Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials

TL;DR

This paper investigates how two- and three-delta potentials, regularized by piecewise constants, converge to one-point interactions under a two-scale squeezing limit, revealing resonance conditions and opaque interactions.

Contribution

It introduces a detailed analysis of the squeezing limit of multi-delta potentials, identifying resonance sets and the conditions for non-zero transmission in the limit.

Findings

Resonance sets form curves and surfaces in parameter space.

Transmission is non-zero at specific resonance parameter values.

Opaque interactions split the system into independent subsystems.

Abstract

Several families of one-point interactions are derived from the system consisting of two and three $\delta$-potentials which are regularized by piecewise constant functions. In physical terms such an approximating system represents two or three extremely thin layers separated by some distance. The two-scale squeezing of this heterostructure to one point as both the width of $\delta$-approximating functions and the distance between these functions simultaneously tend to zero is studied using the power parameterization through a squeezing parameter $\varepsilon \to 0$, so that the intensity of each $\delta$-potential is $c_j =a_j \varepsilon^{1-\mu}$, $a_j \in {\mathbb{R}}$, $j=1,2,3$, the width of each layer $l =\varepsilon$ and the distance between the layers $r = c\varepsilon^\tau$, $c >0$. It is shown that at some values of intensities $a_1$, $a_2$ and $a_3$, the transmission across the limit point interactions is non-zero, whereas outside these (resonance) values the one-point interactions are opaque splitting the system at the point of singularity into two independent subsystems. Within the interval $1 < \mu < 2$, the resonance sets consist of two curves on the $(a_1,a_2)$-plane and three disconnected surfaces in the $(a_1,a_2,a_3)$-space. While approaching the parameter $\mu $ to the critical value $\mu =2$, three types of splitting these sets into countable families of resonance curves and surfaces are observed.

Equations

$$-d^2\psi (x)/d{x}^2+V_{\epsilon }(x)\psi (x)=E\psi (x)$$

$$V_{\epsilon }(x)=\sum _{j=1}^N{c}_j(\epsilon )\delta (x-r_j(\epsilon )),$$

$$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\Lambda_{0}=\left(\begin{array}[]{cc}~{}\cos(kr)~{}~{}~{}~{}~{}k^{-1}\sin(kr)\\ -\,k\sin(kr)~{}~{}~{}~{}~{}\cos(kr)\end{array}\right),~{}~{}~{}~{}\Lambda_{j}=\left(\begin{array}[]{cc}1~{}~{}0\\ c_{j}~{}~{}1\end{array}\right),~{}~{}k:=\sqrt{E}\,.$$

$$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\left.\begin{array}[]{ll}\psi_{s}(0):=\psi(+0)-\psi(-0),{}{}&\psi^{\prime}_{s}(0):=\psi^{\prime}(+0)-\psi^{\prime}(-0),\\ \psi_{r}(0):=\eta\psi(+0)+(1-\eta)\psi(-0),{}{}&\psi^{\prime}_{r}(0):=\eta\psi^{\prime}(-0)+(1-\eta)\psi^{\prime}(+0),\end{array}\right.$$

$$\displaystyle\Lambda=\left(\begin{array}[]{cc}1~{}~{}~{}~{}0\\ \alpha~{}~{}~{}~{}1\end{array}\right).$$

$$\displaystyle\Lambda=\left(\begin{array}[]{cc}1~{}~{}~{}~{}\beta\\ 0~{}~{}~{}~{}1\end{array}\right).$$

$${\delta }^{\prime }(x)\psi (x)=\left[(1-\eta )\psi (-0)+\eta \psi (+0)\right]{\delta }^{\prime }(x)+\left[\eta {\psi }^{\prime }(-0)+(1-\eta ){\psi }^{\prime }(+0)\right]\delta (x).$$

$$\displaystyle\Lambda=\left(\begin{array}[]{cc}\theta~{}~{}~{}~{}0{}{}\\ 0~{}~{}~{}\theta^{-1}\end{array}\right)$$

$$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!V_{lr}(x)=\left\{\begin{array}[]{ll}0&\mbox{for}~{}~{}-\infty<x<0,~{}l<x<l+r,\\ &~{}~{}~{}~{}~{}~{}2l+r<x<2(l+r),~{}3l+2r<x<\infty,\\ v_{j}&\mbox{for}~{}~{}(j-1)(l+r)<x<j(l+r)-r,~{}j=1,2,3,\end{array}\right.$$

$$\displaystyle\Lambda_{j,l}=\left(\begin{array}[]{cc}\cos(k_{j}l)~{}~{}~{}~{}~{}~{}k_{j}^{-1}\sin(k_{j}l)\\ -\,k_{j}\sin(k_{j}l)~{}~{}~{}~{}~{}~{}~{}~{}\cos(k_{j}l)\end{array}\right),$$

$$k_j:=\sqrt{k^2-v_j},j=1,2,3.$$

$$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\left(\begin{array}[]{cc}\psi(x_{2})\\ \psi^{\prime}(x_{2})\end{array}\right)=\Lambda_{lr}\left(\begin{array}[]{cc}\psi(x_{1})\\ \psi^{\prime}(x_{1})\end{array}\right),~{}~{}\Lambda_{lr}=\Lambda_{3,l}\Lambda_{r}\Lambda_{2,l}\Lambda_{0}\Lambda_{1,l}=:\left(\begin{array}[]{cc}\bar{\lambda}_{11}~{}~{}\bar{\lambda}_{12}\\ \bar{\lambda}_{21}~{}~{}\bar{\lambda}_{22}\end{array}\right),$$

$${\bar{\lambda }}_{11}$$

$${\bar{\lambda }}_{12}$$

$${\bar{\lambda }}_{21}$$

$${\bar{\lambda }}_{22}$$

$${\bar{\lambda }}_{11}\to \cos (k_{1}l)\cos (k_{2}l)\cos (k_{3}l)-(k_1/k_2)\sin (k_{1}l)\sin (k_{2}l)\cos (k_{3}l)$$

$$-(k_1/k_3)\sin (k_{1}l)\cos (k_{2}l)\sin (k_{3}l)-(k_2/k_3)\cos (k_{1}l)\sin (k_{2}l)\sin (k_{3}l)$$

