact that the pointlike
interactions built in this paper fall out of the previously mentioned
classification does not result from a choice of an unusual inner product in
the first place. Being the zero-range limit of regular potentials, they are on
the same footing as the three-parameter family presented above.

We also note that energy-dependent potentials in quantum mechanics have
attracted interest for a long time, because relativistic equations can be seen
sometimes as stationary Schrödinger equations with energy-dependent couplings,
see e.g. [38]. Pointlike interactions with an energy dependence have also been
introduced before, in the form of higher-derivatives $\delta$ potentials [39,
40, 41, 42]. However their definition therein is only formal and it is known
that the connection conditions obtained by integrating the Schrödinger
equation with higher derivatives $\delta$ functions are often incorrect, see
e.g. [16] and the appendix of [14]. To the best of our knowledge, our paper is
the first to define energy-dependent potentials constructed from usual,
regular and hermitian potentials in non-relativistic quantum mechanics.

###### Contents

  1. 1 Definitions and reminders
    1. 1.1 Pointlike interaction
    2. 1.2 Energy functional point of view
    3. 1.3 Standard pointlike interactions
  2. 2 Energy-dependent pointlike interaction between bosons
    1. 2.1 The potential
    2. 2.2 The resulting inner product
    3. 2.3 The resulting Hamiltonian
    4. 2.4 Two particles interacting via the pointlike potential
    5. 2.5 Dynamics
    6. 2.6 Energy-dependent potentials with a finite support
  3. 3 Energy-dependent pointlike interaction between fermions
    1. 3.1 The potential
    2. 3.2 The resulting inner product and Hamiltonian

##  1 Definitions and reminders

###  1.1 Pointlike interaction

The wave function $\psi(x,t)$ of a particle in a potential $V(x)$ in a ring
$[-L/2,L/2]$ of size $L$ with periodic boundary condition satisfies the
Schrödinger equation

$i\partial_{t}\psi=-\partial^{2}_{x}\psi+V(x)\psi\,.$ (1)

Let $V_{a}(x)$ a family of potentials parametrized by $a>0$. We say that it
describes a pointlike interaction (or pointlike potential) in the limit $a\to
0$ if

$\underset{a\to 0}{\lim}\,\int_{-L/2}^{L/2}f(x)V_{a}(x){\rm d}x=0\,,$ (2)

for any integrable function $f$ for which there exists $\epsilon>0$ such that
$f(x)=0$ for $x\in[-\epsilon,\epsilon]$ 111Imposing $V_{a}(x)\to 0$ at fixed
$x\neq 0$ when $a\to 0$ is not enough. For example, $V_{a}(x)=\frac{1}{a}$ for
$1<x<1+a$ satisfies this property, but produces a $\delta$ interaction at
$x=1$ and not at $x=0$.. The family $V_{a}(x)$ is then called a regularization
of this pointlike interaction. Given a periodic initial condition
$\psi(x,t=0)$, we define the time evolution of the wave function of the
particle in this pointlike potential as the limit $a\to 0$ of the solution of
(1) for $a>0$.

###  1.2 Energy functional point of view

The previous definition of pointlike interactions is a physical definition,
since a pointlike potential is just an idealised view of a very short-range
potential convenient to the theoretician. However, it does not provide a
description of the system once the zero-range limit is taken. In this section,
we present a simple and flexible framework for this.

Let us first reformulate the previous problem. We consider $\mathcal{H}$ the
Hilbert space of smooth functions on $[-L/2,L/2]$ satisfying periodic boundary
conditions, equipped with the standard inner product

$\langle\phi|\psi\rangle=\int_{-L/2}^{L/2}\phi^{*}(x)\psi(x){\rm d}x\,,$ (3)

and define the following energy functional for a real potential $V(x)$

$\displaystyle H(\psi)$
$\displaystyle=\int_{-L/2}^{L/2}|\psi^{\prime}(x)|^{2}{\rm
d}x+\int_{-L/2}^{L/2}V(x)|\psi(x)|^{2}{\rm d}x\,.$ (4)

We recall that the local minima of $H(\psi)$ at fixed norm
$\langle\psi|\psi\rangle=1$ satisfy the stationary Schrödinger equation

$-\partial_{x}^{2}\psi(x)+V(x)\psi(x)=E\psi(x)\,,$ (5)

with $E$ constrained by the boundary conditions. It is then the value taken by
the energy functional on this local minimum $H(\psi)=E$. We can then define a
dynamics on the linear space $S$ spanned by the local minima of the energy
functional $H$ the following way. If $\psi(x)$ is a local minimum with energy
$H(\psi)=E$, then we define its time evolution $\psi(x,t)$ by

$\psi(x,t)=\psi(x)e^{-iEt}\,,$ (6)

and extend this definition by linearity to the full vector space $S$. It
follows that in $S$, the time evolution of wave functions is governed by the
Schrödinger equation (1).

#### Example: $\delta$ interaction

Let us illustrate now how pointlike interactions can be described in this
framework. To that end, we consider the textbook example of the $\delta$
interaction. It is a pointlike interaction defined by the regularization
$V_{a}(x)=\frac{c}{a}\,\boldsymbol{1}_{|x|<a/2}$ in (4), for $c$ real. The
Schrödinger equation (5) can be straightforwardly solved for $a>0$ since the
potential is constant by part, from which one can deduce the form of the
wavefunction in the pointlike limit $a\to 0$. It is well-known that the
eigenstates in the limit $a\to 0$ satisfy

$\displaystyle\psi^{\prime\prime}(x)+E\psi(x)$
$\displaystyle=0\,,\qquad\text{for }x\neq 0$ (7)
$\displaystyle\psi(0+)-\psi(0-)$ $\displaystyle=0$
$\displaystyle\psi^{\prime}(0+)-\psi^{\prime}(0-)$ $\displaystyle=c\psi(0)\,,$

with $E$ constrained by the boundary conditions.

Let us show now that these boundary conditions exactly correspond to the local
minima of the energy functional

$H(\psi)=\int_{-L/2}^{L/2}|\psi^{\prime}(x)|^{2}{\rm d}x+c|\psi(0)|^{2}\,,$
(8)

at fixed unit norm $\langle\psi|\psi\rangle=1$. Introducing a Lagrange
multiplier $E$, the problem is equivalent to minimizing

$F(\psi,E)=H(\psi)-E(\langle\psi|\psi\rangle-1)\,,$ (9)

without constraining $\psi$. Indeed, minimizing with respect to $E$ imposes
$\langle\psi|\psi\rangle=1$, while minimizing with respect to $\psi$ minimizes
then $H(\psi)$. We have for a continuous perturbation $\delta\psi$

$\displaystyle F(\psi+\delta\psi,E)=$ $\displaystyle
F(\psi,E)+2\Re\int_{-L/2}^{L/2}(\psi^{\prime}(x))^{*}\delta\psi^{\prime}(x){\rm
d}x-2E\Re\int_{-L/2}^{L/2}\psi(x)^{*}\delta\psi(x){\rm d}x$ (10)
$\displaystyle+2c\Re\psi(0)^{*}\delta\psi(0)+\mathcal{O}(\delta\psi^{2})\,.$

