The Propagators for δ and δ′ Potentials With Time-Dependent Strengths

1. Introduction

Dirac delta potentials in quantum mechanics have been used to model different physical systems almost since the beginning of quantum mechanics. Kronig Penney model [1] is the well-known example of these models. These potentials are the particular cases of much more general class of potentials, namely point interactions. In one dimension, one rigorous way of defining the point interaction at the origin is based on the self-adjoint extension of the free symmetric Hamiltonian defined on ℝ \ {0}. In this approach, the initially ill-defined formal δ and δ′ function potentials appear naturally as two special cases of point interactions constructed from the self-adjoint extension theory. In general one has a 4 -parameter family of self-adjoint extension in one dimension. The monograph [2] includes a great deal of all the details and summarize the history of the literature about the δ interactions. The review article [3] and the book [4] are also good reference sources about the δ potentials from the physical point of view.

The δ′ perturbation of free Hamiltonian $H_0=-\frac{d^2}{d{x}^2}$ is defined as a limit of short range potentials in the distributional sense [5–7]. Although there are some controversial issues about δ′ interactions (see e.g., [8–11]), they are also getting considerable amount of interest. The ambiguities about δ′ interactions have been summarized in a very recent article [11], where the integral form of the Schrödinger equation for δ′ potential has been studied based on the work of Kurasov [12]. We also adopt the distributional approach developed by Kurasov [12] for the functions having a discontinuity at the point of δ′. It is possible to overcome these ambiguities by considering different choices, as different type of δ′ interactions [13]. Therefore, the different results on the spectrum of δ′ potential obtained in [2] and in [8] using the Kurasov's approach -as a special case of −aδ(x) + bδ′(x) potential- can be interpreted consistently. In other words, the Kurasov's approach corresponds to different self-adjoint extension of the free Hamiltonian H₀. These self-adjoint extensions are given by matching conditions at the origin (or at the point supporting the perturbation) and two of these matching conditions maybe identified as a δ′ interaction and receive the names of non-local and local δ′ interaction, respectively [13]. Both approaches are used to see how the spectrum of the V-shaped potential changes when it is perturbed by −aδ(x) + bδ′(x) perturbation in [9, 13]. The scattering, and resonant tunneling for δ′ potential in one dimension is also a controversial issue because different results are obtained in the literature [8, 10], depending on whether the δ′ interaction is the non-local or the local one. The results obtained in [14–16] for the non-local δ′ potential, show that it is opaque for all energies of an incoming beam. However, other authors [17–20] claim that there are discrete energy values in the spectrum of δ′ potential which lead to resonant tunneling. As an another physical application, δ′ interactions are used to model Casimir effect in [21, 22].

The exact expression of the propagator for one dimensional single Dirac delta potentials have been found in different ways [23–25]. The generalization to two center case have been studied in Cacciari and Moretti [26]. The propagator for general four parameter family of point interactions have been given in Albeverio et al. [27]. Propagators for systems involving δ potentials are also studied from various points of view in references [28–34]. The propagator for derivatives of Dirac delta distribution for constant strengths has been recently studied in Lange [11]. The Cauchy problem for the non-local δ′ potential with a time dependent strength has also been studied rigorously in detail [35]. Moreover, time dependent one dimensional point interactions have been studied in Campbell [36] and the exact solution to the initial value problem for Schrödinger equation has been given for some particular form of strength λ(t) of the Dirac delta potential. Later on, the system has been investigated in Hmidi et al. [37] more rigorously and the regularity assumptions on λ(t) is determined for which the initial value problem defines a unitary strongly continuous dynamical system on L²(ℝ). Such time-dependent point interactions have been studied rigorously in order to model asymptotic complete ionization and suitable conditions on the function λ(t) has been determined for ionization problem [38–41]. The higher dimensional generalizations of the problem have been studied in great detail and summarized in the thesis by Correggi [42]. Transmission properties of a monochromatic beam and wave packets by studying the scattering from the time-dependent δ potential are studied in Martinez and Reichl [43] and Kuhn et al. [32]. Utilizing δ potential with a time dependent coefficient in an infinite well, Baek et al. [44] showed that it is possible to split a wave function which may have applications in statistical mechanics and condensed matter physics. However, it may also lead to philosophical problems [44]. The time dependence of the Dirac delta potentials could also be expressed through the motion of its support [45]. As a physical application, a moving Dirac δ potential is used to describe particle displacement using a standard tunneling microscope [46].

The paper is organized as follows: In section 2, we obtain a formal expression of the propagator for a finite number of Dirac δ potentials with time dependent strengths and solve the time dependent Schrödinger equation for this system. In the following subsections, we investigate one δ potential with time dependent strength in more detail and find the propagator for N Dirac δ potential in the limit that centers are infinitely separated. In section 3, we get an expression of the propagator for a finite number of δ′ potentials with time dependent strengths and solve the time dependent Schrödinger equation for this potential. Finally, we elaborate on one δ′ case.

2. The Propagator for N Dirac Delta Potentials With Time Dependent Strengths

We begin with a one dimensional model in which a free Hamiltonian of the type $H_0=-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}$ is perturbed with a time dependent potential

$\begin{array}{ll}V(x,t)={\displaystyle \sum _{j=1}^N}{\lambda }_j(t)\delta (x-x_j).& (1)\end{array}$

The initial value problem of the time-dependent Schrödinger equation for this potential is

$\begin{array}{ll}i\frac{\partial }{\partial t}\psi (x,t)=\left[-\frac{d^2}{d{x}^2}+{\displaystyle \sum _{j=1}^N}{\lambda }_j(t)\delta (x-x_j)\right]\psi (x,t),& (2)\end{array}$

with the given sufficiently smooth function ψ(x, 0). Here we have used the units such that ℏ = 2m = 1 for simplicity. It is well-known that the Laplace transform is a very useful tool to solve initial value problems so we first take the Laplace transformation of Equation (2) with respect to time variable t and get

$\begin{array}{ll}{\bar{\psi }}_{xx}(x,s)-i\psi (x,0)+is\bar{\psi }(x,s)={\displaystyle \sum _{j=1}^N}\delta (x-x_j){L}\left\{{\lambda }_j(t)\psi (x_j,t)\right\},& (3)\end{array}$

where $\bar{\psi }(x,s)={L}\left\{\psi (x,t)\right\}$ and ${\bar{\psi }}_{xx}(x,s)={L}\left\{\frac{{\partial }^2}{\partial x^2}\psi (x,t)\right\}$.

After this, we take the Fourier transformation of both sides of Equation (3) with respect to the coordinate variable x and get:

$\begin{array}{ll}-k^2\widehat{\bar{\psi }}(k,s)-i\widehat{\psi }(k,0)+is\widehat{\bar{\psi }}(k,s)={\displaystyle \sum _{j=1}^N}{e}^{-ik{x}_j}{L}\left\{{\lambda }_j(t)\psi (x_j,t)\right\},& (4)\end{array}$

where $\widehat{\bar{\psi }}(k,s)={F}(\bar{\psi }(x,s))$ denotes the Fourier transform of $\bar{\psi }(x,s)$ with respect to the variable x. This equation is an algebraic equation for the unknown wave function $\widehat{\bar{\psi }}(k,s)$ and the solution is easily obtained as

$\begin{array}{ll}\widehat{\bar{\psi }}(k,s)=-\frac{1}{k^2-is}\left[i\widehat{\psi }(k,0)+{\displaystyle \sum _{j=1}^N}{e}^{-ik{x}_j}{L}\left\{{\lambda }_j(t)\psi (x_j,t)\right\}\right].& (5)\end{array}$

Now, we immediately find the inverse Fourier transform of the $\widehat{\bar{\psi }}(k,s)$:

$\begin{array}{l}\bar{\psi }(x,s)=-i{\displaystyle {\int }_{-\infty }^{\infty }}\frac{dk}{2\pi }\frac{e^{ikx}}{k^2-is}\widehat{\psi }(k,0)\\ -{\displaystyle \sum _{j=1}^N}{L}\left\{{\lambda }_j(t)\psi (x_j,t)\right\}{\displaystyle {\int }_{-\infty }^{\infty }}\frac{dk}{2\pi }\frac{e^{ik(x-x_j)}}{k^2-is}.& (6)\end{array}$

Using $\widehat{\psi }(k,0)={\int }_{-\infty }^{\infty }d{x}^{\prime }{e}^{-ik{x}^{\prime }}\psi (x^{\prime },0)$, the Equation (6) becomes