$$-2k_{1}r\sin (k_{1}l)\cos (k_{2}l)\cos (k_{3}l)-k_{2}r\cos (k_{1}l)\sin (k_{2}l)\cos (k_{3}l)$$

$$+k_1{k}_2{r}^2\sin (k_{1}l)\sin (k_{2}l)\cos (k_{3}l)+(k_1{k}_{2}r/k_3)\sin (k_{1}l)\sin (k_{2}l)\sin (k_{3}l),$$

$${\bar{\lambda }}_{12}\to -k_2{r}^2\cos (k_{1}l)\sin (k_{2}l)\cos (k_{3}l),$$

$${\bar{\lambda }}_{21}\to -k_1\sin (k_{1}l)\cos (k_{2}l)\cos (k_{3}l)-k_2\cos (k_{1}l)\sin (k_{2}l)\cos (k_{3}l)$$

$$-k_3\cos (k_{1}l)\cos (k_{2}l)\sin (k_{3}l)+k_1{k}_{2}r\sin (k_{1}l)\sin (k_{2}l)\cos (k_{3}l)$$

$$+2k_1{k}_{3}r\sin (k_{1}l)\cos (k_{2}l)\sin (k_{3}l)+k_2{k}_{3}r\cos (k_{1}l)\sin (k_{2}l)\sin (k_{3}l)$$

$$+k_1{k}_3(k_2^{-1}-k_2{r}^2)\sin (k_{1}l)\sin (k_{2}l)\sin (k_{3}l)$$

$$+k^2{r}^2\cos (k_{2}l)[k_1\sin (k_{1}l)\cos (k_{3}l)+k_3\cos (k_{1}l)\sin (k_{3}l)],$$

$${\bar{\lambda }}_{22}\to \cos (k_{1}l)\cos (k_{2}l)\cos (k_{3}l)-(k_2/k_1)\sin (k_{1}l)\sin (k_{2}l)\cos (k_{3}l)$$

$$-(k_3/k_1)\sin (k_{1}l)\cos (k_{2}l)\sin (k_{3}l)-(k_3/k_2)\cos (k_{1}l)\sin (k_{2}l)\sin (k_{3}l)$$

$$-2k_{3}r\cos (k_{1}l)\cos (k_{2}l)\sin (k_{3}l)-k_{2}r\cos (k_{1}l)\sin (k_{2}l)\cos (k_{3}l)$$

$$+k_2{k}_3{r}^2\cos (k_{1}l)\sin (k_{2}l)\sin (k_{3}l)+(k_2{k}_{3}r/k_1)\sin (k_{1}l)\sin (k_{2}l)\sin (k_{3}l).$$

$$v_j=a_j{\epsilon }^{-\mu },l=\epsilon \text{mbox}andr=c{\epsilon }^{\tau },c>0.$$

$$k_j\to \sqrt{-a_j}{\epsilon }^{-\mu /2},k_{j}l\to \sqrt{-a_j}{\epsilon }^{1-\mu /2},k_j^{2}l\to -a_j{\epsilon }^{1-\mu }.$$

Full text

Keywords: one-point interactions, single- and multiple-resonant tunnelling, resonance curves and surfaces

1 Introduction

The models described by the Schrödinger operators with singular zero-range potentials have widely been discussed in both the physical and mathematical literature (see books [1, 2, 3, 4] for details and references). These models admit exact closed analytical solutions which describe realistic situations using different approximations via Hamiltonians describing point interactions [5, 6, 7, 8, 9, 10]. Currently, because of the rapid progress in fabricating nanoscale quantum devices, of particular importance is the point modelling of different structures like quantum waveguides [11, 12], spectral filters [13, 14] or infinitesimally thin sheets [15, 16, 17]. A whole body of literature (see, e.g., [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], a few to mention), including the very recent studies [31, 32, 33, 34, 35, 36, 37, 38] with references therein, has been published where the one-dimensional Schrödinger operators with potentials given in the form of distributions are shown to exhibit a number of peculiar features with possible applications to quantum physics. A detailed list of references on this subject can also be found in the recent review [39]. On the other hand, using some particular regular approximations of the potential expressed in the form of the derivative of Dirac’s delta function, a number of interesting resonance properties of quantum particles tunnelling through this point potential has been observed [8, 40, 41, 42]. Particularly, it was found that at some values of the potential strength of the $\delta^{\prime}$-potential the transmission across this barrier is non-zero, whereas outside these values the barrier is fully opaque. In general terms, the existence of such resonance sets in the space of potential intensities has rigorously been established for a whole class of approximations of the derivative delta potential by Golovaty with coworkers [43, 44, 45, 46, 47, 48, 49]. This type of point interactions may be referred to as ‘resonant-tunnelling $\delta^{\prime}$-potentials’. These results differ from those obtained within Kurasov’s theory [21] which was developed for the distributions defined on the space of functions discontinuous at the point of singularity. Here the limit point interaction is also called a $\delta^{\prime}$-potential. The common feature of Kurasov’s point potential and a resonant-tunnelling $\delta^{\prime}$-potential is that the transmission matrices of both these interactions are of the diagonal form, but the elements of these matrices are different. It is of interest therefore to find a way where it would be possible to describe both these types in a unique regularization scheme starting from the same initial regularized potential profile.

In the present work we address the problem on the relation between the point interactions realized within Kurasov’s theory and the resonant-tunnelling $\delta^{\prime}$-potentials studied in [8, 40, 41, 42, 43, 44, 46, 48, 49, 50]. We show that Kurasov’s $\delta^{\prime}$-potential emerges from the realistic heterostructure consisting of two or three extremely thin parallel plane layers separated by some distance. This system is studied in the limit as both the width of layers and the distance between them simultaneously tend to zero. In such a squeezing limit, the limit one-point interactions are proved to depend crucially on relative approaching zero the width and the distance. As a result, different types of one-point interactions appear in this limit depending on the way of convergence. Surprisingly, within the same regularization scheme, it is possible to realize both the Kurasov $\delta^{\prime}$-potential and the $\kappa\delta^{\prime}$-potentials with countable sets in the $\kappa$-space at which a non-zero resonant tunnelling occurs. Another surprising point is that the $\delta$-potential discovered by Šeba in [18] can also be realized together with its countable splitting at some critical point.