Let us assume that there are points $y_{j}$ where $\psi^{\prime}$ is
discontinuous. Then performing an integration by part

$\displaystyle F(\psi+\delta\psi,E)$
$\displaystyle-F(\psi,E)=-2\Re\int_{-L/2}^{L/2}(\psi^{\prime\prime}(x)+E\psi(x))^{*}\delta\psi(x){\rm
d}x$ (11)
$\displaystyle+2c\Re\psi(0)^{*}\delta\psi(0)-2\Re\sum_{j}(\psi^{\prime}(y_{j}+)-\psi^{\prime}(y_{j}-))^{*}\delta\psi(y_{j})$
$\displaystyle+\mathcal{O}(\delta\psi^{2})\,.$

It follows that in order to make vanish all the $\delta\psi$ terms, one
requires a continuous $\psi^{\prime}(x)$ when $x\neq 0$, and

$\psi^{\prime}(0+)-\psi^{\prime}(0-)=c\psi(0)\,,$ (12)

as well as for $x\neq 0$

$\psi^{\prime\prime}(x)+E\psi(x)=0\,.$ (13)

###  1.3 Standard pointlike interactions

We briefly recall the reasoning made in the introduction. The eigenstates of a
particle interacting with a pointlike potential at $x=0$ according to the
definition of Section 1.1, for any $\epsilon>0$ should satisfy a free
Schrodinger equation $\psi^{\prime\prime}(x)+E\psi(x)=0$ for $|x|>\epsilon$.
Hence the only possible effect of the pointlike interaction is to impose some
boundary conditions for $x>0$ and $x<0$. Since the Hamiltonian is hermitian
for $a>0$, it is tempting to look for connection conditions that will make the
resulting pointlike interaction hermitian. We will call such settings a
standard pointlike interaction at $x=0$. In the math terminology, this
corresponds to self-adjoint extensions of the operator $\partial_{x}^{2}$ on
the space of smooth functions on $[-L/2,L/2]\setminus\\{0\\}$ [26]. It is
known that the only such possible connection conditions are [18, 19, 20]

$\begin{cases}\text{either }\qquad\left(\begin{matrix}\psi(0+)\\\
\psi^{\prime}(0+)\end{matrix}\right)=e^{i\theta}\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\left(\begin{matrix}\psi(0-)\\\
\psi^{\prime}(0-)\end{matrix}\right)\\\ \text{or
}\qquad\psi^{\prime}(0+)=h_{+}\psi(0+)\,,\qquad\psi^{\prime}(0-)=h_{-}\psi(0-)\,,\end{cases}$
(14)

with $\theta,a,b,c,d,h_{\pm}$ real parameters satisfying $ad-bc=1$. The second
possibility in (14) can be obtained as particular limits of the first
possibility when some parameters go to $\pm\infty$. This leaves thus a four-
parameter family. If now time-reversal invariance is imposed, it constrains
$\theta=0$, leaving a three-parameter family [20]. Besides, the connection
conditions (14) are indeed pointlike interactions in the sense of Section 1.1,
i.e. they can be obtained as limits of regular potentials [28].

In the following Sections we precisely show that not all pointlike
interactions are standard pointlike interactions, i.e. there exist pointlike
interactions with more general connection conditions than (14).

##  2 Energy-dependent pointlike interaction between bosons

###  2.1 The potential

The first energy-dependent pointlike interaction that we present acts like a
$\delta$ potential with an energy-dependent coupling for bosons, and is
transparent for fermions. Its construction is given by the following Theorem.

######  Theorem 1.

Let $c\geq 0,\nu\geq 0$. Let $\sigma(x)$ a smooth odd increasing function such
that $\sigma(x)\to 1$ when $x\to\infty$, and $\Delta(x)$ a smooth positive
even function that satisfies

$\displaystyle\int_{-\infty}^{\infty}\Delta(x){\rm
d}x=1\,,\qquad\int_{-\infty}^{\infty}\sqrt{\Delta(x)}{\rm
d}x<\infty\,,\qquad\max(x^{2}\Delta(x))<\infty\,.$ (15)

[For example, one can take $\sigma(x)=\tanh(x)$ and
$\Delta(x)=\tfrac{1}{2}\tanh^{\prime}(x)$.] For $E$ real and $a>0$, we define
$\psi_{a}(x)$ as the solution to

$-\psi_{a}^{\prime\prime}(x)+V_{a}(x)\psi_{a}(x)=E\psi_{a}(x)\,,$ (16)

with some fixed $a$-independent boundary conditions at $x=1$, and where the
potential is

$V_{a}(x)=\frac{v_{a}^{\prime\prime}(x)}{v_{a}(x)}\,,\qquad
v_{a}(x)=1+\frac{c}{2}x\sigma(\tfrac{x}{a})+\nu\sqrt{\tfrac{1}{a}\Delta(\tfrac{x}{a})}\,.$
(17)

The limit $\psi(x)\equiv\underset{a\to 0}{\lim}\,\psi_{a}(x)$ satisfies

$\displaystyle\psi^{\prime\prime}(x)+E\psi(x)=0\quad{\rm for}\,\,x\neq 0$ (18)
$\displaystyle\left(\begin{matrix}\psi(0+)\\\
\psi^{\prime}(0+)\end{matrix}\right)=\left(\begin{matrix}1&0\\\
c-E\nu^{2}&1\end{matrix}\right)\left(\begin{matrix}\psi(0-)\\\
\psi^{\prime}(0-)\end{matrix}\right)\,.$

Before entering the proof, let us make the following comment on the connection
conditions (18). If the wave function describes two bosons interacting via
$V_{a}(x)$ in the center of mass frame, then $\psi(x)$ has to be even, so
$\psi(0+)=\psi(0-)$ and $\psi^{\prime}(0+)=-\psi^{\prime}(0-)$. This leads to
the jump condition

$\displaystyle\psi^{\prime}(0+)-\psi^{\prime}(0-)$
$\displaystyle=(c-E\nu^{2})\psi(0)\,,$ (19)

which is exactly like a $\delta$ potential with an energy-dependent coupling.
Now, if the wave function describes fermions, it has to be odd and so
$\psi(0+)=-\psi(0-)$ and $\psi^{\prime}(0+)=\psi^{\prime}(0-)$. The connection
conditions (18) are then automatically satisfied, meaning that this pointlike
potential is transparent for fermions, i.e. has no effect.

Besides, we give a plot of the potential $V_{a}(x)$ in the right panel of
Figure 1.