$\begin{array}{l}\bar{\psi }(x,s)=-i{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }\psi (x^{\prime },0){\displaystyle {\int }_{-\infty }^{\infty }}\frac{dk}{2\pi }\frac{e^{ik(x-x^{\prime })}}{k^2-is}\\ -{\displaystyle \sum _{j=1}^N}{L}\left\{{\lambda }_j(t)\psi (x_j,t)\right\}{\displaystyle {\int }_{-\infty }^{\infty }}\frac{dk}{2\pi }\frac{e^{ik(x-x_j)}}{k^2-is}.& (7)\end{array}$

The integrals in Equation (7) are easily taken using residue theorem

$\begin{array}{ll}{\displaystyle {\int }_{-\infty }^{\infty }}\frac{dk}{2\pi }\frac{e^{ikx}}{k^2-is}=i\frac{e^{i\sqrt{is}\left|x\right|}}{2\sqrt{is}},& (8)\end{array}$

so we obtain

$\begin{array}{l}\bar{\psi }(x,s)=\frac{1}{2\sqrt{is}}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }{e}^{i\sqrt{is}|x-x^{\prime }|}\psi (x^{\prime },0)\\ -\frac{i}{2\sqrt{is}}{\displaystyle \sum _{j=1}^N}{e}^{i\sqrt{is}|x-x_j|}{L}\left\{{\lambda }_j(t)\psi (x_j,t)\right\}.& (9)\end{array}$

Now we need to take inverse Laplace transform to obtain the solution of the time dependent equation for the potential λ(t)δ(x). Using the Bromwich contour [47], one can easily find the inversion

$\begin{array}{ll}{{L}}^{-1}\left\{\frac{e^{i\sqrt{is}\left|x\right|}}{\sqrt{is}}\right\}=\frac{1}{\sqrt{i\pi t}}exp\left(\frac{i{x}^2}{4t}\right),& (10)\end{array}$

and using the convolution theorem we get the formal solution as

$\begin{array}{l}\psi (x,t)={\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }\frac{1}{\sqrt{4\pi it}}exp\left[\frac{i{(x-x^{\prime })}^2}{4t}\right]\psi (x^{\prime },0)\\ -i{\displaystyle \sum _{j=1}^N}{\displaystyle {\int }_0^t}d{t}^{\prime }\frac{{\lambda }_j(t^{\prime })\psi (x_j,t^{\prime })}{\sqrt{4\pi i(t-t^{\prime })}}exp\left[\frac{i{(x-x_j)}^2}{4(t-t^{\prime })}\right].& (11)\end{array}$

Although this is an explicit formal expression for ψ(x, t), it is not completely expressed in terms of the initial condition ψ(x, 0) and includes the unknown factors ψ(x_j, t). These can be found by simply inserting x = x_j in the formal solution and then solving the resulting coupled Volterra type integral equations

$\begin{array}{l}\psi (x_j,t)={\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }\frac{1}{\sqrt{4\pi it}}exp\left[\frac{i{\left(x_j-x^{\prime }\right)}^2}{4t}\right]\psi (x^{\prime },0)\\ -i{\displaystyle {\int }_0^t}d{t}^{\prime }\frac{{\lambda }_j\left(t^{\prime }\right)\psi \left(x_j,t^{\prime }\right)}{\sqrt{4\pi i\left(t-t^{\prime }\right)}}\\ -i{\displaystyle \sum _{\begin{array}{c}k=1\\ k\ne j\end{array}}^N}{\displaystyle {\int }_0^t}d{t}^{\prime }\frac{{\lambda }_k\left(t^{\prime }\right)\psi \left(x_k,t^{\prime }\right)}{\sqrt{4\pi i\left(t-t^{\prime }\right)}}exp\left[\frac{i{\left(x_j-x_k\right)}^2}{4\left(t-t^{\prime }\right)}\right],& (12)\end{array}$

where we split the term k = j in the summation over k. Since this is not an easy problem for a generic function λ(t), we will first investigate for particular cases, where λ is constant and λ is inversely proportional to t. All these results we present in the next subsection is previously obtained by Campbell [36] using a slightly different method, where only one integral transformation with the boundary conditions at the position of δ potential was used.

2.1. Single δ Potential With a Time Dependent Strength

As a particular case of (1), we consider a single delta potential with time-dependent strength

$\begin{array}{ll}V(x,t)=\lambda (t)\delta (x),& (13)\end{array}$

where N = 1 and x₁ = 0. We can formally obtain the solution of the time dependent Schrödinger equation for this case using Equation (11)

$\begin{array}{l}\psi (x,t)={\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }\frac{1}{\sqrt{4\pi it}}exp\left[\frac{i{(x-x^{\prime })}^2}{4t}\right]\psi (x^{\prime },0)\\ -i{\displaystyle {\int }_0^t}d{t}^{\prime }\frac{\lambda (t^{\prime })\psi (0,t^{\prime })}{\sqrt{4\pi i(t-t^{\prime })}}exp\left[\frac{i{x}^2}{4(t-t^{\prime })}\right].& (14)\end{array}$

Actually this result can be directly obtained from the Duhamel's formula [48] for time-dependent Schrödinger equation associated with the Hamiltonian H = H₀ + V, where H₀ is self-adjoint free Hamiltonian and V is bounded (or relatively H₀ -bounded with relative bound <1):

$\begin{array}{l}{e}^{-itH}|{\psi }_0\rangle =e^{-it{H}_0}|{\psi }_0\rangle +\\ (-i){\displaystyle {\int }_0^t}d{t}^{\prime }{e}^{-i(t-t^{\prime })H_0}V{e}^{-i{t}^{\prime }H}|{\psi }_0\rangle & (15)\end{array}$

for every |ψ₀〉 = |ψ(t = 0)〉. This shows that even if we formally take V = λ(t′)|0〉〈0| which corresponds to our Dirac delta potential, one immediately sees that the Duhamel's formula is still formally valid for such singular interactions.

Given the initial condition ψ(x, 0) and the function λ(t), the function ψ(0, t) can be determined by solving the following integral equation:

$\begin{array}{l}\psi (0,t)={\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }\frac{1}{\sqrt{4\pi it}}exp\left[\frac{i{x^{\prime }}^2}{4t}\right]\psi (x^{\prime },0)\\ -i{\displaystyle {\int }_0^t}d{t}^{\prime }\frac{\lambda (t^{\prime })\psi (0,t^{\prime })}{\sqrt{4\pi i(t-t^{\prime })}}.& (16)\end{array}$

However, this is in general hard to solve and one usually applies some approximation techniques, e.g., Dyson series [48]. Nevertheless, as shown in Campbell [36], there are cases where one can calculate the Green's function explicitly. We will show two such cases here explicitly. Although Green's functions for these cases are derived in Campbell [36], we repeat these results here for the sake of completeness. Instead of directly solving the above integral equation for particular cases, it is convenient to start with the Laplace transformed wave function Equation (9) for a single δ potential centered at the origin:

$\begin{array}{l}\bar{\psi }(x,s)=\frac{1}{2\sqrt{is}}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }{e}^{i\sqrt{is}|x-x^{\prime }|}\psi (x^{\prime },0)\\ -\frac{i}{2\sqrt{is}}{e}^{i\sqrt{is}\left|x\right|}{L}\left\{\lambda (t)\psi (0,t)\right\}.& (17)\end{array}$

The cases for constant λ and $\lambda \propto \frac{1}{t}$ are reviewed in Appendices A and B in detail.

2.2. The Propagator for N Dirac δ Potentials in a Limiting Case

When λ_j's are constant, the Laplace transformed wave function $\bar{\psi }(x,s)$ given by (9) becomes

$\begin{array}{l}\bar{\psi }(x,s)=\frac{1}{2\sqrt{is}}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }{e}^{i\sqrt{is}|x-x^{\prime }|}\psi (x^{\prime },0)\\ -\frac{i}{2\sqrt{is}}{\displaystyle \sum _{k=1}^N}{\lambda }_k{e}^{i\sqrt{is}|x-x_k|}\bar{\psi }(x_k,s).& (18)\end{array}$

The unknown functions $\bar{\psi }(x_k,s)$ can be found by evaluating the above expression at x = x_j:

$\begin{array}{ll}\bar{\psi }(x_k,s)={\displaystyle \sum _{j=1}^N}{\left[{\Phi }^{-1}(s)\right]}_{kj}\bar{\rho }(x_j,s),& (19)\end{array}$

where

$\begin{array}{ll}{\Phi }_{kj}(s)=\{\begin{array}{ll}1+\frac{i{\lambda }_j}{2\sqrt{is}}& \text{if }j=k\\ \frac{i{\lambda }_k}{2\sqrt{is}}{e}^{i\sqrt{is}|x_j-x_k|}& \text{if }j\ne k\end{array},& (20)\end{array}$

and

$\begin{array}{ll}\bar{\rho }(x_k,s)=\frac{1}{2\sqrt{is}}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }{e}^{i\sqrt{is}|x_k-x^{\prime }|}\psi (x^{\prime },0).& (21)\end{array}$

Substituting (19) into (18), we obtain

$\begin{array}{ll}\bar{\psi }(x,s)=\bar{\rho }(x,s)-\frac{i}{2\sqrt{is}}{\displaystyle \sum _{j=1}^N}{\displaystyle \sum _{k=1}^N}{\lambda }_k{e}^{i\sqrt{is}|x-x_k|}{\left[{\Phi }^{-1}(s)\right]}_{kj}\bar{\rho }(x_j,s).& (22)\end{array}$

Although we have obtained the Laplace transformed wave function $\bar{\psi }(x,s)$, it is not explicitly given since one has to invert the matrix Φ and find the inverse Laplace transform of the resulting expression to get the final solution ψ(x, t). In general, it is difficult to find the inverse Laplace transforms so one may apply some approximation schemes [49]. Moreover, one could use some numerical computations, but we will here simply show the limiting case, where the centers are infinitely far away from each other.