Consider the system consisting of $N$ parallel sheets arranged successively with their planes perpendicular to the $x$-axis. The sheets are assumed to be homogeneous, so that one can explore the one-dimensional stationary Schrödinger equation

[TABLE]

where $\psi(x)$ is the wavefunction and $E$ the energy of a particle. The potential $V_{\varepsilon}(x)$ with a squeezing parameter $\varepsilon>0$ shrinks to one point, say $x=0$, as $\varepsilon\to 0$. One of the ways to realize limit point interactions is to choose the potential $V_{\varepsilon}(x)$ in the form of a sum of several Dirac’s delta functions as [31, 51, 52]

[TABLE]

where the constants $c_{j}\in{\mathbb{R}}$ and the distances between the $\delta$-functions $r_{j}(\varepsilon)$ tend to zero as $\varepsilon\to 0$. The particular case of the three-delta ($N=3$) spatially symmetric potential (2), in the squeezing limit has been studied by Cheon and Shigehara [51], and Albeverio and Nizhnik [52]. In this limit a whole four-parameter family of point interactions has been constructed, independently on whether or not potential (2) has a distributional limit. Here we follow the approach developed by Exner, Neidhardt and Zagrebnov [7], who have approximated the $\delta$-potentials by regular functions and constructed a one-point $\delta^{\prime}$-interaction. In particular, they have proved that the limit takes place if the distances between the ‘centers’ of regularized potentials tend to zero sufficiently slow relatively to shrinking the $\delta$-approximating potentials. A similar research [6] concerns about the convergence of regularized $\delta$-approximating structures to point potentials in higher dimensions.

In this paper we focus on the two cases when potential (2) consists of two ($N=2$) and three ($N=3$) $\delta$-potentials separated equidistantly by a distance $r(\varepsilon)$ that tends to zero as $\varepsilon\to 0$. The transmission matrices for the two- and three-delta potentials are the products $\Lambda_{\varepsilon}=\Lambda_{2}\Lambda_{0}\Lambda_{1}$ and $\Lambda_{\varepsilon}=\Lambda_{3}\Lambda_{0}\Lambda_{2}\Lambda_{0}\Lambda_{1}$, respectively, where ($j=1,2,3$)

[TABLE]

We restrict ourselves to the most simple approximation of the $\delta$-potentials by piecewise constant functions resulting in a three- (for $N=2$) and a five- (for $N=3$) layered potential profile. In the limit as both the width of $\delta$-approximating functions and the distance between them tends to zero simultaneously we obtain different families of one-point interactions. We observe that, starting from the same profile of the three- and five-layered structure that approximates potential (2), the limit point interactions crucially depend on the relative rate of tending the width of layers and the distance between them to zero.

We follow the notations and the classification of one-point interactions given by Brasche and Nizhnik [31]. Thus, we denote

[TABLE]

where $\eta\in{\mathbb{R}}$ is an arbitrary parameter (this is a generalization of the generally accepted case with $\eta=1/2$, see, e.g., [20, 21, 31, 33]). Then the $\delta$-interaction, or $\delta$-potential, with intensity $\alpha$ is defined by the boundary conditions $\psi_{s}(0)=0$ and $\psi^{\prime}_{s}(0)=\alpha\psi_{r}(0)$, so that the $\Lambda$-matrix in this case has the form

[TABLE]

The dual interaction is called a $\delta^{\prime}$-interaction (the notation has been suggested in [3, 19] and adopted in the literature). This point interaction with intensity $\beta$ defined by the boundary conditions $\psi^{\prime}_{s}(0)=0$ and $\psi_{s}(0)=\beta\psi^{\prime}_{r}(0)$ has the $\Lambda$-matrix of the form

[TABLE]

As follows from formulae (11) and (14), the usage of the parameter $\eta$ for both the $\delta$- and $\delta^{\prime}$-interactions does not play any role. However, for the $\delta^{\prime}$-potential with intensity $\gamma$ the potential part in equation (1) is given by $\gamma\delta^{\prime}(x)\psi(x)$ where the wavefunction $\psi(x)$ must be discontinuous at $x=0$. Therefore, due to the ambiguity of the product $\delta^{\prime}(x)\psi(x)$, one can suppose the following generalized (asymmetric) averaging in the form

[TABLE]

This suggestion is also motivated by the studies [53, 54, 55] which demonstrate that the plausible averaging with $\eta=1/2$ at the point of singularity in general does not work. The $\delta^{\prime}$-potential with intensity $\gamma$ is defined by the boundary conditions $\psi_{s}(0)=\gamma\psi_{r}(0)$ and $\psi^{\prime}_{s}(0)=-\gamma\psi^{\prime}_{r}(0)$ [31]. An equivalent form of these conditions is given by the $\Lambda$-matrix in the diagonal form

[TABLE]

with $\theta=[1+(1-\eta)\gamma]/(1-\eta\gamma)$.

Finally, instead of the fourth type of point interactions defined in [31] as $\delta$-magnetic potentials, in this paper we shall be dealing with potentials which at some (resonant) values of intensities are fully transparent, whereas outside these values they are completely opaque satisfying the Dirichlet boundary conditions $\psi(\pm 0)=0$. At the resonance sets the boundary conditions are given by one of the unit matrices $\Lambda=\pm\,I$, I:=\left(\begin{array}[]{cc}1~{}~{}0\\ 0~{}~{}1\end{array}\right).