###### Proof of Theorem 1.

Two independent solutions of (16) for $E=0$ are $v_{a}(x)$ which is even, and

$w_{a}(x)=v_{a}(x)\int_{0}^{x}\frac{{\rm d}t}{v_{a}(t)^{2}}\,,$ (20)

which is odd. Hence the solutions to

$-\psi_{a}^{\prime\prime}(x)+V_{a}(x)\psi_{a}(x)=f(x)\,,$ (21)

for $f(x)$ an arbitrary even smooth function, are

$\psi_{a}(x)=v_{a}(x)\int_{0}^{x}f(y)w_{a}(y){\rm
d}y-w_{a}(x)\int_{0}^{x}f(y)v_{a}(y){\rm d}y+A_{a}v_{a}(x)+B_{a}w_{a}(x)\,.$
(22)

with $A_{a},B_{a}$ integration constants depending on the boundary conditions.
It follows that $\psi_{a}(x)$ satisfies the self-consistency equation

$\displaystyle\psi_{a}(x)=Ev_{a}(x)\int_{0}^{x}\psi_{a}(y)w_{a}(y){\rm
d}y-Ew_{a}(x)\int_{0}^{x}\psi_{a}(y)v_{a}(y){\rm
d}y+A_{a}v_{a}(x)+B_{a}w_{a}(x)\,.$ (23)

Let us argue that one can assume that $A_{a},B_{a}$ are independent of $a$.
Indeed, by linearity of this integral equation, one can focus on the two cases
$A_{a}=0$ (odd solution) and $B_{a}=0$ (even solution). Then, focusing e.g. on
the case $B_{a}=0$, if we define $\tilde{\psi}_{a}$ as the solution to this
integral equation with an $A$ fixed independent of $a$, one has
$\psi_{a}(x)=\tfrac{A_{a}}{A}\tilde{\psi}_{a}(x)$. Hence this provides a
solution to the problem provided
$A_{a}=A\tfrac{\psi_{a}(1)}{\tilde{\psi}_{a}(1)}$ has a finite limit when
$a\to 0$.

We will thus assume from now on that $A_{a}=A$ and $B_{a}=B$ are independent
of $a$. Defining then

$\bar{\psi}_{a}(x)=\psi_{a}(x)-A\nu\sqrt{\tfrac{1}{a}\Delta(\tfrac{x}{a})}\,,$
(24)

it satisfies

$\displaystyle\bar{\psi}_{a}(x)$
$\displaystyle=Ev_{a}(x)\int_{0}^{x}\bar{\psi}_{a}(y)w_{a}(y){\rm
d}y-Ew_{a}(x)\int_{0}^{x}\bar{\psi}_{a}(y)\bar{v}_{a}(y){\rm d}y$ (25)
$\displaystyle-E\nu
w_{a}(x)\int_{0}^{x}\bar{\psi}_{a}(y)\sqrt{\tfrac{1}{a}\Delta(\tfrac{y}{a})}{\rm
d}y$ $\displaystyle+EA\nu
v_{a}(x)\int_{0}^{x}\sqrt{\tfrac{1}{a}\Delta(\tfrac{y}{a})}w_{a}(y){\rm
d}y-EA\nu
w_{a}(x)\int_{0}^{x}\sqrt{\tfrac{1}{a}\Delta(\tfrac{y}{a})}\bar{v}_{a}(y){\rm
d}y$ $\displaystyle-
EA\nu^{2}w_{a}(x)\int_{0}^{x}\tfrac{1}{a}\Delta(\tfrac{y}{a}){\rm
d}y+A\bar{v}_{a}(x)+Bw_{a}(x)\,,$

with

$\bar{v}_{a}(x)=1+\frac{c}{2}x\sigma(\tfrac{x}{a})\,.$ (26)

To go further, we prove the following Lemma.

######  Lemma 1.

If $|f_{a}(x)|$ is bounded independently of $a$ and $x\in[-L/2,L/2]$, then

$\underset{a\to
0}{\lim}\,\int_{-L/2}^{L/2}f_{a}(x)\sqrt{\tfrac{1}{a}\Delta(\tfrac{x}{a})}{\rm
d}x=0\,.$ (27)

###### Proof.

Denoting $M$ the bound on $f_{a}$, we have

$\left|\int_{-L/2}^{L/2}f_{a}(x)\sqrt{\tfrac{1}{a}\Delta(\tfrac{x}{a})}{\rm
d}x\right|\leq M\sqrt{a}\int_{-L/(2a)}^{L/(2a)}\sqrt{\Delta(x)}{\rm d}x\,,$
(28)

which goes to zero when $a\to 0$ from the assumption
$\int_{-\infty}^{\infty}\sqrt{\Delta(x)}{\rm d}x<\infty$. ∎

This will be useful to prove the following Lemma.

######  Lemma 2.

The functions $w_{a}(x)$ and $\bar{\psi}_{a}(x)$ are bounded independently of
$a<1$ and $x\in[-L/2,L/2]$.

###### Proof.

Let us start with $w_{a}(x)$, and restrict to $x\geq 0$ by oddness of
$w_{a}(x)$. Since $v_{a}(x)\geq 1$ by assumptions on $c,\nu,\sigma$, we have

$0\leq w_{a}(x)\leq v_{a}(x)\int_{0}^{x}{\rm d}t=xv_{a}(x)\,.$ (29)

We then write

$xv_{a}(x)=x\bar{v}_{a}(x)+\nu\sqrt{a}\sqrt{\tfrac{x^{2}}{a^{2}}\Delta(\tfrac{x}{a})}\,.$
(30)

We have $|\bar{v}_{a}(x)|\leq 1+cL/4$ for all $x\in[-L/2,L/2]$. Since there
exists $M>0$ such that $x^{2}\Delta(x)<M$ for all $x\geq 0$ by assumption, it
follows that for $a<1$

$|w_{a}(x)|\leq\tfrac{L}{2}(1+cL/4)+\nu\sqrt{M}\,,$ (31)

which is a bound independent of $x$ and $a<1$. We note that from this, we are
allowed to use uniform convergence theorem in the expression for $w_{a}(x)$
and take the limit $a\to 0$ under the integral. This yields the limits $a\to
0$ for $x\neq 0$

$v_{a}(x)\to 1+\tfrac{c}{2}|x|\,,\qquad w_{a}(x)\to x\,.$ (32)

Let us now focus on $\bar{\psi}_{a}(x)$ and (25), and consider again $x>0$.
Using Lemma 1 and the uniform bound on $w_{a}(x),\bar{v}_{a}(x)$ we have for
all $x\in[-L/2,L/2]$

$\underset{a\to
0}{\lim}\,Ev_{a}(x)\int_{0}^{x}\sqrt{\tfrac{1}{a}\Delta(\tfrac{y}{a})}w_{a}(y){\rm
d}y-Ew_{a}(x)\int_{0}^{x}\sqrt{\tfrac{1}{a}\Delta(\tfrac{y}{a})}\bar{v}_{a}(y){\rm
d}y=0\,,$ (33)

so this term can be bounded independently of $x\in[-L/2,L/2]$ and $a<1$. We
have as well