When all the centers are infinitely separated from each other, that is, |x_j − x_k| → ∞, we expect that the off-diagonal elements of the matrix Φ given in (20) vanish, so that

$\begin{array}{ll}{\left[{\Phi }^{-1}(s)\right]}_{kj}=\left(\frac{1}{1+\frac{i{\lambda }_k}{2\sqrt{is}}}\right){\delta }_{kj}.& (23)\end{array}$

Then, $\bar{\psi }(x,s)$ can be explicitly found as

$\begin{array}{ll}\bar{\psi }(x,s)=\bar{\rho }(x,s)-i{\displaystyle \sum _{j=1}^N}{e}^{i\sqrt{is}|x-x_j|}\left(\frac{{\lambda }_j}{2\sqrt{is}+i{\lambda }_j}\right)\bar{\rho }(x_j,s),& (24)\end{array}$

from which the propagator G(x, x′, s) reads

$\begin{array}{ll}G(x,x^{\prime },s)=\frac{e^{i\sqrt{is}|x-x^{\prime }|}}{2\sqrt{is}}-{\displaystyle \sum _{j=1}^N}{e}^{i\sqrt{is}|x-x_j|}\left(\frac{{\lambda }_j}{2\sqrt{s}+\sqrt{i}{\lambda }_j}\right)\frac{e^{i\sqrt{is}|x_j-x^{\prime }|}}{2\sqrt{s}}.& (25)\end{array}$

Hence,

$\begin{array}{l}G(x,x^{\prime },t)=\frac{1}{2\sqrt{i\pi t}}exp\left[\frac{i{(x-x^{\prime })}^2}{4t}\right]\\ -{\displaystyle \sum _{j=1}^N}\frac{{\lambda }_j}{4}exp\left[\frac{{\lambda }_j}{2}(|x-x_j|+|x_j-x^{\prime }|)+i\frac{{\lambda }_j^2}{4}t\right]\\ \text{erfc }\left[\frac{(|x-x_j|+|x_j-x^{\prime }|)}{2\sqrt{it}}+\sqrt{it}\frac{{\lambda }_j}{2}\right].& (26)\end{array}$

This is actually the superposition of the individual propagators associated with single delta centers. This is expected since there is no correlation among the centers when they are far away from each other. Another limiting case is the case where all the centers coincide.

3. The Propagator for N Dirac δ′ Potentials With Time Dependent Strengths

In this section we first obtain a formal solution of the time-dependent Schrödinger equation, where the potential term is chosen formally as

$\begin{array}{ll}V(x,t)={\displaystyle \sum _{i=1}^N}{\lambda }_i(t){\delta }^{\prime }(x-x_i).& (27)\end{array}$

The time dependent Schrödinger equation for this potential is

$\begin{array}{ll}i\frac{\partial }{\partial t}\psi (x,t)=\left[-\frac{d^2}{d{x}^2}+{\displaystyle \sum _{j=1}^N}{\lambda }_j(t){\delta }^{\prime }(x-x_j)\right]\psi (x,t).& (28)\end{array}$

As we mentioned in the introduction we adopt the distributional approach given by [12] for the definition of δ′(x), for functions having discontinuity at the point of δ′

$\begin{array}{ll}{\delta }^{\prime }(x)f(x)=\langle f(0)\rangle {\delta }^{\prime }(x)-\langle f_x(0)\rangle \delta (x),& (29)\end{array}$

where we define $\langle f(y)\rangle =\frac{f(y^{+})+f(y^{-})}{2}$ and $\langle f_x(y)\rangle =\frac{f_x(y^{+})+f_x(y^{-})}{2}$ and f(y^±) denote the limits $\underset{x\to y^{\pm }}{lim}f(x)$. Note that this definition reduces to the well-known property of δ′(x) [50]

$\begin{array}{ll}{\delta }^{\prime }(x)f(x)=f(0){\delta }^{\prime }(x)-f^{\prime }(0)\delta (x).& (30)\end{array}$

for continuous functions. As in the previous section we proceed by taking the Laplace transform of all the terms in the Equation (28) for time variable t and find

$\begin{array}{l}{\bar{\psi }}_{xx}(x,s)-i\psi (x,0)+is\bar{\psi }(x,s)\\ ={\displaystyle \sum _{j=1}^N}[-{L}\left\{{\lambda }_j(t)\langle \psi (x_j)\rangle \right\}{\delta }^{\prime }(x-x_j)\\ +{L}\left\{{\lambda }_j(t)\langle {\psi }_x(x_j)\rangle \right\}\delta (x-x_j)].& (31)\end{array}$

Now, we take the Fourier transform with respect to the variable x and solve $\widehat{\bar{\psi }}(k,s)$ to get:

$\begin{array}{l}\widehat{\bar{\psi }}(k,s)=-\frac{1}{k^2-is}(i\widehat{\psi }(k,0)+{\displaystyle {\sum }_{j=1}^N}-ik{e}^{-ik{x}_j}{L}\left\{{\lambda }_j(t)\langle \psi (x_j)\rangle \right\}\\ +e^{-ik{x}_j}{L}\left\{{\lambda }_j(t)\langle \psi (x_j)\rangle \right\}).& (32)\end{array}$

Before taking the inverse Fourier transform of this equation we write $\widehat{\psi }(k,0)={\int }_{-\infty }^{\infty }d{x}^{\prime }{e}^{-ik{x}^{\prime }}\psi (x^{\prime },0)$ then take the inverse Fourier transform and get

$\begin{array}{l}\bar{\psi }(x,s)=\frac{1}{2\sqrt{is}}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }{e}^{i\sqrt{is}|x-x^{\prime }|}\psi (x^{\prime },0)\\ -\frac{1}{2}{\displaystyle \sum _{j=1}^N}\text{sgn}(x-x_j)e^{i\sqrt{is}|x-x_j|}{L}\left\{{\lambda }_j(t)\langle \psi (x_j,t)\rangle \right\}\\ -i\frac{1}{2\sqrt{is}}{\displaystyle \sum _{j=1}^N}{e}^{i\sqrt{is}|x-x_j|}{L}\left\{{\lambda }_j(t)\langle {\psi }_x(x_j,t)\rangle \right\}.& (33)\end{array}$

where sgn(x) denotes sign function $\text{sgn}(x)=\{\begin{array}{cc}1& x>0\\ -1& x<0\end{array}$. We have also used ${\int }_{-\infty }^{\infty }\frac{dk}{2\pi }\frac{e^{ikx}}{k^2-is}=-\frac{i}{2\sqrt{is}}{e}^{i\sqrt{is}\left|x\right|}$ and ${\int }_{-\infty }^{\infty }\frac{dk}{2\pi }\frac{ik{e}^{ikx}}{k^2-is}=-\frac{1}{2\sqrt{is}}{e}^{i\sqrt{is}\left|x\right|}\mathrm{\text{ sgn}}(x)$. Now using ${L}\left\{\frac{exp\left[i\frac{x^2}{4t}\right]}{2\sqrt{i\pi t}}\right\}=\frac{e^{i\sqrt{is}\left|x\right|}}{2\sqrt{is}}$ and ${L}\left\{{(-1)}^{3/4}\frac{exp\left[i\frac{x^2}{4t}\right]x}{2\sqrt{\pi }{t}^{3/2}}\right\}=e^{i\sqrt{is}\left|x\right|}\mathrm{\text{ sgn}}(x)$ and convolution theorem for Laplace transform we get the formal expression of the wave function