2 A piecewise constant approximation of the $\delta$-potentials

Let us approximate the $\delta$-potentials in (2) by piecewise constant functions. Then potential (2) is replaced by the rectangular function

[TABLE]

and, as a result, all the matrices $\Lambda_{j}$, $j=1,2,3$, in the product for $\Lambda_{\varepsilon}$ are replaced by

[TABLE]

where

[TABLE]

In other words, the regularized transmission matrix $\Lambda_{lr}$ defined by the relations

[TABLE]

connects the boundary conditions for the wavefunction $\psi(x)$ and its derivative $\psi^{\prime}(x)$ at $x=x_{1}=0$ and $x=x_{2}=3l+2r$ ($N=3$). For the case of the two-delta potential ($N=2$) we set $a_{3}\equiv 0$ in potential (19), so that the boundary conditions are $x_{1}=0$ and $x_{2}=2l+r$. The matrix elements in (30), denoted by overhead bars, depend on the shrinking parameters $l$ and $r$, whereas in the limit matrix elements, if they exist, the bars are omitted, i.e., we write \lim_{l,r\to 0}\Lambda_{lr}=:\Lambda=\left(\begin{array}[]{cc}{\lambda}_{11}~{}~{}{\lambda}_{12}\\ {\lambda}_{21}~{}~{}{\lambda}_{22}\end{array}\right). Having accomplished the limit procedure, we set $x_{1}=-0$ and $\lim_{l,r\to 0}x_{2}=+0$.

One can compute the matrix products $\Lambda_{lr}$ explicitly both for $N=2$ and $N=3$. Using that $k_{j}\to\infty$, $k_{j}l$ and $k_{i}/k_{j}$, $i,j=1,2,3$, are finite and $r\to 0$, it is sufficient to write their truncated expressions. As a result, we find the asymptotic behaviour of the elements of the matrix $\Lambda_{lr}=\Lambda_{2,l}\Lambda_{0}\Lambda_{1,l}$ ($N=2$):

[TABLE]

Similarly, for the three-delta potential the $\bar{\lambda}_{ij}$-asymptotes of the matrix product $\Lambda_{lr}=\Lambda_{3,l}\Lambda_{0}\Lambda_{2,l}\Lambda_{0}\Lambda_{1,l}$ are as follows

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The convergence of these elements in the limit as both the parameters $l$ and $r$ tend to zero will be analyzed below. Depending on the relative degree of coming up these parameters to zero, quite different limit transmission matrices will be obtained realizing various one-point interactions.

3 The power parameterization of the $\Lambda_{lr}$-matrix

The convergence of the transmission matrix $\Lambda_{lr}$ as $l,r\to 0$ both for $N=2$ and $N=3$ can be parameterized through the parameter $\varepsilon>0$ using the positive powers $\mu$ and $\tau$ from the $\{\mu>0,\,\tau>0\}$-quadrant as follows

[TABLE]

Here each coefficient $a_{j}\in{\mathbb{R}}$ may be called an ‘intensity’ or a ‘charge’ of the $j$th $\delta$-approximating layer. According to (23), we have the following asymptotic relations:

[TABLE]

Using these relations in which $\mu\in(0,2)$ and $\tau\in(0,\infty)$, we obtain that in the limit as $\varepsilon\to 0$ asymptotic relations (31), (33)-(35), (37) and (38) are reduced to

[TABLE]

[TABLE]

[TABLE]

Next, in the case when $\mu=2$ and $0<\tau<\infty$, asymptotes (31), (33) and (34) become

[TABLE]

[TABLE]

for $N=2$. Similarly, in the case with $N=3$ asymptotes (35), (37) and (38) are transformed to

[TABLE]

[TABLE]

[TABLE]

4 Admissible sets of the parameters $\mu$ and $\tau$ for realizing one-point

interactions

For the realization of (both connected and separated) interactions in the squeezing limit, the elements $\bar{\lambda}_{11}$ and $\bar{\lambda}_{22}$ given by asymptotes (41), (43), (45) and (47) must be finite as $\varepsilon\to 0$. It follows from these asymptotes that the region $Q:=\{0<\mu\leq 1,\,0<\tau<\infty\}\cup\{1<\mu\leq 2,\,\mu-1\leq\tau<\infty\}$ is admissible for the finiteness of the limit elements $\lambda_{11}$ and $\lambda_{22}$. Concerning the asymptote for $\bar{\lambda}_{12}$ in the case with $N=3$, we have in the region $Q$ the limits $\bar{\lambda}_{12}\to c^{2}a_{2}\varepsilon^{1-\mu+2\tau}\to 0$ for $0<\mu<2$ and $\bar{\lambda}_{12}\to-\,c^{2}\sqrt{-a_{2}}\cos\!\sqrt{-a_{1}}\sin\!\sqrt{-a_{2}}\cos\!\sqrt{-a_{3}}\,\varepsilon^{2\tau-1}\to 0$ for $\mu=2$. Therefore the convergence of the matrix $\Lambda_{lr}$ in the limit as $\varepsilon\to 0$ has to be analyzed in the region $Q$ shadowed in figure 1. While the matrix element $\bar{\lambda}_{12}$ for both $N=2$ and $N=3$ has the zero limit, the elements $\bar{\lambda}_{21}$ given by asymptotes (42), (44), (46) and (48) are in general divergent as $\varepsilon\to 0$. In the region $Q$, the highest divergence is of the order $\varepsilon^{1-\mu}$, $1<\mu\leq 2$. However, on some sets of $Q$ and at some conditions on the intensities $a_{j}$, the coefficients at the divergent term $\varepsilon^{1-\mu}$ can be zero. These conditions will be analyzed below in detail, but now we single out those sets of $Q$ where this analysis will be carried out. Thus, we split the region $Q$ into the sets as shown in figure 1

using the following definitions:

[TABLE]

In the region $Q_{0}$, the transmission is perfect because each layer is fully transparent in the squeezing limit, so that this point interaction is trivial. At the boundary of this region, i.e., on the line $L_{1}$, the squeezing limit of each layer leads to the $\delta$-potential with the total intensity equal the algebraic sum of the layer intensities, i.e., $\alpha=a_{1}+a_{2}+a_{3}$ in $\Lambda$-matrix (11).