$\left|-EA\nu^{2}w_{a}(x)\int_{0}^{x}\tfrac{1}{a}\Delta(\tfrac{y}{a}){\rm
d}y+A\bar{v}_{a}(x)+Bw_{a}(x)\right|\leq\frac{1}{2}EA\nu^{2}\max
w+|A|\max\bar{v}+|B|\max w\,,$ (34)

where $\max w,\bar{v}$ are the uniform bounds of $w_{a}(x),\bar{v}_{a}(x)$. It
follows then from (25) that there exist constants $\lambda,\mu,\eta>0$
independent of $x\in[-L/2,L/2]$ and $a<1$ such that

$|\bar{\psi}_{a}(x)|\leq\int_{0}^{x}|\bar{\psi_{a}}(y)|(\lambda+\mu\sqrt{\tfrac{1}{a}\Delta(\tfrac{y}{a})}){\rm
d}y+\eta\,.$ (35)

Let us introduce

$g_{a}(x)=\lambda+\mu\sqrt{\tfrac{1}{a}\Delta(\tfrac{y}{a})}\,,$ (36)

and

$G_{a}(x)=e^{\int_{0}^{x}g_{a}(t){\rm d}t}\,,$ (37)

as well as $h_{a}(x)$ defined by

$h_{a}(x)G_{a}(x)=\int_{0}^{x}|\bar{\psi}_{a}(y)|g_{a}(y){\rm d}y\,.$ (38)

With these notations, (35) translates into

$h_{a}^{\prime}(x)\leq\eta\frac{g_{a}(x)}{G_{a}(x)}\,.$ (39)

Hence

$h_{a}(x)\leq\eta\int_{0}^{x}\frac{g_{a}(t)}{G_{a}(t)}{\rm d}t\,.$ (40)

Noting that $G_{a}(x)\geq 1$ we have then

$0\leq h_{a}(x)\leq\eta\int_{0}^{x}g_{a}(t){\rm d}t\,.$ (41)

The right-hand side has a finite limit $\eta\lambda x$ when $a\to 0$ according
to Lemma 1, so it can be bounded independently of $a<1$ and $x\in[-L/2,L/2]$.
∎

Lemma 2 allows one to use dominated convergence theorem and swap the
integration and limits $a\to 0$ in (25). Denoting for $x\geq 0$,
$\bar{\psi}(x)=\underset{a\to 0}{\lim}\,\bar{\psi}_{a}(x)$, and noting that

$\underset{a\to 0}{\lim}\,\int_{0}^{x}\tfrac{1}{a}\Delta(\tfrac{y}{a}){\rm
d}y=\frac{1}{2}\,{\rm sgn}\,(x)\,,$ (42)

we obtain

$\displaystyle\bar{\psi}(x)$
$\displaystyle=E(1+\tfrac{c}{2}|x|)\int_{0}^{x}\bar{\psi}(y)y{\rm
d}y-Ex\int_{0}^{x}\bar{\psi}(y)(1+\tfrac{c}{2}|y|){\rm d}y$ (43)
$\displaystyle-\frac{1}{2}EA\nu^{2}|x|+A\left(1+\tfrac{c}{2}|x|\right)+Bx\,.$

From this one finds $\bar{\psi}(0+)=\bar{\psi}(0-)=A$ and
$\bar{\psi}^{\prime}(0\pm)=B\pm\tfrac{A}{2}(c-E\nu^{2})$. Differentiating
twice for $x\neq 0$, one finds moreover
$\bar{\psi}^{\prime\prime}(x)+E\bar{\psi}(x)=0$. Hence this is exactly
equivalent to

$\displaystyle\bar{\psi}^{\prime\prime}(x)+E\bar{\psi}(x)$
$\displaystyle=0\quad{\rm for}\,\,x\neq 0$ (44)
$\displaystyle\bar{\psi}(0+)-\bar{\psi}(0-)$ $\displaystyle=0$
$\displaystyle\bar{\psi}^{\prime}(0+)-\bar{\psi}^{\prime}(0-)$
$\displaystyle=(c-E\nu^{2})\bar{\psi}(0)\,.$

Since for all $x\neq 0$ we have $\psi(x)=\bar{\psi}(x)$, this concludes the
proof of the Theorem.

∎

###  2.2 The resulting inner product

Because of the energy dependence in the amplitude of the cusp in (18), this
cannot be put into the form of the standard pointlike interactions whose
connection conditions are classified by (14). This could seem contradictory,
since (18) are obtained as the zero-range limit of hermitian Hamiltonians and
(14) are precisely obtained by imposing hermiticity. To resolve this
contradiction, let us consider a particle in a ring $[-L/2,L/2]$ of size $L$
with periodic boundary conditions, interacting with this pointlike potential
at position $0$. One finds that the even eigenstates are

$\psi^{[k]}(x)=\frac{1}{Z_{k}}\left(\cos(kx)+\alpha(k)\sin(k|x|)\right)\,,$
(45)

with

$\alpha(k)=\frac{c}{2k}-\frac{k\nu^{2}}{2}\,,$ (46)

with $Z_{k}$ a normalization factor. They have energy $E_{k}=k^{2}$, and
periodic boundary conditions impose the following quantization condition on
$k$

$\tan(kL/2)=\alpha(k)\,.$ (47)

As for the odd eigenstates, they are

$\phi^{[k]}(x)=\frac{\sin(kx)}{Z^{\prime}_{k}}\,,$ (48)

with $Z^{\prime}_{k}$ a normalization factor, and where the quantization
condition is $\sin(kL/2)=0$. Crucially, one checks that the eigenstates (45)
are not orthogonal with respect to the standard inner product

$\langle\psi^{[k]}|\psi^{[q]}\rangle\neq 0\quad\text{if }k\neq q\,.$ (49)

Yet, for $a>0$, denoting $\psi^{[k]}_{a}(x)$ the corresponding eigenstate, we
certainly have

$\langle\psi^{[k]}_{a}|\psi^{[q]}_{a}\rangle=0\quad\text{if }k\neq q\,,$ (50)

since the $\psi^{[k]}_{a}$’s are eigenstates of a hermitian Hamiltonian with
respect to the standard inner product. In fact, writing
$\psi^{[k]}_{a}=\bar{\psi}^{[k]}_{a}+A^{[k]}\nu\sqrt{\tfrac{1}{a}\Delta(\tfrac{1}{a})}$
and noting that $\bar{\psi}^{[k]}_{a}$ converges uniformly from Lemma 2, we
have

$\int_{-L/2}^{L/2}\bar{\psi}^{[k]*}(x)\bar{\psi}^{[q]}(x){\rm
d}x+A^{[k]*}A^{[q]}\nu^{2}=0\,.$ (51)

Since $A^{[k]}=\psi^{[k]}(0)$, it follows that the eigenstates $\psi^{[k]}$
are orthogonal with respect to the inner product

$[\phi|\psi]=\int_{-L/2}^{L/2}\phi^{*}(x)\psi(x){\rm
d}x+\nu^{2}\phi^{*}(0)\psi(0)\,.$ (52)

Namely, with the normalization factor $Z_{k}$

$Z_{k}^{2}=\frac{L}{2}(1+\alpha(k)^{2})+\frac{\alpha(k)}{k}+\nu^{2}\,,$ (53)

we have

$[\psi^{[k]}|\psi^{[q]}]=\delta_{kq}\,.$ (54)

The physical meaning of this inner product is that in the zero-range limit
$a\to 0$, the particle has a finite probability (and not probability density)
to be located at the position of the potential $x=0$, given by

$\nu^{2}|\psi(0)|^{2}\,.$ (55)

The pointlike interaction can thus be thought as an idealized version of a
potential well that has a finite probability of capturing a particle.