$\begin{array}{l}\psi (x,t)={\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }\frac{1}{\sqrt{4\pi it}}{e}^{i\frac{{(x-x^{\prime })}^2}{4t}}\psi (x^{\prime },0)\\ -{\displaystyle {\int }_0^t}d{t}^{\prime }\{{\displaystyle \sum _{j=1}^N}\frac{{(-1)}^{3/4}}{4\sqrt{\pi }}{\lambda }_j(t^{\prime })\langle \psi (x_j)\rangle \frac{exp\left[i\frac{{(x-x_j)}^2}{4(t-t^{\prime })}\right]}{{(t-t^{\prime })}^{3/2}}(x-x_j)\\ +\frac{i}{2\sqrt{i\pi }}{\displaystyle \sum _{j=1}^N}{\lambda }_j(t^{\prime })\langle {\psi }_x(x_j)\rangle \frac{exp\left[i\frac{{(x-x_j)}^2}{4(t-t^{\prime })}\right]}{\sqrt{(t-t^{\prime })}}\}.& (34)\end{array}$

Similar to the time dependent δ potential case, this formal equation contains unknown functions $\psi (x_j^{\pm })$'s and ${\psi }_x(x_j^{\pm })$'s. One can find equations for $\psi (x_j^{\pm })$'s by inserting $x=x_j^{\pm }$ to the Equation (34). In order to find equations for ${\psi }_x{(x_j)}^{\pm }$'s, the derivative of the Equation (34) with respect to the variable x has to be calculated. By taking this derivative we get

$\begin{array}{l}{\psi }_x(x,t)=\frac{1}{4\sqrt{i\pi }{t}^{3/2}}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }i(x-x^{\prime })e^{i\frac{{(x-x^{\prime })}^2}{4t}}\psi (x^{\prime },0)\\ -{\displaystyle {\int }_0^t}d{t}^{\prime }\{-\frac{1}{8}\sqrt{\frac{i}{\pi }}{\displaystyle \sum _{j=1}^N}{\lambda }_j(t^{\prime })\langle \psi (x_j)\rangle i{(x-x_j)}^2\frac{exp\left[i\frac{{(x-x_j)}^2}{4(t-t^{\prime })}\right]}{{(t-t^{\prime })}^{5/2}}\\ +\frac{{(-1)}^{3/4}}{4\sqrt{\pi }}{\displaystyle \sum _{j=1}^N}{\lambda }_j(t^{\prime })\langle \psi (x_j)\rangle \frac{exp\left[i\frac{{(x-x_j)}^2}{4(t-t^{\prime })}\right]}{{(t-t^{\prime })}^{3/2}}\\ -\frac{1}{4\sqrt{i\pi }}{\displaystyle \sum _{j=1}^N}{\lambda }_j(t^{\prime })\langle {\psi }_x(x_j)\rangle (x-x_j)\frac{exp\left[i\frac{{(x-x_j)}^2}{4(t-t^{\prime })}\right]}{{(t-t^{\prime })}^{3/2}}\}.& (35)\end{array}$

Now putting $x=x_j^{\pm }$ in the Equation (35) one can get also integral equations for ${\psi }_x(x_j^{\pm })$'s. Finally one has to solve the system of integral equations for $\psi (x_j^{\pm })$'s and ${\psi }_x(x_j^{\pm })$'s to get the complete solution.

3.1. Single δ′ Potential With a Time Dependent Strength

Now we will elaborate more on the single δ′ interaction which is described by the potential

$\begin{array}{ll}V(x,t)=\lambda (t){\delta }^{\prime }(x)& (36)\end{array}$

The formal solution of the time dependent Schrödinger equation for single δ′ can be obtained from the above section for N = 1, x₁ = 0 and λ₁ = λ

$\begin{array}{l}\psi (x,t)={\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }\frac{1}{\sqrt{4\pi it}}{e}^{i\frac{{(x-x^{\prime })}^2}{4t}}\psi (x^{\prime },0)\\ -{\displaystyle {\int }_0^t}d{t}^{\prime }\{\frac{{(-1)}^{3/4}}{4\sqrt{\pi }}\lambda (t^{\prime })\langle \psi (0)\rangle x\frac{exp\left[i\frac{x^2}{4(t-t^{\prime })}\right]}{{(t-t^{\prime })}^{3/2}}\\ +\frac{i}{2\sqrt{i\pi }}\lambda (t^{\prime })\langle {\psi }_x(0)\rangle \frac{exp\left[i\frac{x^2}{4(t-t^{\prime })}\right]}{\sqrt{(t-t^{\prime })}}\}.& (37)\end{array}$

Similarly the expression for the ψ_x(x, t) is obtained from the Equation (35)

$\begin{array}{l}{\psi }_x(x,t)=\frac{1}{4\sqrt{i\pi }{t}^{3/2}}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }i(x-x^{\prime })e^{i\frac{{(x-x^{\prime })}^2}{4t}}\psi (x^{\prime },0)\\ -{\displaystyle {\int }_0^t}d{t}^{\prime }\{-\frac{1}{8}\sqrt{\frac{i}{\pi }}\lambda (t^{\prime })\langle \psi (0)\rangle x^2\frac{exp\left[i\frac{x^2}{4(t-t^{\prime })}\right]}{{(t-t^{\prime })}^{5/2}}\\ +\frac{{(-1)}^{3/4}}{4\sqrt{\pi }}\lambda (t^{\prime })\langle \psi (0)\rangle \frac{exp\left[i\frac{x^2}{4(t-t^{\prime })}\right]}{{(t-t^{\prime })}^{3/2}}\\ -\frac{1}{4\sqrt{i\pi }}\lambda (t^{\prime })\langle {\psi }_x(0)\rangle x\frac{exp\left[i\frac{x^2}{4(t-t^{\prime })}\right]}{{(t-t^{\prime })}^{3/2}}\}.& (38)\end{array}$

For a given λ(t) one gets integral equations for ψ(x, 0) and ψ_x(x, 0) by inserting x = 0 in Equations (37) and (38), respectively:

$\begin{array}{l}\psi (0,t)={\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }\frac{1}{\sqrt{4\pi it}}{e}^{i\frac{{x^{\prime }}^2}{4t}}\psi (x^{\prime },0)\\ -{\displaystyle {\int }_0^t}d{t}^{\prime }\{\frac{{(-1)}^{3/4}}{4\sqrt{\pi }}\lambda (t^{\prime })\langle \psi (0)\rangle \frac{1}{{(t-t^{\prime })}^{3/2}}\\ +\frac{i}{2\sqrt{i\pi }}\lambda (t^{\prime })\langle {\psi }_x(0)\rangle \frac{1}{\sqrt{(t-t^{\prime })}}\}.& (39)\end{array}$

$\begin{array}{l}{\psi }_x(0,t)=\frac{1}{4\sqrt{i\pi }{t}^{3/2}}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }(-x^{\prime })e^{i\frac{{x^{\prime }}^2}{4t}}\psi (x^{\prime },0)\\ -\frac{{(-1)}^{3/4}}{4\sqrt{\pi }}{\displaystyle {\int }_0^t}d{t}^{\prime }\lambda (t^{\prime })\langle \psi (0)\rangle \frac{1}{{(t-t^{\prime })}^{3/2}}.& (40)\end{array}$

The Equations (39) and (40) constitute an equation system for ψ(0, t) and ψ_x(0, t). Solving this system one can determine ψ(0, t) and ψ_x(0, t) and insert them to Equation (37) to get the wave function for all times. When studying special cases the expression of the Laplace transform of the wave function is necessary. Therefore, utilizing Equation (33) we write the general formula of the Laplace transform of the wave function for a delta prime at x₁ = 0 and λ₁(t) = λ(t):

$\begin{array}{l}\bar{\psi }(x,s)=\frac{1}{2\sqrt{is}}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }{e}^{i\sqrt{is}|x-x^{\prime }|}\psi (x^{\prime },0)\\ -\frac{1}{2}\text{sgn}(x)e^{i\sqrt{is}|x|}{L}\left\{\lambda (t)\langle \psi (0,t)\rangle \right\}\\ -i\frac{1}{2\sqrt{is}}{e}^{i\sqrt{is}|x|}{L}\left\{\lambda (t)\langle {\psi }_x(0,t)\rangle \right\}.& (41)\end{array}$

Now will investigate some special cases.