There exists a possibility to consider the line $\tau=3(\mu-1)/2$, $1<\mu<2$, if both the equations $a_{1}+a_{2}+a_{3}=0$ and $a_{1}a_{2}+2a_{1}a_{3}+a_{2}a_{3}=0$ are fulfilled simultaneously. Excluding $a_{3}$ from these equations, we find the condition $a_{1}^{2}+(a_{1}+a_{2})^{2}=0$ which is valid only if $a_{1}=a_{2}=0$ and therefore $a_{3}=0$. Similarly, we have to analyze the case $\tau=3/2$ at $\mu=2$ in (48). Here the $\varepsilon\to 0$ limit of $\bar{\lambda}_{21}$ will be finite if both the coefficients at $\varepsilon^{-1}$ and $\varepsilon^{\tau-2}$ equal zero simultaneously resulting in two equations. Excluding from these equations the term $\sqrt{a_{3}}\tan\!\sqrt{a_{3}}$, we find the condition $a_{1}\tan^{2}\!\sqrt{a_{1}}\cos^{2}\!\sqrt{a_{2}}+\left(\sqrt{a_{1}}\tan\!\sqrt{a_{1}}+\sqrt{a_{2}}\tan\!\sqrt{a_{2}}\right)^{2}=0$ which can also be satisfied if $a_{1}=a_{2}=a_{3}=0$ and therefore the case with $\tau=3/2$ does not produce connected point interactions for $1<\mu\leq 2$.

Finally, as follows from asymptotes (45)-(48), on the whole line $L_{2}$, there are the point subsets of intensities for which $\sin\!\sqrt{-a_{j}}=0$, $j=1,2$ ($N=2$) and $j=1,2,3$ ($N=3$). For this case we have $\Lambda=\pm\,I$ and therefore the limit point interactions are reflectionless. They are ‘non-interacting’ wells because in the case of a single well ($N=1$) the full transmission across this well occurs if $\sin\!\sqrt{-a_{1}}=0$. Thus, so far we have examined the one-point interactions with full transmission (non-resonant in the region $Q_{0}$ and resonant on the line $L_{2}$) and non-resonant $\delta$-potentials realizing on the line $L_{1}$.

5 One-point interactions at the line $L_{K}$

One of the ways to remove the divergence of the element $\bar{\lambda}_{21}$ in (44) as $\varepsilon\to 0$ is the requirement that the total group of terms at the divergent term $\varepsilon^{1-\mu}$, $1<\mu<2$, has to be zero, i.e., $\lim_{\varepsilon\to 0}\left(\varepsilon^{\mu-1}\bar{\lambda}_{21}\right)_{\tau=\mu-1}=0$. Indeed, this requirement can be satisfied on the line $L_{K}$ if the conditions

[TABLE]

are fulfilled. At fixed coefficient $c$, these conditions may be considered as equations with respect to the intensities $a_{1}$, $a_{2}$ and $a_{3}$. The first of these equations ($N=2$) has been obtained earlier by Brasche and Nizhnik [31]. In what follows we shall also be dealing with other conditions like (50). They may be referred to as ‘resonance equations’ and their solutions ‘resonance sets’. The solutions of equations (50) are plotted in figures 2 and 3, respectively for $N=2$ and 3. In each of these two cases, the curve and surface marked in these figures by 0 appear to be ‘pinned’ to the origins $a_{1}=a_{2}=0$ and $a_{1}=a_{2}=a_{3}=0$. They are considered as background branches of the corresponding resonance sets and therefore we call them the ‘zeroth resonance’ curve ($N=2$, figure 2) and surface ($N=3$, figure 3). In the limit as $c\to 0$, the curve 1’ in figure 2 and the surfaces 1’ and 2’ in figure 3 vanish ‘escaping’ to infinity. At the same time, the zeroth branches 0 straighten to the line $a_{1}+a_{2}=0$ and the plane $a_{1}+a_{2}+a_{3}=0$, respectively.

Using next equations (50) in asymptotes (41) and (43), we obtain the diagonal elements of the limit $\Lambda$-matrix:

[TABLE]

In virtue of equations (50), we have $\rho=\theta^{-1}$ and therefore on the $L_{K}$-line the limit transmission matrix becomes of diagonal form (18). Setting here $\theta=(2+\gamma)/(2-\gamma)$ and $\rho=(2-\gamma)/(2+\gamma)$, we obtain Kurasov’s $\delta^{\prime}$-potential with intensity $\gamma\in{\mathbb{R}}\setminus\{\pm 2\}$ defined in the distributional sense on the space of discontinuous at $x=0$ test functions. Next, we find the resonance values of $a_{1}$, $a_{2}$ and $a_{3}$ as functions of the strength $\gamma$:

[TABLE]

for $N=2$ and

[TABLE]

with arbitrary $a_{2}\in{\mathbb{R}}\setminus\{-a_{2}/c\}$, for $N=3$. In particular, for $N=2$ the barrier-well structure corresponds to the interval $-2<\gamma<2$ ($a_{1}>0,~{}a_{2}<0$ for $-2<\gamma<0$ and $a_{1}<0,~{}a_{2}>0$ for $0<\gamma<2$), whereas beyond this interval ($2<|\gamma|<\infty$), we have the double-well configuration. We call the vectors $(a_{1},a_{2})\in{\mathbb{R}}^{2}$ and $(a_{1},a_{2},a_{3})\in{\mathbb{R}}^{3}$ that satisfy equations (50) the resonance sets ${\cal K}_{2}$ and ${\cal K}_{3}$, respectively. Next, the point interactions realized on the line $L_{K}$ are referred in the following to as ‘resonant-tunnelling $\delta^{\prime}$-potentials of the ${\cal K}$-type’, despite that the double-well case is involved here as well, together with a barrier-well structure.

Finally, it should be noticed that there exists a particular subfamily of the intensities $a_{1}$ $a_{2}$ and $a_{3}$ from the ${\cal K}_{2,3}$-sets for which $\theta=\pm\,1$ in (18), realizing the point interactions with full transmission. Thus, for $N=2$ these values are $a_{1}=a_{2}=-2/c$ (this point is indicated in figure 2) resulting in the matrices $\Lambda=-\,I$. In the three-delta case the two conditions $a_{1}=a_{3}$ and $2a_{1}+a_{2}+ca_{1}a_{2}=0$ provide the matrix $\Lambda=I$, whereas the other two conditions $a_{1}+a_{3}=-2/c$ (in general, an asymmetric structure) and $2a_{1}+a_{2}+ca_{1}a_{2}=-2/c$ lead to the matrix $\Lambda=-\,I$.