###  2.3 The resulting Hamiltonian

Let us show that the local minima of the energy functional

$H(\psi)=\int_{-L/2}^{L/2}|\psi^{\prime}(x)|^{2}{\rm d}x+c|\psi(0)|^{2}\,,$
(56)

at fixed unit norm $[\psi|\psi]=1$ where this non-standard inner product is
defined in (52), satisfy the connection conditions (18). Introducing a
Lagrange multiplier $E$, we minimize

$F(\psi,E)=H(\psi)-E([\psi|\psi]-1)\,.$ (57)

We have up to $(\delta\psi)^{2}$ corrections

$\displaystyle F(\psi+\delta\psi,E)=$ $\displaystyle
F(\psi,E)-\int_{-L/2}^{L/2}(\psi^{\prime\prime}(x)+E\psi(x))^{*}\delta\psi(x){\rm
d}x$ (58)
$\displaystyle+2\Re[(c\psi(0)-\psi^{\prime}(0+)+\psi^{\prime}(0-))^{*}\delta\psi(0)]$
$\displaystyle-2E\nu^{2}\Re[\psi(0)^{*}\delta\psi(0)]\,.$

For this to vanish for all $\delta\psi$, one requires the conditions (18)
indeed.

###  2.4 Two particles interacting via the pointlike potential

The problem of two particles interacting via a potential straightforwardly
reduces to a one-particle problem in the center of mass frame. However, in our
case the zero-range limit brings some additional peculiarities that need to be
discussed. We thus consider two particles that interact via $V_{a}(x)$ in
(17), i.e. with Hamiltonian 222We omitted the factor $1/2$ in (56) compared to
(59) and in the definition in Section 1.1 to anticipate the two-particle case
with a factor $1/2$ which in the center of mass frame reduces to (1) without
factor $1/2$.

$\displaystyle H_{a}=$
$\displaystyle\frac{1}{2}\iint_{-L/2}^{L/2}\left(|\partial_{x_{1}}\psi(x_{1},x_{2})|^{2}+|\partial_{x_{2}}\psi(x_{1},x_{2})|^{2}\right){\rm
d}x_{1}{\rm d}x_{2}$ (59)
$\displaystyle+\iint_{-L/2}^{L/2}V_{a}(x_{1}-x_{2})|\psi(x_{1},x_{2})|^{2}{\rm
d}x_{1}{\rm d}x_{2}\,.$

The local minima $\psi_{a}(x_{1},x_{2})$ of this Hamiltonian satisfy the
Schrödinger equation

$-\frac{1}{2}(\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2})\psi_{a}+V_{a}(x_{1}-x_{2})\psi_{a}=E\psi_{a}\,,$
(60)

with $E$ to determine from the boundary conditions. Writing

$\psi_{a}(x_{1},x_{2})=\phi_{a}(x)e^{iky}\,,$ (61)

with

$x=x_{2}-x_{1}\,,\qquad y=x_{1}+x_{2}\,,$ (62)

we have that $\phi_{a}(x)$ satisfies

$-\phi_{a}^{\prime\prime}(x)+V_{a}(x)\phi_{a}(x)=(E-k^{2})\phi_{a}(x)\,.$ (63)

Hence Theorem 1 applies and in the limit $a\to 0$,
$\phi(x)\equiv\underset{a\to 0}{\lim}\phi_{a}(x)$ satisfies

$\phi^{\prime}(0+)-\phi^{\prime}(0-)=(c-\nu^{2}(E-k^{2}))\phi(0)\,.$ (64)

Equivalently, this jump condition can be written

$\phi^{\prime}(0+)-\phi^{\prime}(0-)=c\phi(0)+\nu^{2}\phi^{\prime\prime}(0)\,,$
(65)

which is, in terms of the original function

$\left.\left[\partial_{x_{2}}-\partial_{x_{1}}-c-\frac{\nu^{2}}{4}(\partial_{x_{2}}-\partial_{x_{1}})^{2}\right]\psi(x_{1},x_{2})\right|_{x_{2}=x_{1}+}=0\,.$
(66)

In this limit, the eigenstates of the Hamiltonian are orthogonal with respect
to the inner product

$[\phi|\psi]=\iint_{-L/2}^{L/2}\phi^{*}(x_{1},x_{2})\psi(x_{1},x_{2}){\rm
d}x_{1}{\rm d}x_{2}+\nu^{2}\int_{-L/2}^{L/2}\phi^{*}(x,x)\psi(x,x){\rm d}x\,.$
(67)

Physically, the additional piece proportional to $\nu^{2}$ in this inner
product corresponds to a finite probability that the two particles form a
pointlike pair of particles. However this bound state is not stable as we will
see in Section 2.5.

The expression of the Hamiltonian brings however an important difference
compared to the one-particle case, as is not the same as for a simple $\delta$
interaction (56). Let us show that the local minima of the following
Hamiltonian

$\displaystyle H(\psi)$
$\displaystyle=\frac{1}{2}\iint_{-L/2}^{L/2}\left(|\partial_{x_{1}}\psi(x_{1},x_{2})|^{2}+|\partial_{x_{2}}\psi(x_{1},x_{2})|^{2}\right){\rm
d}x_{1}{\rm d}x_{2}$ (68)
$\displaystyle+\frac{\nu^{2}}{4}\int_{-L/2}^{L/2}|(\partial_{x_{1}}+\partial_{x_{2}})\psi(x,x)|^{2}{\rm
d}x$ $\displaystyle+c\int_{-L/2}^{L/2}|\psi(x,x)|^{2}{\rm d}x$

at fixed unit norm $[\psi|\psi]=1$ implement the boundary conditions (66). To
that end, we introduce a Lagrange multiplier $E$ and minimize

$F(\psi,E)=H(\psi)-E([\psi|\psi]-1)\,.$ (69)

We have up to $\mathcal{O}(\delta\psi^{2})$ corrections, with simplified
notations $\partial_{j}=\partial_{x_{j}}$ and where the integrations are on
$x_{1},x_{2}$ in $[-L/2,L/2]\times[-L/2,L/2]$

$\displaystyle F(\psi+\delta\psi,E)-F(\psi,E)=$
$\displaystyle-2\Re\int\left[E\psi^{*}\delta\psi-\frac{1}{2}\partial_{1}\psi^{*}\partial_{1}\delta\psi-\frac{1}{2}\partial_{2}\psi^{*}\partial_{2}\delta\psi\right]$
(70)
$\displaystyle+2\Re\int\delta(x_{1}-x_{2})\left[(c-E\nu^{2})\psi^{*}\delta\psi+\frac{\nu^{2}}{4}(\partial_{1}+\partial_{2})\psi^{*}(\partial_{1}+\partial_{2})\delta\psi\right]\,.$