3.1.1. Case 1: λ Is Constant

When the strength of the single δ′ interaction is constant, we obtain from Equation (33)

$\begin{array}{l}\bar{\psi }(x,s)=\frac{1}{2\sqrt{is}}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }{e}^{i\sqrt{is}|x-x^{\prime }|}\psi (x^{\prime },0)\\ -\frac{\lambda }{2}\text{sgn}(x)e^{i\sqrt{is}|x|}\langle \psi (\bar{0},s)\rangle \\ -i\frac{\lambda }{2\sqrt{is}}{e}^{i\sqrt{is}|x|}\langle {\psi }_x(\bar{0},s)\rangle ,& (42)\end{array}$

where $\langle \bar{\psi }(0,s)\rangle ={L}\left\{\langle \psi (x,t)\rangle \right\}$ and $\langle {\bar{\psi }}_x(0,s)\rangle ={L}\left\{\langle {\psi }_x(x,t)\rangle \right\}$. We need also derivative of the Laplace transformed wave function $\bar{\psi }(x,s)$ with respect to x.

$\begin{array}{l}{\bar{\psi }}_x(x,s)=\frac{i}{2}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }\text{ sgn}(x-x^{\prime })e^{i\sqrt{is}|x-x^{\prime }|}\psi (x^{\prime },0)\\ -\frac{i\lambda \sqrt{is}}{2}{\text{sgn}}^2(x)e^{i\sqrt{is}|x|}\langle \psi (\bar{0},s)\rangle \\ -\frac{\lambda }{2}\frac{d\text{ sgn}(x)}{dx}{e}^{i\sqrt{is}|x|}\langle \psi (\bar{0},s)\rangle +\frac{\lambda \text{ sgn}(x)e^{i\sqrt{is}|x|}}{2}\langle {\psi }_x(\bar{0},s)\rangle ,& (43)\end{array}$

where we have used $\frac{d\left|x\right|}{dx}=\mathrm{\text{sgn}}(x)$. Now we find wave function and its derivative at 0^± by choosing x = 0^± in Equations (42) and (43):

$\begin{array}{l}\bar{\psi }(0^{\pm },s)=\frac{1}{2\sqrt{is}}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }{e}^{i\sqrt{is}|x^{\prime }|}\psi (x^{\prime },0)\mp \frac{\lambda }{2}\langle \psi (\bar{0},s)\rangle \\ -\frac{i\lambda }{2\sqrt{is}}\langle {\psi }_x(\bar{0},s)\rangle ,& (44)\end{array}$

where we take sgn(0^±) = ±1. In order to find $\bar{\psi }(x,s)$ we need to calculate $\langle \bar{\psi }(0,s)\rangle$ and $\langle {\bar{\psi }}_x(0,s)\rangle$. From Equation (44) we get

$\begin{array}{ll}\langle \bar{\psi }(0,s)\rangle +\frac{i\lambda }{2\sqrt{is}}\langle {\bar{\psi }}_x(0,s)\rangle =\frac{1}{2\sqrt{is}}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }{e}^{i\sqrt{is}|x^{\prime }|}\psi (x^{\prime },0).& (45)\end{array}$

In order to find another equation for $\langle \bar{\psi }(0,s)\rangle$ and $\langle {\bar{\psi }}_x(0,s)\rangle$ we calculate ${\bar{\psi }}_x(0^{\pm },s)$

$\begin{array}{l}{\bar{\psi }}_x(0^{\pm },s)=\frac{i}{2}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }\text{ sgn}(-x^{\prime })e^{i\sqrt{is}|x^{\prime }|}\psi (x^{\prime },0)\\ -\frac{i\lambda \sqrt{is}}{2}\langle \bar{\psi }(0,s)\rangle \mp \frac{\lambda }{2}\langle {\bar{\psi }}_x(0,s)\rangle .& (46)\end{array}$

where we take $\frac{d\mathrm{\text{ sgn}}(x)}{dx}{|}_{x=0^{\pm }}=0$. Using this equation we get

$\begin{array}{ll}\frac{i\lambda \sqrt{is}}{2}\langle \bar{\psi }(0,s)\rangle +\langle {\bar{\psi }}_x(0,s)\rangle =\frac{i}{2}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }\text{ sgn}(-x^{\prime })e^{i\sqrt{is}|x^{\prime }|}\psi (x^{\prime },0).& (47)\end{array}$

Solving Equations (45) and (47) we get

$\begin{array}{l}\langle \bar{\psi }(0,s)\rangle =\frac{1}{2\sqrt{is}}{\left(1+\frac{{\lambda }^2}{4}\right)}^{-1}\\ {\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }\left(1-\frac{\lambda }{2}\text{sgn}(x^{\prime })\right)e^{i\sqrt{is}|x^{\prime }|}\psi (x^{\prime },0),& (48)\end{array}$

and

$\begin{array}{l}\langle {\bar{\psi }}_x(0,s)\rangle ={\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }[-\frac{i}{2}\text{sgn}(x^{\prime })-\frac{i\lambda }{4}{\left(1+\frac{{\lambda }^2}{4}\right)}^{-1}\\ (1-\frac{\lambda }{2}\text{sgn}(x^{\prime }))]e^{i\sqrt{is}|x^{\prime }|}\psi (x^{\prime },0),& (49)\end{array}$

respectively. Substituting the expressions in Equations (48) and (49) into Equation (42) we get

$\begin{array}{l}\bar{\psi }(x,s)={\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }\{\frac{e^{i\sqrt{is}|x-x^{\prime }|}}{2\sqrt{is}}+{\left(1+\frac{{\lambda }^2}{4}\right)}^{-1}[-\frac{\lambda }{4}\text{sgn}(x^{\prime })\\ -\frac{{\lambda }^2}{8}-\frac{\lambda }{4}\text{sgn}(x)+\frac{{\lambda }^2}{8}\text{sgn}(x)\text{ sgn}(x^{\prime })]\frac{e^{i\sqrt{is}(|x|+|x^{\prime }|)}}{\sqrt{is}}\}\psi (x^{\prime },0).& (50)\end{array}$

The factor that multiplies ψ(x′, 0) in the integral of the equation above is the Green's function:

$\begin{array}{l}Ḡ(x,x^{\prime },s)=\{\frac{e^{i\sqrt{is}|x-x^{\prime }|}}{2\sqrt{is}}+{\left(1+\frac{{\lambda }^2}{4}\right)}^{-1}[-\frac{\lambda }{4}\text{sgn}(x^{\prime })-\frac{{\lambda }^2}{8}\\ -\frac{\lambda }{4}\text{sgn}(x)+\frac{{\lambda }^2}{8}\text{sgn}(x)\text{ sgn}(x^{\prime })]\frac{e^{i\sqrt{is}(|x|+|x^{\prime }|)}}{\sqrt{is}}\}.& (51)\end{array}$

The Green's function in terms of s variable can be converted using Equation (10) to get

$\begin{array}{l}G(x,x^{\prime },t)=\frac{1}{\sqrt{4\pi it}}\{e^{i\frac{{(x-x^{\prime })}^2}{4t}}+{\left(1+\frac{{\lambda }^2}{4}\right)}^{-1}[-\frac{\lambda }{2}\text{sgn}(x^{\prime })\\ -\frac{{\lambda }^2}{4}-\frac{\lambda }{2}\text{sgn}(x)+\frac{{\lambda }^2}{4}\text{sgn}(x)\text{ sgn}(x^{\prime })]e^{i\frac{{(|x|+|x^{\prime }|)}^2}{4t}}\}.& (52)\end{array}$

So, we can write the wave function for all times as:

$\begin{array}{l}\psi (x,t)=\frac{1}{\sqrt{4\pi it}}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }\{e^{i\frac{{(x-x^{\prime })}^2}{4t}}+{\left(1+\frac{{\lambda }^2}{4}\right)}^{-1}\\ [-\frac{\lambda }{2}\text{sgn}(x^{\prime })-\frac{{\lambda }^2}{4}-\frac{\lambda }{2}\text{sgn}(x)+\frac{{\lambda }^2}{4}\text{sgn}(x)\text{ sgn}(x^{\prime })]\\ e^{i\frac{{(|x|+|x^{\prime }|)}^2}{4t}}\}\psi (x^{\prime },0).& (53)\end{array}$

This result is completely consistent with the one given in Lange [11] except the convention λ = −c.