6 Šeba’s transition at the line $L_{S}$

Other types of one-point interactions can be realized in the regions $Q_{1}$ and $Q_{2}$ including the line $L_{S}$ that separates these regions (see figure 1). In fact, as shown below, the point interactions on these three sets are related to those studied by Šeba in [18] (see Theorem 3 therein). First, we note that in the limit as $\varepsilon\to 0$, the divergence of the elements $\bar{\lambda}_{21}$ given by (42) and (44) can be excluded at the line $L_{S}$ and in the region $Q_{2}$ if the following (resonance) conditions

[TABLE]

hold true, being just a ‘linearized’ version of equations (50). In what follows we refer the line $a_{1}+a_{2}=0$ (green in figure 2) and the plane $a_{1}+a_{2}+a_{3}=0$ (figure 3) to as ${\cal L}_{2}$- and ${\cal L}_{3}$-sets, respectively. Next, in virtue of (41) and (43), we have $\bar{\lambda}_{11},\,\bar{\lambda}_{22}\to 1$, whereas $\bar{\lambda}_{21}\to I$ in the region $Q_{2}$ and $\bar{\lambda}_{21}\to\alpha$ with

[TABLE]

at the line $L_{S}$. Below this line, in the region $Q_{1}$, the $\bar{\lambda}_{21}$-term is divergent as $\varepsilon\to 0$. Since the limits of $\bar{\lambda}_{11}$ and $\bar{\lambda}_{22}$ are finite, we get in this region the separated interactions satisfying the Dirichlet conditions $\psi(\pm 0)=0$.

Thus, the point interactions realized in the region $Q_{1}\cup Q_{2}\cup L_{S}$ exhibit the transition of transmission that occurs on the resonance ${\cal L}_{2,3}$-sets while varying the rate of increasing the distance $r$ between the $\delta$-approximating potentials in (19). For sufficiently slow squeezing [$\mu-1<\tau<2(\mu-1)$] this distance, the limit point interactions are opaque, for intermediate shrinking [$\tau=2(\mu-1)$] the interactions become partially transparent ($\delta$-well) and for fast shrinking [$2(\mu-1)<\tau<\infty$] the interactions appear to be fully transparent. In other words, the line $L_{S}$ separates the regions $Q_{1}$ of full reflection and $Q_{2}$ of perfect transmission.

Finally, note that, contrary to the point interactions of the $\delta$-potential type realized on the line $L_{1}$ where the intensities $a_{1}$ and $a_{2}$ ($N=2$) or $a_{1}$, $a_{2}$ and $a_{3}$ ($N=3$) are arbitrary and the intensity of the total $\delta$-potential is just the algebraic sum $a_{1}+a_{2}$ or $a_{1}+a_{2}+a_{3}$, on the line $L_{S}$, the non-zero transmission across the limit $\delta$-well potential, but now with intensity (55), occurs only on the resonance line and plane $L_{2,3}=0$ referred in the following to as ${\cal L}_{2,3}$-sets. Similarly, the interactions realized at the line $L_{S}$ are called ‘resonant-tunnelling $\delta$-potentials of the ${\cal L}$-type’ and ones in the region $Q_{2}$ ‘resonant-tunnelling reflectionless potentials of the ${\cal L}$-type’.

Let us consider now potential (2) with $N=3$ rewritten in the form

[TABLE]

with a new squeezing parameter $\epsilon>0$. For the case with $N=2$ [$a_{3}\equiv 0$ in (56)], Šeba has proved (see Theorem 3 in [18]) that at $\sigma=1/2$ the limit point interaction is the $\delta$-potential described by the $\Lambda$-matrix of form (11) with the intensity $\alpha$ given by the first equation (55) if $a_{1}+a_{2}=0$, being in fact a resonance condition for intensities $a_{1}$ and $a_{2}$. At this condition for all $\sigma\in(0,1/2)$ the limit $\Lambda$-matrix is the unit, while for $\sigma\in(1/2,1)$ the limit point interactions are separated satisfying the two-sided Dirichlet conditions $\psi(\pm 0)=0$. In physical terms, the value $\sigma=1/2$ is a ‘transition’ point (at which the transmission is partial) separating the opaque potentials from those with full transmission.

In fact, Šeba’s theorem can be extended to the case with $N=3$. Indeed, by a straightforward calculation of the matrix product $\Lambda_{\epsilon}=\Lambda_{3}\Lambda_{0}\Lambda_{2}\Lambda_{0}\Lambda_{1}$ with matrices (7) in which $c_{j}=a_{j}(c/\epsilon)^{\sigma}$ and $r=\epsilon$, we find the limits $\bar{\lambda}_{11},\,\bar{\lambda}_{22}\to 1$ and $\bar{\lambda}_{12}\to 0$ for $\sigma\in(0,1)$. The singular element $\bar{\lambda}_{21}$ has the asymptote

[TABLE]

as $\epsilon\to 0$. It follows from this asymptote that under the resonance condition $a_{1}+a_{2}+a_{3}=0$ the limit $\delta$-potential is realized at $\sigma=1/2$ with the intensity $\alpha$ given by the second formula (55).

The value $\sigma=2/3$ which could result in a finite $\epsilon\to 0$ limit of $\bar{\lambda}_{21}$ is not admissible because the equations $a_{1}+a_{2}+a_{3}=0$ and $a_{1}a_{2}+2a_{1}a_{3}+a_{2}a_{3}=0$ can be fulfilled simultaneously only if $a_{1}=a_{2}=a_{3}=0$. However, as follows from (57), at $\sigma=1$ we have $\bar{\lambda}_{21}\to 0$ if $(a_{1},a_{2},a_{3})\in{\cal K}_{3}$. Moreover, in this case the $\epsilon\to 0$ limits of $\bar{\lambda}_{11}$ and $\bar{\lambda}_{22}$ are given by the second formulae (51).