Performing integration by parts to remove all derivatives of $\delta\psi$, we
have

$\displaystyle F(\psi+\delta\psi,E)-F(\psi,E)=$
$\displaystyle-2\Re\int\left[E\psi^{*}+\frac{1}{2}\partial^{2}_{1}\psi^{*}+\frac{1}{2}\partial^{2}_{2}\psi^{*}\right]\delta\psi$
(71)
$\displaystyle+\Re\int\partial_{1}\psi^{*}\delta\psi[\delta(x_{1}-(x_{2}+))-\delta(x_{1}-(x_{2}-))]$
$\displaystyle+\Re\int\partial_{2}\psi^{*}\delta\psi[\delta(x_{2}-(x_{1}+))-\delta(x_{2}-(x_{1}-))]$
$\displaystyle+2\Re\int\delta(x_{1}-x_{2})\left[(c-E\nu^{2})\psi^{*}-\frac{\nu^{2}}{4}(\partial_{1}+\partial_{2})^{2}\psi^{*}\right]\delta\psi\,.$

Using that $\psi$ is symmetric in its arguments, this is

$\displaystyle F(\psi+\delta\psi,E)$
$\displaystyle-F(\psi,E)=-2\Re\int\left[E\psi^{*}+\frac{1}{2}\partial^{2}_{1}\psi^{*}+\frac{1}{2}\partial^{2}_{2}\psi^{*}\right]\delta\psi$
(72)
$\displaystyle+2\Re\int\delta(x_{1}-x_{2})\left[(\partial_{1}-\partial_{2})\psi^{*}+(c-E\nu^{2})\psi^{*}-\frac{\nu^{2}}{4}(\partial_{1}+\partial_{2})^{2}\psi^{*}\right]\delta\psi\,.$

Hence imposing the cancellation of the $\delta\psi$ terms yields

$\displaystyle-\frac{1}{2}(\partial^{2}_{1}+\partial^{2}_{2})\psi=E\psi$ (73)
$\displaystyle\left.\left[\partial_{2}-\partial_{1}-c+E\nu^{2}+\frac{\nu^{2}}{4}(\partial_{2}+\partial_{1})^{2}\right]\psi(x_{1},x_{2})\right|_{x_{2}=x_{1}+}=0\,.$

Using the first line, the second line is exactly the connection condition
(66).

We see that there is an additional kinetic term proportional to $\nu^{2}$ in
the Hamiltonian. Physically, this term corresponds to the kinetic energy of
the pointlike pair of particles that can be formed.

###  2.5 Dynamics

In order to illustrate some properties of the pointlike interaction
constructed in Theorem 1, we consider the simple problem of the dynamics of a
particle in a ring $[-L/2,L/2]$ interacting with the pointlike potential at
$x=0$, or equivalently two particles interacting with each other via the
pointlike potential.

Given an initial condition $\psi^{\rm ini}(x)$, the usual way to compute its
time evolution is to decompose it along the eigenstates of the Hamiltonian.
However in our case the unusual inner product requires some precautions. The
safe way to understand the decomposition with this unusual inner product is to
work at fixed $a>0$ and take the limit $a\to 0$. Let us thus consider an
$a$-dependent initial condition $\psi^{\rm ini}_{a}(x)$, that we will assume
to be even for simplicity of the discussion, and that we decompose along the
eigenstates at $a>0$

$\psi^{\rm
ini}_{a}(x)=\sum_{k}\psi^{[k]}_{a}(x)\langle\psi^{[k]}_{a}|\psi^{\rm
ini}_{a}\rangle\,.$ (74)

The question is then the limit $a\to 0$ of the overlap
$\langle\psi^{[k]}_{a}|\psi^{\rm ini}_{a}\rangle$. As seen from Lemma 2,
$\psi^{[k]}_{a}(x)$ contains a piece
$\psi^{[k]}(0)\nu\sqrt{\tfrac{1}{a}\Delta(\tfrac{x}{a})}$ that does not
converge uniformly. If $\psi^{\rm ini}_{a}$ itself converges uniformly to
$\psi^{\rm ini}$, then from Lemma 1 we have

$\underset{a\to 0}{\lim}\,\langle\psi^{[k]}_{a}|\psi^{\rm
ini}_{a}\rangle=\langle\psi^{[k]}|\psi^{\rm ini}\rangle\,.$ (75)

This leads to the time evolution in the $a\to 0$ limit

$\psi(x,t)=\sum_{k}\psi^{[k]}(x)\langle\psi^{[k]}|\psi^{\rm ini}\rangle
e^{-itE_{k}}\,,$ (76)

with an inner product that is not the one for which the eigenstates
$\psi^{[k]}$ are orthogonal. However, if the initial condition $\psi^{\rm
ini}_{a}$ contains itself a piece of the form $\psi^{\rm
ini}(0)\nu\sqrt{\tfrac{1}{a}\Delta(\tfrac{x}{a})}$, then

$\underset{a\to 0}{\lim}\,\langle\psi^{[k]}_{a}|\psi^{\rm
ini}_{a}\rangle=[\psi^{[k]}|\psi^{\rm ini}]\,,$ (77)

and the time evolution is written in terms of the modified inner product

$\psi(x,t)=\sum_{k}\psi^{[k]}(x)[\psi^{[k]}|\psi^{\rm ini}]e^{-itE_{k}}\,.$
(78)

Let us illustrate the second situation with the following problem. We would
like to consider the case in which the particle is located at $x=0$ with
probability $1$ at time $t=0$. This can be achieved with $\psi^{\rm
ini}_{a}(x)=\sqrt{\tfrac{1}{a}\Delta(\tfrac{x}{a})}$. One finds then

$\underset{a\to 0}{\lim}\,\langle\psi^{[k]}_{a}|\psi^{\rm
ini}_{a}\rangle=\frac{\nu}{Z_{k}}\equiv[\psi^{[k]}|\psi^{\rm ini}]\,,$ (79)

where $Z_{k}$ is given in (53). For finite $L$, the non-trivial quantization
condition (47) has to be solved numerically and (78) evaluated numerically.
This way one can check for example the completeness relation
$\sum_{k}|[\psi^{\rm ini}|\psi^{[k]}]|^{2}=1$ that holds for the non-standard
inner product $[.|.]$. As an illustration, we plot in the right panel of
Figure 1 the probability $p(t)$ that the particle is exactly located at $x=0$
(and not just probability density) given by

$p(t)=\nu^{2}|\psi(0,t)|^{2}\,.$ (80)

The case $L\to\infty$ deserves attention since it is tractable. In this limit
the quantization condition becomes

$k=\frac{2\pi(n+1/2)}{L}+\mathcal{O}(L^{-1})\,,$ (81)

with $n\geq 0$ an integer. The time evolution of the wave function can then be
analytically computed in a straightforward way and is given by

$\psi(x,t)=\frac{\nu}{\pi}\int_{0}^{\infty}\frac{e^{-itk^{2}}}{1+\alpha(k)^{2}}\left(\cos(kx)+\alpha(k)\sin(k|x|)\right){\rm
d}k\,,$ (82)

where $\alpha(k)$ is defined in (46). In particular, the probability $p(t)$ of
finding the particle exactly at $x=0$ is

$p(t)=\frac{\nu^{4}}{\pi^{2}}\left|\int_{0}^{\infty}\frac{e^{-itk^{2}}}{1+\alpha(k)^{2}}{\rm
d}k\right|^{2}\,.$ (83)

We see that it goes to $0$ when $t\to\infty$ as $t^{-3}$. This means that the
pointlike pair of particles formed by two particles interacting via this
pointlike potential is unstable and eventually decays with probability $1$.