3.1.2. Case 2: λ(t) = α/t

The general expression (41) for the Laplace transform of the wave function in this particular case $\lambda (t)=\frac{\alpha }{t}$, where α is a constant, becomes

$\begin{array}{l}\bar{\psi }(x,s)=\frac{1}{2\sqrt{is}}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }{e}^{i\sqrt{is}|x-x^{\prime }|}\psi (x^{\prime },0)-\frac{1}{2}\text{sgn}(x)e^{i\sqrt{is}|x|}\\ {L}\left\{\frac{\alpha }{t}\langle \psi (0,t)\rangle \right\}-i\frac{1}{2\sqrt{is}}{e}^{i\sqrt{is}|x|}{L}\left\{\frac{\alpha }{t}\langle {\psi }_x(0,t)\rangle \right\}.& (54)\end{array}$

Using the identity given in Equation (B.2) in Appendix B this equation becomes

$\begin{array}{l}\bar{\psi }(x,s)=\frac{1}{2\sqrt{is}}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }{e}^{i\sqrt{is}|x-x^{\prime }|}\psi (x^{\prime },0)-\frac{1}{2}\text{sgn}(x)e^{i\sqrt{is}|x|}\\ \alpha {\displaystyle {\int }_s^{\infty }}d{s}^{\prime }\langle \bar{\psi }(0,s^{\prime })\rangle -i\frac{1}{2\sqrt{is}}{e}^{i\sqrt{is}|x|}\alpha {\displaystyle {\int }_s^{\infty }}d{s}^{\prime }\langle {\bar{\psi }}_x(0,s^{\prime })\rangle .& (55)\end{array}$

In order to calculate this expression we need ψ(0^±, s) and ${\psi }_x(0^{\pm },s)$. The first of this is easily calculated by choosing x = 0^± in Equation (55)

$\begin{array}{l}\bar{\psi }(0^{\pm },s)=\frac{1}{2\sqrt{is}}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }{e}^{i\sqrt{is}|x^{\prime }|}\psi (x^{\prime },0)\\ \mp \frac{\alpha }{2}{\displaystyle {\int }_s^{\infty }}d{s}^{\prime }\langle \bar{\psi }(0,s^{\prime })\rangle -\frac{i\alpha }{2\sqrt{is}}{\displaystyle {\int }_s^{\infty }}d{s}^{\prime }\langle {\bar{\psi }}_x(0,s^{\prime })\rangle .& (56)\end{array}$

In order to find ${\psi }_x(0^{\pm },s)$ we take the derivative of the Equation (54) with respect to x variable and get

$\begin{array}{l}{\bar{\psi }}_x(x,s)=\frac{i}{2}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }sgn(x-x^{\prime })e^{i\sqrt{is}|x-x^{\prime }|}\psi (x^{\prime },0)\\ -\frac{i\sqrt{is}\alpha }{2}{\text{sgn}}^2(x)e^{i\sqrt{is}|x|}{\displaystyle {\int }_s^{\infty }}d{s}^{\prime }\langle \bar{\psi }(0,s^{\prime })\rangle \\ -2e^{i\sqrt{is}|x|}\alpha \frac{d\text{ sgn}(x)}{dx}{\displaystyle {\int }_s^{\infty }}d{s}^{\prime }\langle \bar{\psi }(0,s^{\prime })\rangle \\ +\frac{\alpha }{2}\text{sgn}(x)e^{i\sqrt{is}|x|}{\displaystyle {\int }_s^{\infty }}d{s}^{\prime }\langle {\bar{\psi }}_x(0,s^{\prime })\rangle .& (57)\end{array}$

From this equation we get

$\begin{array}{l}{\bar{\psi }}_x(0^{\pm },s)=-\frac{i}{2}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }\text{ sgn}(x^{\prime })e^{i\sqrt{is}|x^{\prime }|}\psi (x^{\prime },0)\\ -\frac{i\sqrt{is}\alpha }{2}{\displaystyle {\int }_s^{\infty }}d{s}^{\prime }\langle \bar{\psi }(0,s^{\prime })\rangle \pm \frac{\alpha }{2}{\displaystyle {\int }_s^{\infty }}d{s}^{\prime }\langle {\bar{\psi }}_x(0,s^{\prime })\rangle .& (58)\end{array}$

We denote $u_1(s)={\int }_s^{\infty }d{s}^{\prime }\langle \bar{\psi }(0,s^{\prime })\rangle$ and $u_2(s)={\int }_s^{\infty }d{s}^{\prime }\langle {\bar{\psi }}_x(0,s^{\prime })\rangle$. We obtain from Equations (56) and (58)

$\begin{array}{ll}\langle \bar{\psi }(0,s)\rangle =\frac{I_0(s)}{2\sqrt{is}}-\frac{i\alpha }{2\sqrt{is}}{u}_1(s)& (59)\end{array}$

and

$\begin{array}{ll}\langle {\bar{\psi }}_x(0,s)\rangle =-i\frac{I_1(s)}{2}-\frac{i\sqrt{is}\alpha }{2}{u}_2(s).& (60)\end{array}$

Here $I_0(s)={\int }_{-\infty }^{\infty }d{x}^{\prime }{e}^{i\sqrt{is}\left|x^{\prime }\right|}\psi (x^{\prime },0)$ and $I_1(s)={\int }_{-\infty }^{\infty }d{x}^{\prime }\mathrm{\text{ sgn }}{x}^{\prime }{e}^{i\sqrt{is}\left|x^{\prime }\right|}\psi (x^{\prime },0)$. Note that $\langle \bar{\psi }(0,s)\rangle =-\frac{d{u}_1(s)}{ds}$ and $\langle {\bar{\psi }}_x(0,s)\rangle =-\frac{d{u}_2(s)}{ds}$. Inserting these equalities into Equations (59) and (60) we obtain two coupled differential equations:

$\begin{array}{ll}\frac{d{u}_1(s)}{ds}-\frac{i\alpha }{2\sqrt{is}}{u}_2(s)=-\frac{I_0(s)}{2\sqrt{is}}& (61)\end{array}$

and

$\begin{array}{ll}\frac{d{u}_2(s)}{ds}-\frac{i\sqrt{is}\alpha }{2}{u}_1(s)=i\frac{I_1(s)}{2\sqrt{is}}.& (62)\end{array}$

Although the coefficients of the unknown functions are not constants in these coupled equations, by taking the derivative of the Equations (61) and (62) with respect to the variable s one can uncouple these equations and get second order differential equations for u₁(s) and u₂(s):

$\begin{array}{ll}\frac{d^2{u}_1(s)}{d{s}^2}+\frac{1}{2s}\frac{d{u}_1(s)}{ds}+\frac{{\alpha }^2}{4}{u}_1(s)=\frac{\alpha }{4\sqrt{is}}{I}_1(s)-\frac{1}{2\sqrt{is}}\frac{d{I}_0(s)}{ds}& (63)\end{array}$

and

$\begin{array}{ll}\frac{d^2{u}_2(s)}{d{s}^2}-\frac{1}{2s}\frac{d{u}_2(s)}{ds}+\frac{{\alpha }^2}{4}{u}_2(s)=\frac{1}{4is}{I}_1(s)-\frac{i\alpha }{4}{I}_0(s)+\frac{i}{2}\frac{d{I}_1(s)}{ds}.& (64)\end{array}$

The solutions of these equations are elementary and easily obtained after some algebra as

$\begin{array}{l}{u}_1(s)=-\frac{\pi }{4\sqrt{2}}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }\psi (x^{\prime },0)\{{\displaystyle {\int }_0^s}d{s}^{\prime }\frac{s^{1/4}{s^{\prime }}^{3/4}}{\sqrt{i{s}^{\prime }}}{g}_1(s,s^{\prime })\\ \left[\alpha \text{ sgn}(x^{\prime })+\frac{|x^{\prime }|}{\sqrt{i{s}^{\prime }}}\right]e^{i\sqrt{i{s}^{\prime }}|x^{\prime }|}\}& (65)\end{array}$

where

$\begin{array}{ll}{g}_1(s,s^{\prime })=J_{\frac{1}{4}}\left(\frac{\alpha s^{\prime }}{2}\right)J_{-\frac{1}{4}}\left(\frac{\alpha s}{2}\right)-J_{\frac{1}{4}}\left(\frac{\alpha s}{2}\right)J_{-\frac{1}{4}}\left(\frac{\alpha s^{\prime }}{2}\right).& (66)\end{array}$

Here and in the following expressions J_a(x), stands for the first kind of Bessel's function. The solution of the differential equation in Equation (64) is

$\begin{array}{l}{u}_2(s)=-\frac{\pi }{4\sqrt{2}}{\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }\psi (x^{\prime },0)\{{\displaystyle {\int }_0^s}d{s}^{\prime }{s}^{3/4}{s^{\prime }}^{1/4}{g}_2(s,s^{\prime })\\ \left[\frac{\text{sgn}(x^{\prime })}{i{s}^{\prime }}-i\alpha -\frac{i{x}^{\prime }}{\sqrt{i{s}^{\prime }}}\right]e^{i\sqrt{i{s}^{\prime }}|x^{\prime }|}\}& (67)\end{array}$

where

$\begin{array}{ll}{g}_2(s,s^{\prime })=J_{\frac{3}{4}}\left(\frac{\alpha s^{\prime }}{2}\right)J_{-\frac{3}{4}}\left(\frac{\alpha s}{2}\right)-J_{\frac{3}{4}}\left(\frac{\alpha s}{2}\right)J_{-\frac{3}{4}}\left(\frac{\alpha s^{\prime }}{2}\right).& (68)\end{array}$