All these results obtained above for potential (56) appear to be in agreement with those obtained for both the lines $L_{K}$ and $L_{S}$. Indeed, the comparison of expression (56) with potential (2) in which $N=2,3$ leads to the equations $c_{j}=a_{j}\varepsilon^{1-\mu}$ and $r=c\varepsilon^{\tau}$. As a result, we find the relation $\tau=\sigma^{-1}(\mu-1)$, so that at $\sigma=1$ we have the resonance sets defined by equations (50) on the line $L_{K}$, while at $\sigma=1/2$, i.e., on the line $L_{S}$, we find that formulae (55) hold true.

Thus, all the interactions realized at $\sigma=1$ or the same on the line $L_{K}$ are of Kurasov’s $\delta^{\prime}$-potential type defined via the test functions which are discontinuous at $x=0$. It is of interest the problem whether or not the standard $\delta^{\prime}$-potentials defined on the $C_{0}^{\infty}$ functions can be found among the ${\cal K}$-families defined by conditions (50). This problem has been examined by Brasche and Nizhnik [31] including also the case when the fourth $\delta$-potential, i.e., the term $a_{4}\delta(x-3\epsilon)$ is added in the square brackets of (56). In our notations these results read as follows. For $N=2$, in virtue of the first equation (50), the distributional limit $V_{\epsilon}(x)\to\kappa\delta^{\prime}(x)$ cannot exist because it is necessary that $a_{1}+a_{2}=0$. However, for $N=3$ and 4, such a limit can be realized on the planes $a_{1}+a_{2}+a_{3}=0$ and $a_{1}+a_{2}+a_{3}+a_{4}=0$, respectively. For instance, in the case with $N=3$ we have $\kappa=c(a_{1}-a_{3})$ where $a_{1}$ and $a_{2}$ are connected through the quadratic equation

[TABLE]

Solving the last equation with respect to $a_{1}$ or $a_{3}$, one can be convinced that there exists a non-empty set of solutions to this equation. The existence of these solutions is illustrated in figure 3 as the intersection of the resonance surface 1’ with the ${\cal L}_{3}$-plane.

7 Splitting of resonance sets at the critical point $\mu=2$

Consider now the realization of point interactions at the limiting right-hand points of the lines $L_{K}$ and $L_{S}$ ($P_{1}$ and $P_{2}$, respectively), and the limiting right-hand line ($L_{0}$) of the region $Q_{2}$ (see figure 1). Thus, at $\tau=1$, the total coefficients at the divergent term in the singular matrix elements $\bar{\lambda}_{21}$ given by (46) and (48) become zero if the resonance equations

[TABLE]

for $N=2$ and

[TABLE]

[TABLE]

for $N=3$ hold true. Note that $F_{2}(a_{1},a_{2})=F_{2}(a_{2},a_{1})$ and $F_{3}(a_{1},a_{2},a_{3})=F_{3}(a_{3},a_{2},a_{1})$. The solutions of the equation $F_{2}(a_{1},a_{2};c)=0$ for two values of the coefficient $c$ are plotted in figure 4. In order to display in the figure the maximum number of resonance curves, instead of $a_{1}$ and $a_{2}$, the new rescaled coordinates $X:=\mbox{sign}(a_{1})|a_{1}|^{1/2}$ and $Y:=\mbox{sign}(a_{2})|a_{2}|^{1/2}$ have been introduced.

For the sake of brevity of the formulae that will appear in the further analysis of the point interactions to be realized on the limiting sets $P_{1}$, $P_{2}$ and $L_{0}$, we introduce the following functions:

[TABLE]

[TABLE]

for any $c\geq 0$. By straightforward calculations one can prove that, under resonance conditions (59) and (60), $f_{2}(a_{1},a_{2};c)=g_{2}(a_{1},a_{2})$ and $f_{3}(a_{1},a_{2},a_{3};c)=g_{3}(a_{1},a_{2},a_{3};c)$ as well as

[TABLE]

Thus, in virtue of properties (63), the family of point interactions realized at the point $P_{1}$ is described by the $\Lambda$-matrix of type (18) with

[TABLE]

where the intensities $a_{j}$, $j=1,2,3$, satisfy the resonance equations (59) and (60), respectively.

The solutions of transcendental equations (59) and (60) determine the countable sets of resonance curves on the $(a_{1},a_{2})$-plane and resonance surfaces in the $(a_{1},a_{2},a_{3})$-space which are plotted in figures 4 and 5, respectively. We refer these resonance curves and surfaces to as ${\cal F}_{2}$- and ${\cal F}_{3}$-sets, respectively. The limit transmission matrix on these sets is of diagonal form (18) with the element $\theta$ given by equations (62)-(64). The point interactions of this countable family may be called ‘multiple-resonant-tunnelling $\delta^{\prime}$-potentials of the ${\cal F}$-type’.

At the point $P_{2}$, with $\tau=2$ in (46) and (48), we find that these divergent terms become finite in the $\varepsilon\to 0$ limit under conditions (59) and (60) in which $c$ is formally set zero, i.e., if the equations

[TABLE]

are fulfilled. Similarly to the ${\cal F}_{2,3}$-sets, the solutions of these equations yield curves on the $(a_{1},a_{2})$-plane (see green curves in figure 4) and surfaces in the $(a_{1},a_{2},a_{3})$-space that may be referred to as ${\cal J}_{2}$- and ${\cal J}_{3}$-sets, respectively. Under resonance conditions (65), the $\varepsilon\to 0$ limits of (46) and (48) become finite producing a family of one-point interactions described by the transmission matrix of the type

[TABLE]

where

[TABLE]

and the diagonal elements are given by formulae (64) in which again $c$ is formally set zero, i.e.,

[TABLE]

The elements $\theta$ and $\alpha$ in $\Lambda$-matrix (68) are determined by the countable sets of solutions to resonance equations (65). Therefore, this family of one-point interactions may be called ‘resonant-tunnelling ($\delta^{\prime}+\delta$)-potentials of the ${\cal J}$-type’.

Finally, for $\tau>2$ (at the line $L_{0}$) the divergent terms in (46) and (48) vanish at all at the same resonance conditions (65). Next, from expressions (45) and (47) we conclude that the limit $\Lambda$-matrix is of form (18) with the element $\theta$ defined by formulae (70). Similarly, this family of point interactions may be called ‘resonant-tunnelling $\delta^{\prime}$-potentials of the ${\cal J}$-type’. For $N=2$ this type of one-point interactions coincides with that examined earlier in [8, 43].