$-0.1$$-5\cdot 10^{-2}$$0$$5\cdot 10^{-2}$$0.1$$-1.0$$-0.5$$0.0$$0.5$$\cdot
10^{4}$$x$$V_{a}(x)$

$0$$5\cdot 10^{-2}$$0.1$$0.15$$0.2$$0.0$$0.2$$0.4$$0.6$$0.8$$1.0$$t$$p(t)$

Figure 1: Left: plot of the regulated potential (17) with $c=1$, $\nu=1$,
$a=0.01$ and with the functions $\sigma(x)=\tanh(x)$,
$\Delta(x)=\tfrac{1}{2}\tanh^{\prime}(x)$. Right: probability of finding the
particle at $x=0$ as a function of time, with $c=1$, $\nu=1$, for a system
size $L=1$ (plain) and $L=\infty$ (dashed) [see Section 2.5].

###  2.6 Energy-dependent potentials with a finite support

Let us now consider a particle interacting with $N$ pointlike interactions of
the type of Theorem 1 centered at positions $x_{n}$ with coupling strength
$c_{n}$ and energy-dependence $\nu_{n}$, for $n=1,...,N$. The stationary
states $\psi_{N}$ are continuous functions that satisfy

$\psi_{N}^{\prime\prime}(x)+E\psi_{N}(x)=0\,,\qquad\text{for
}x\notin\\{x_{1},...,x_{N}\\}\,,$ (84)

and

$\psi^{\prime}_{N}(x_{n}+)-\psi_{N}^{\prime}(x_{n}-)=(c_{n}-E\nu_{n}^{2})\psi_{N}(x_{n})\,.$
(85)

We consider now $x_{n}=-L/2+(n-1/2)\frac{L}{N}$, couplings
$c_{n}=\frac{L}{N}\mathcal{V}(x_{n})$ where $\mathcal{V}(x)\geq 0$ is a fixed
arbitrary potential, and $\nu^{2}_{n}=\frac{L}{N}\nu^{2}\mathcal{V}(x_{n})$
with some $\nu\geq 0$. Between two points $x_{n},x_{n+1}$, $\psi_{N}$
satisfies (84) so that

$\psi^{\prime}_{N}(x_{n+1}-)=\psi_{N}^{\prime}(x_{n}+)-E\psi_{N}(x_{n})\frac{L}{N}+\mathcal{O}(N^{-2})\,.$
(86)

Hence

$\frac{\psi^{\prime}_{N}(x_{n+1}-)-\psi^{\prime}_{N}(x_{n}-)}{L/N}=-E\psi_{N}(x_{n})+\mathcal{V}(x_{n})(1-E\nu^{2})\psi_{N}(x_{n})+\mathcal{O}(N^{-1})\,.$
(87)

It follows that in the limit $N\to\infty$, $\psi(x)\equiv\psi_{\infty}(x)$
satisfies

$-\psi^{\prime\prime}(x)+\mathcal{V}(x)(1-E\nu^{2})\psi(x)=E\psi(x)\,,$ (88)

which is a Schrödinger equation in a (non pointlike) potential
$\mathcal{V}(x)$ with an energy-dependent coupling. This construction,
together with Theorem 1, provides a regularization of energy-dependent
potentials in terms of hermitian, regular potentials in non-relativistic 1D
quantum mechanics.

##  3 Energy-dependent pointlike interaction between fermions

###  3.1 The potential

The potential presented in Section 2 is transparent for fermions. We recall
that the only non-trivial standard pointlike interaction between fermions
induces a discontinuity in the wave function $\psi$ itself, namely

$\psi(0+)-\psi(0-)=2\beta\psi^{\prime}(0)\,,$ (89)

with $\beta$ some coupling constant. The continuity of the derivative at $0$
is ensured by the oddness of the wave function for fermions. We will call
$\eta$ this pointlike interaction. Depending on the regularization, this
$\eta$ potential can act like an infinite wall for bosons [14, 25] or be
transparent [28].

The second energy-dependent pointlike interaction that we present in this
paper acts like an infinite wall for bosons, and like a $\eta$ potential with
an energy-dependent coupling for fermions. Its construction is given by the
following Theorem.

######  Theorem 2.

We consider $c,\nu,\sigma,\Delta$ as defined in Theorem 1, with moreover
$\sigma^{\prime}(0)>0$. For $E$ real and $a>0$, we define $\psi_{a}(x)$ as the
solution to

$-\psi_{a}^{\prime\prime}(x)+V_{a}(x)\psi_{a}(x)=E\psi_{a}(x)\,,$ (90)

with some fixed $a$-independent boundary conditions at $x=1$, and where the
potential is

$V_{a}(x)=\frac{v_{a}^{\prime\prime}(x)}{v_{a}(x)}\,,\qquad
v_{a}(x)=x+\frac{2}{c}\sigma(\tfrac{x}{a})+\nu\sigma(\tfrac{x}{a})\sqrt{\tfrac{1}{a}\Delta(\tfrac{x}{a})}\,.$
(91)

The limit $\psi(x)\equiv\underset{a\to 0}{\lim}\,\psi_{a}(x)$ satisfies

$\displaystyle\psi^{\prime\prime}(x)+E\psi(x)=0\quad{\rm for}\,\,x\neq 0$ (92)
$\displaystyle\left(\begin{matrix}\psi(0+)\\\
\psi^{\prime}(0+)\end{matrix}\right)=-\left(\begin{matrix}1&0\\\
c-E\nu^{2}&1\end{matrix}\right)\left(\begin{matrix}\psi(0-)\\\
\psi^{\prime}(0-)\end{matrix}\right)$

Before entering the proof, let us make the following comment on the connection
conditions (92). If the wave function describes two fermions interacting via
$V_{a}(x)$ in the center of mass frame, then $\psi(x)$ has to be odd, so
$\psi(0+)=-\psi(0-)$ and $\psi^{\prime}(0+)=\psi^{\prime}(0-)$. This leads to
the jump condition

$\displaystyle\psi(0+)-\psi(0-)$
$\displaystyle=\frac{4}{c-E\nu^{2}}\psi^{\prime}(0)\,,$ (93)

which is exactly (89) with an energy-dependent coupling. For bosons, $\psi$
has to be even, so $\psi(0+)=\psi(0-)$ and
$\psi^{\prime}(0+)=-\psi^{\prime}(0-)$. The connection condition (92) then
implies $\psi(0+)=\psi(0-)=0$, corresponding to hard core bosons. Hence the
pointlike potential acts like an infinite wall for bosons.