Note that

$\begin{array}{ll}{L}\left\{\frac{1}{t}\langle \psi (0,t)\rangle \right\}=u_1(s)={\displaystyle {\int }_s^{\infty }}d{s}^{\prime }\langle \bar{\psi }(0,s^{\prime })\rangle & (69)\end{array}$

and

$\begin{array}{ll}{L}\left\{\frac{1}{t}\langle {\psi }_x(0,t)\rangle \right\}=u_2(s)={\displaystyle {\int }_s^{\infty }}d{s}^{\prime }\langle {\bar{\psi }}_x(0,s^{\prime })\rangle .& (70)\end{array}$

Thus using Equations (69) and (70) in (41) we get

$\begin{array}{l}\bar{\psi }(x,s)={\displaystyle {\int }_{-\infty }^{\infty }}d{x}^{\prime }\psi (x^{\prime },0)\{\frac{e^{i\sqrt{is}|x-x^{\prime }|}}{2\sqrt{is}}{e}^{i\sqrt{is}|x-x^{\prime }|}\\ +\frac{\text{sgn}(x)\pi \alpha s^{1/4}{e}^{i\sqrt{is}|x|}}{8\sqrt{2i}}{\displaystyle {\int }_s^{\infty }}d{s}^{\prime }{s^{\prime }}^{1/4}{g}_1(s,s^{\prime })\left[\alpha \text{ sgn}(x^{\prime })+\frac{|x^{\prime }|}{\sqrt{i{s}^{\prime }}}\right]\\ e^{i\sqrt{i{s}^{\prime }}|x^{\prime }|}+\frac{\sqrt{i}\pi \alpha s^{1/4}{e}^{i\sqrt{is}|x|}}{8\sqrt{2}}{\displaystyle {\int }_s^{\infty }}d{s}^{\prime }{s^{\prime }}^{1/4}{g}_2(s,s^{\prime })\\ \left[\frac{\text{sgn}(x^{\prime })}{i{s}^{\prime }}-i\alpha -\frac{i{x}^{\prime }}{\sqrt{i{s}^{\prime }}}\right]e^{i\sqrt{i{s}^{\prime }}|x^{\prime }|}\}.& (71)\end{array}$

Thus the Green's function in Laplace transformed space for the Schrödinger equation with a potential $V(x,t)=\frac{\alpha }{t}{\delta }^{\prime }(x)$ is

$\begin{array}{l}G(x,x^{\prime },s)=\{\frac{e^{i\sqrt{is}|x-x^{\prime }|}}{2\sqrt{is}}{e}^{i\sqrt{is}|x-x^{\prime }|}+\frac{\text{sgn}(x)\pi \alpha s^{1/4}{e}^{i\sqrt{is}|x|}}{8\sqrt{2i}}\\ {\displaystyle {\int }_s^{\infty }}d{s}^{\prime }{s^{\prime }}^{1/4}{g}_1(s,s^{\prime })\left[\alpha \text{ sgn}(x^{\prime })+\frac{|x^{\prime }|}{\sqrt{i{s}^{\prime }}}\right]e^{i\sqrt{i{s}^{\prime }}|x^{\prime }|}\\ +\frac{\sqrt{i}\pi \alpha s^{1/4}{e}^{i\sqrt{is}|x|}}{8\sqrt{2}}{\displaystyle {\int }_s^{\infty }}d{s}^{\prime }{s^{\prime }}^{1/4}{g}_2(s,s^{\prime })\\ \left[\frac{\text{sgn}(x^{\prime })}{i{s}^{\prime }}-i\alpha -\frac{i{x}^{\prime }}{\sqrt{i{s}^{\prime }}}\right]e^{i\sqrt{i{s}^{\prime }}|x^{\prime }|}\}.& (72)\end{array}$

This Green' s function cannot be converted in terms of elementary functions but it is possible to use numerical methods to obtain Green' s function in the position-time space.

4. Conclusion

In this work, we have studied some analytically solvable time-dependent point interactions. First, we have obtained a formal expression of the propagator for finite number of Dirac δ potentials with time dependent strengths and solved the time dependent Schrödinger equation for this system. Then we have investigated one δ potential with various time dependent strengths in more detail and found the propagator for N Dirac δ potential in the limit that centers are infinitely separated. Furthermore, we have found an expression of the propagator for finite number of δ′ potentials with time dependent strengths and solved the time dependent Schrödinger equation for this potential. We believe that these results obtained are useful in models of ionization problems, where the particle is initially bound to the time dependent δ or δ′ potentials. Such type of models have been studied (see e.g., [42, 51]) and the results obtained are compared with the experiment [52].

Data Availability Statement

All datasets analyzed for this study are included in the article/Supplementary Material.

Author Contributions

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

Funding

The authors thank TUBITAK for the 2221-Visiting Scientist Fellowship Programme (2016). MG acknowledged partial financial support from the Spanish MINECO (Grant no. MTM2014-57129-C2-1-P) and the Junta de Castilla y León (Grant nos. VA137G18 and BU229P18).

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2020.00065/full#supplementary-material

References

1. Kronig R de L, Penney WG. A quantum mechanics of electrons in crystal lattices. Proc R Soc. (1931) 130:499.

Google Scholar

2. Albeverio S, Gesztesy F, Hoegh-Krohn R, Holden H. Solvable Models in Quantum Mechanics. 2nd ed. Chelsea, RI: American Mathematical Society (2004).

Google Scholar

3. Belloni M, Robinett RW. The infinite well and Dirac delta function potentials as pedagogical, mathematical and physical models in quantum mechanics. Phys Rep. (2014) 540:25. doi: 10.1016/j.physrep.2014.02.005

CrossRef Full Text | Google Scholar

4. Demkov Yu N, Ostrovskii VN. Zero-range Potentials and Their Applications in Atomic Physics. New York, NY: Plenum Press (1988).

Google Scholar

5. Golovaty, YD, Hryniv, RO. Norm resolvent convergence of singularly scaled Schrödinger operators and δ′-potentials. Proc R Soc Edinb A Math. (2013) 143:791–816. doi: 10.1017/S0308210512000194

CrossRef Full Text | Google Scholar

6. Golovaty Y. Two-parametric δ′-interactions: approximation by Schrödinger operators with localized rank-two perturbations. J Phys A Math Theor. (2018) 51:255202. doi: 10.1088/1751-8121/aac110

CrossRef Full Text | Google Scholar

7. Golovaty Y. Schrödinger operators with singular rank-two perturbations and point interactions. Integr Equat Oper Theory. (2018) 90:57. doi: 10.1007/s0002

CrossRef Full Text | Google Scholar

8. Gadella M, Negro J, Nieto LM. Bound states and scattering coefficients of the −aδ(x) + bδ′(x) potential. Phys Lett A. (2009) 373:1310–3. doi: 10.1016/j.physleta.2009.02.025

CrossRef Full Text | Google Scholar

9. Gadella M, Glasser ML, Nieto LM. One dimensional models with a singular potential of the type −αδ(x) + βδ′(x). Int J Theor Phys. (2010) 50:2144–52. doi: 10.1007/s10773-010-0641-6

CrossRef Full Text | Google Scholar

10. Zolataryuk AV. Boundary conditions for the states with resonant tunnelling across the δ′ potential. Phys Lett A. (2010) 374:1636–41. doi: 10.1016/j.physleta.2010.02.005

CrossRef Full Text | Google Scholar

11. Lange RJ. Distribution theory for Schrödinger's integral equation. J Math Phys. (2015) 56:122105. doi: 10.1063/1.4936302

CrossRef Full Text | Google Scholar

12. Kurasov P. Distribution theory for discontinuous test functions and differential operators with generalized coefficients. J Math Anal Appl. (1996) 201:297–323. doi: 10.1006/jmaa.1996.0256

CrossRef Full Text | Google Scholar

13. Fassari S, Gadella M, Nieto LM, Rinaldi F. Spectroscopy of a one-dimensional V-shaped quantum well with a point impurity. Ann Phys. (2018) 389:48–62. doi: 10.1016/j.aop.2017.12.006

CrossRef Full Text | Google Scholar

14. Seba P. Some remarks on the δ′-interaction in one dimension. Rep Math Phys. (1986) 24:111. doi: 10.1016/0034-4877(86)90045-5