We also note that under the Girardeau mapping $\psi(x)=\,{\rm
sgn}\,(x)\phi(x)$ [43, 12, 13], if a wave function $\psi$ satisfies (18), then
$\phi$ satisfies (92). This shows that the pointlike interaction of Theorem 2
is dual to that of Theorem 1 under the Bose-Fermi duality that is the
Girardeau mapping in 1D.

###### Proof of Theorem 2.

The proof goes along the same lines as for Theorem 1, so that we will only
give the main steps.

One first determines two independent solutions to (90) when $E=0$, that are
$v_{a}(x)$ that is odd, and

$w_{a}(x)=\frac{1}{v_{a}^{\prime}(x)}+v_{a}(x)\int_{0}^{x}\frac{v_{a}^{\prime\prime}(y)}{v_{a}(y)v_{a}^{\prime}(y)^{2}}{\rm
d}y\,,$ (94)

that is even. A study of $w_{a}(x)$ using the assumption
$\sigma^{\prime}(0)>0$ shows that it can be bounded independently of $a<1$ and
$x\in[-L/2,L/2]$, and yields the limits when $a\to 0$

$v_{a}(x)\to x+\tfrac{c}{2}\,{\rm sgn}\,(x)\,,\qquad
w_{a}(x)\to-\frac{2}{c}|x|\,.$ (95)

Then using the same arguments, $\psi_{a}$ satisfies the following self-
consistency equation

$\displaystyle\psi_{a}(x)=Ev_{a}(x)\int_{0}^{x}\psi_{a}(y)w_{a}(y){\rm
d}y-Ew_{a}(x)\int_{0}^{x}\psi_{a}(y)v_{a}(y){\rm d}y+Av_{a}(x)+Bw_{a}(x)\,.$
(96)

with $A,B$ integration constants that can be assumed to be independent of $a$
up to a global rescaling. Then one defines

$\bar{\psi}_{a}(x)=\psi_{a}(x)-A\nu\sigma(\tfrac{x}{a})\sqrt{\tfrac{1}{a}\Delta(\tfrac{x}{a})}\,,$
(97)

and shows similarly to Lemma 2 that it can be bounded independently of $a<1$
and $x\in[-L/2,L/2]$. This allows one to take the limit $a\to 0$ in the self-
consistency equation when rewritten in terms of $\bar{\psi}_{a}$. Using then
again (42), one shows straightforwardly that the resulting self-consistency
equation on $\bar{\psi}=\underset{a\to 0}{\lim}\bar{\psi}_{a}$ is equivalent
to the connection conditions (92).

∎

###  3.2 The resulting inner product and Hamiltonian

In a ring $[-L/2,L/2]$ with periodic boundary conditions, the even eigenstates
of the Hamiltonian with a pointlike interaction at $x=0$ are

$\psi^{[k]}(x)=\frac{\sin(k|x|)}{Z_{k}^{\prime}}\,,$ (98)

with $Z^{\prime}_{k}$ a normalization factor, and where the quantization
condition is $\cos(kL/2)=0$. The odd eigenstates are

$\phi^{[k]}(x)=\frac{1}{Z_{k}}(\,{\rm sgn}\,(x)\cos(kx)+\alpha(k)\sin(kx))\,,$
(99)

with $Z_{k}$ a normalization factor and $\alpha(k)$ as in (46), and with

$\tan(kL/2)=-\frac{1}{\alpha(k)}\,.$ (100)

The eigenstates are orthogonal with respect to the inner product

$[\phi|\psi]=\int_{-L/2}^{L/2}\phi^{*}(x)\psi(x){\rm
d}x+\frac{\nu^{2}}{4}(\phi^{*}(0+)-\phi^{*}(0-))(\psi(0+)-\psi(0-))\,.$ (101)

As for the resulting Hamiltonian, one obtains with a similar calculation as in
Section 2.3 that the eigenstates are produced as local minima of the energy
functional

$H(\psi)=\int_{-L/2}^{L/2}|\psi^{\prime}(x)|^{2}{\rm
d}x+\frac{c}{4}|\psi(0+)-\psi(0-)|^{2}\,,$ (102)

at fixed unit norm $[\psi|\psi]=1$.

## Summary and perspectives

In this paper we constructed a potential $V_{a}(x)$ with range $a$ that
converges in the zero-range limit $a\to 0$ to the pointlike Dirac $\delta$
potential with an energy-dependent coupling. We constructed as well a
potential that converges to the fermionic ”$\delta^{\prime}$” potential with
an energy-dependent coupling. These results are noteworthy for several
reasons.

Firstly, it reveals a new class of pointlike interactions in 1D quantum
mechanics, that were previously thought to be classified into a four-parameter
family. The inconsistency is resolved as follows. This classification relied
on the idea that the Hamiltonian for a pointlike potential should be hermitian
for the standard inner product. But we show that our Hamiltonian, despite
being indeed hermitian for the standard inner product when $a>0$, becomes
hermitian with respect to a different inner product in the limit $a\to 0$,
which allows for the pointlike interaction studied in this paper. This raises
thus the question of classifying all the possible pointlike interactions in 1D
that can be obtained through the zero-range limit of regular potentials. For
example, one could ask whether it is possible to obtain a $\delta$ interaction
with a more general $E$ dependence in the coupling than what is constructed in
this paper. One could also ask whether such construction can be performed in
higher dimensions.

Secondly, this pointlike interaction can be interpreted as a pointlike well
that has a finite probability of capturing a particle. Indeed, the resulting
inner product for which it is hermitian attributes a finite probability (and
not just probability density) for the particle to be located at the position
of the potential. This pointlike interaction can thus model an inter-particle
force that allows for the formation of a pair of particles with a finite
probability (although such pairs are unstable). This model could find
applications to describe the effect of formation of diatomic molecules in
actual Bose gases [44]. A natural question in that direction is whether a
Hamiltonian for $N$ particles interacting via this pointlike interaction enjoy
an inner product with a similar physical interpretation.

Thirdly, this potential shows that running couplings, i.e. energy-dependent
couplings are not a specificity of quantum field theories or relativistic
quantum mechanics, but also exist in 1D non-relativistic quantum mechanics. To
the best of our knowledge this observation has not been made before. It is
certainly interesting to investigate whether relations between these pointlike
interactions and quantum field theories can be drawn beyond this observation.

## References

  * [1] R. de L. Kronig and W. G. Penney,  _Quantum mechanics of electrons in crystal lattices_ ,  Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 130(814), 499 (1931).
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