CrossRef Full Text | Google Scholar

15. Patil SH. Schrödinger equation with δ′ and δ″ potentials. Phys Scr. (1994) 49:645. doi: 10.1088/0031-8949/49/6/002

CrossRef Full Text | Google Scholar

16. Coutinho FAB, Nogami Y, Perez JF. Generalized point interactions in one-dimensional quantum mechanics. J Phys A Math Gen. (1997) 30:3937. doi: 10.1088/0305-4470/30/11/021

CrossRef Full Text | Google Scholar

17. Christiansen PL, Arnbak NC, Zolotaryuk AV, Ermakov VN, Gaididei YB. On the existence of resonances in the transmission probability for interactions arising from derivatives of Dirac's delta function. J Phys A Math Gen. (2003) 36:7589. doi: 10.1088/0305-4470/36/27/311

CrossRef Full Text | Google Scholar

18. Zolotaryuk AV, Christiansen PL, Iermakova SV. Scattering properties of point dipole interactions. J Phys A Math Gen. (2006) 36:9329. doi: 10.1088/0305-4470/39/29/023

CrossRef Full Text | Google Scholar

19. Toyama FN, Nogami Y. Transmission-reflection problem with a potential of the form of the derivative of the delta function. J Phys A Math Theor. (2007) 40:F685. doi: 10.1088/1751-8113/40/29/F05

CrossRef Full Text | Google Scholar

20. Zolotaryuk AV. Two-parametric resonant tunneling across the δ′(x) potential. Adv Sci Lett. (2008) 1:187. doi: 10.1166/asl.2008.019

CrossRef Full Text | Google Scholar

21. Muñoz-Castañeda JM and Mateos-Guilarte J. Delta-delta' generalized Robin boundary conditions and quantum vacuum fluctuations. Phys Rev D. (2015) 91:025028. doi: 10.1103/PhysRevD.91.025028

CrossRef Full Text | Google Scholar

22. Silva, JDL, Braga, AN, Alves, DT. Dynamical Casimir effect with delta-delta' mirrors. Phys Rev D. (2016) 94:105009. doi: 10.1103/PhysRevD.94.105009

CrossRef Full Text | Google Scholar

23. Gaveau B, Schulman LS. Explicit time-dependent Schrödinger propagators. J Phys A Math Gen. (1986) 19:1833–46. doi: 10.1088/0305-4470/19/10/024

CrossRef Full Text | Google Scholar

24. Manoukian EB. Explicit derivation of the propagator for a Dirac delta potential. J Phys A Math Gen. (1989) 22:67. doi: 10.1088/0305-4470/22/1/013

CrossRef Full Text | Google Scholar

25. Blinder SM. Greens function and propagator for the one-dimensional delta function potential. Phys Rev A. (1988) 37:973. doi: 10.1103/PhysRevA.37.973

CrossRef Full Text | Google Scholar

26. Cacciari I, Moretti P. Propagator for the double delta potential. Phys Lett A. (2006) 359:396–401. doi: 10.1016/j.physleta.2006.06.061

CrossRef Full Text | Google Scholar

27. Albeverio S. Brzezniak Z, Dabrowski L. Time-dependent propagator with point interaction. J Phys A Math Gen. (1994) 27:4933–43.

Google Scholar

28. Elberfeld W, Kleber M. Time-dependent tunneling through thin barriers: a simple analytical solution. Am J Phys. (1988) 56:154. doi: 10.1119/1.15695

CrossRef Full Text | Google Scholar

29. Crandall RE. Combinatorial approach to Feynman path integration. J Phys A Math Gen. (1993) 26:3627. doi: 10.1088/0305-4470/26/14/024

CrossRef Full Text | Google Scholar

30. Cheng BK, da Luz MGE. Propagator for the δ-function potential moving with constant velocity. Phys Rev A. (1993) 47:4720–4. doi: 10.1103/PhysRevA.47.4720

PubMed Abstract | CrossRef Full Text | Google Scholar

31. Scheitler G, Kleber M. Propagator for two dispersing δ-function potentials. Phys Rev A. (1990) 42:55–60. doi: 10.1103/PhysRevA.42.55

PubMed Abstract | CrossRef Full Text | Google Scholar

32. Kuhn J, Zanetti FM, Azevedo AL, Schmidt AGM, Cheng BK, da Luz MGE. Time-dependent point interactions and infinite walls: some results for wavepacket scattering. J Opt B Quant Semiclass Opt. (2005) 7:S77–85. doi: 10.1088/1464-4266/7/3/011

CrossRef Full Text | Google Scholar

33. Laissaoui A, Chetouani L. Propagator for symmetric double delta potential using path decomposition method. J Math Phys. (2018) 59:022106. doi: 10.1063/1.5008516

CrossRef Full Text | Google Scholar

34. Refaei A, Kheirandish F. Quantum propagator and characteristic equation in the presence of a chain of δ potentials. Int J Mod Phys B. (2015) 29:1550099. doi: 10.1142/S021797921550099X

CrossRef Full Text | Google Scholar

35. Cacciapuoti C, Mantile A, Posilicano A. Time dependent delta-prime interactions in dimension one. Nanosyst Phys Chem Math. (2016) 2:303–14. doi: 10.17586/2220-8054-2016-7-2-303-314

CrossRef Full Text | Google Scholar

36. Campbell J. Some exact results for the Schrödinger wave equation with a time dependent potential. J Phys A Math Theor. (2009) 42:365212. doi: 10.1088/1751-8113/42/36/365212

CrossRef Full Text | Google Scholar

37. Hmidi T, Mantile A, Nier F. Time-dependent delta-interactions for 1D Schrödinger Hamiltonians. Math Phys Anal Geom. (2010) 13:83–103. doi: 10.1007/s11040-009-9070-2

CrossRef Full Text | Google Scholar

38. Costin O, Costin RD, Lebowitz JL, Rokhlenko A. Evolution of a model quantum system under time periodic forcing: conditions for complete ionization. Comm Math Phys. (2001) 221:1–26. doi: 10.1007/s002200100

CrossRef Full Text | Google Scholar

39. Costin O, Lebowitz JL, Rokhlenko A. Decay versus survival of a localized state subjected to harmonic forcing: exact results. J Phys A Math Gen. (2002) 35:8943–51. doi: 10.1088/0305-4470/35/42/305

CrossRef Full Text | Google Scholar

40. Costin O, Lebowitz JL, Rokhlenko A. Exact results for the ionization of a model quantum system. J Phys A Math Gen. (2000) 33:6311–9. doi: 10.1088/0305-4470/33/36/303

CrossRef Full Text | Google Scholar

41. Costin O, Costin RD, Lebowitz JL. Transition to the continuum of a particle in time-periodic potentials. In: Karpeshina Y, Stolz G, Weikard R, Zeng Y, editors. Advances in Differential Equations and Mathematical Physics, AMS Contemporary Mathematics Series. Providence, RI: AMS. (2003). p. 75–86.

Google Scholar

42. Correggi M. Time-dependent singular interactions (Ph.D. thesis), Mathematical Physics Sector, International School for Advanced Studies (2004).

Google Scholar

43. Martinez DF, Reichl LE. Transmission properties of the oscillating delta-function potential. Phys Rev B. (2001) 64:245315. doi: 10.1103/PhysRevB.64.245315

CrossRef Full Text | Google Scholar

44. Baek SK, Yi SD, Kim M. Particle in a box with a time dependent delta-function potential. Phys Rev A. Upper Saddle River: Prentice Hall (2016) 96:052124. doi: 10.1103/PhysRevA.94.052124

CrossRef Full Text | Google Scholar

45. Griffiths DJ. Introduction to Quantum Mechanics. 2nd ed. Printice Hall (2005).

46. Granot E, Marchewka A. Quantum particle displacement by a moving localized potential trap. Europhys Lett. (2009) textbf47:20007. doi: 10.1209/0295-5075/86/20007

CrossRef Full Text | Google Scholar

47. Appel W. Mathematics for Physics and Physicists. Princeton, NJ: Princeton University Press (2007).

Google Scholar

48. Reed M, Simon B. Methods of Modern Mathematical Physics. II. New York, NY: Academic Press (1975).

Google Scholar

49. Akin JE, Counts J. On rational approximation to the inverse laplace transform. SIAM J Appl Math. (1969) 17:1035–40.

Google Scholar

50. Hoskins RF. Delta Functions, Introduction to Generalized Functions. 2nd ed. Cambridge: Woodhead Publishing (2011).

Google Scholar

51. Qiu Z. Study of ionization of quantum systems with delta potentials in damped and undamped time periodic fields (Ph.D. thesis), The Ohio State University, Columbus, OH, United States (2009).

Google Scholar

52. Koch PM, Van Leeuwen KAH. The importance of resonances in microwave “ionization” of excited hydrogen atoms. Phys Rep. (1995) 255:289–403.

Google Scholar