Thus, we have defined in this section the resonance ${\cal F}_{2}$-, ${\cal F}_{3}$- and ${\cal J}_{2,3}$-sets as solutions of transcendental equations (59), (60) and (65), respectively. These sets consist of the infinite number of curves (for $N=2$) and surfaces (for $N=3$) which are numbered by $0,1,2,\dots$. On the other hand, in the open sets $L_{K}$, $L_{S}$ and $Q_{2}$, we have defined above the resonance ${\cal K}_{2,3}$- and ${\cal L}_{2,3}$-sets as solutions of equations (50) and (54), respectively, consisting of two curves ($N=2$, numbered by 0 and 1’) and three surfaces ($N=3$, numbered by 0, 1’ and 2’). The phenomenon of ‘splitting’ these resonance sets is observed while approaching the right-hand limiting sets: the point $P_{1}$ and the line $P_{2}\cup L_{0}$. The mechanism of this splitting can be explained as follows. The curve 1’ in figure 2 and the surfaces 1’ and 2’ in figure 3, which are ‘unpinned’ to the origins $a_{1}=a_{2}=0$ and $a_{1}=a_{2}=a_{3}=0$, vanish while approaching the limiting sets. At the same time, the zeroth (background) resonance subsets marked in figures 4 and 5 by 0 remain to be ‘pinned’ to the origins, modifying their shape and becoming to be embedded in the angles with the vertices at $a_{1}=a_{2}=-\,b_{0}$ where $b_{0}$ is a solution of the equation $\cot\!\sqrt{b}=c\sqrt{b}$ and $a_{1}=a_{2}=a_{3}=-(\pi/2)^{2}$, respectively for $N=2$ and 3. Instead of the curve 1’ and the surfaces 1’ and 2’, the detachment of the resonances numbered by $n=1,2,\ldots$ from the zeroth resonances 0 occurs forming countable sets. Thus, figure 4 illustrates the resonance curves with $n=0,1,2,3,4$ for the two cases with $c>0$ (${\cal F}_{2}$-set) and $c=0$ (${\cal J}_{2}$-set). In figure 5 the three resonance surfaces with $n=0,1,2$ from the ${\cal J}_{3}$-set are plotted.

To conclude this section, it should be noticed that everywhere beyond all the resonance sets described above on the sets $L_{0}$, $L_{K}$, $L_{S}$, $P_{1}$, $P_{2}$ and $Q_{2}$, the point interactions are separated, similarly to the region $Q_{1}$, where they satisfy the Dirichlet conditions $\psi(\pm 0)=0$ and act as a fully reflecting wall.

8 Concluding remarks

In this paper, the system consisting of two or three $\delta$-potentials (with intensities $a_{j}\in{\mathbb{R}}$, $j=1,2$ if $N=2$ and $j=1,2,3$ if $N=3$) has been approximated in the most simple way, namely by piecewise constant functions and then the convergence of the corresponding transmission matrices has been studied in the squeezing limit as both the width of $\delta$-approximating functions $l$ and the distance between them $r$ tend to zero. The admissible rates of shrinking the parameters $l$ and $r$ have been controlled through power parameterization (39), involving the two powers $\mu$ and $\tau$ as well as the squeezing parameter $\varepsilon\to 0$. For convenience of the presentation, we have used the diagram of admissible values for $\mu$ and $\tau$ plotted in figure 1. Using this parameterization as well as the piecewise constant approximation of the $\delta$-functions in potential (2) with $N=2,3$, it was possible to get the explicit expressions for the corresponding $\Lambda$-matrices and to treat thus the reflection-transmission properties of the one-point interactions directly.

Starting from the same three-layer (for $N=2$) and five-layer (for $N=3$) potential profile given by (19), a whole family of limit one-point interactions with resonant-tunnelling behaviour has been realized. For the interactions realized at the line $L_{K}$, the resonance sets referred to as ${\cal K}_{2,3}$ consist of two curves on the $(a_{1},a_{2})$-plane ($N=2$) and three surfaces in the $(a_{1},a_{2},a_{3})$-space ($N=3$). In its turn, for the interactions obtained in the region $L_{S}\cup Q_{2}$, the resonance sets called ${\cal L}_{2,3}$ appear to be the line $a_{1}+a_{2}=0$ ($N=2$) and the plane $a_{1}+a_{2}+a_{3}=0$ ($N=3$). While approaching the right-hand limiting sets ($L_{K}\to P_{1}$, $L_{S}\to P_{2}$ and $Q_{2}\to L_{0}$), the ${\cal K}$- and ${\cal L}$-sets split into infinite but countable sets resulting in the set transformations ${\cal K}_{2,3}\to{\cal F}_{2,3}$ and ${\cal L}_{2,3}\to{\cal J}_{2,3}$. These transformations are illustrated by the figures from 2 to 5. In other words, at the limiting sets, the detachment of the countable ${\cal F}$- and ${\cal J}$-sets from the ${\cal K}$- and ${\cal L}$-sets takes place, respectively. Accordingly, the point interactions realized on these sets belong to ${\cal K}$-, ${\cal L}$-, ${\cal F}$- and ${\cal J}$-families and their $\Lambda$-matrices are given by (18) with (51), (11) with (55), (18) with (64), and (68) with (69) and (70) as well as (18) with (70). Outside all these resonance sets as well as for all $a_{j}$, $j=1,2,3$, in the region $Q_{1}$, the point interactions are separated with the Dirichlet boundary conditions $\psi(\pm 0)=0$.

In principle, a similar straightforward analysis could be carried out for higher $N$ resulting in the same types of one-point interactions with resonance sets ${\cal K}_{N}$, ${\cal L}_{N}$, ${\cal F}_{N}$ and ${\cal J}_{N}$ being $(N-1)$-dimensional hypersurfaces, however, the corresponding formulae appear to be quite complicated. To conclude, it should be noticed that the approach developed in this paper can be a starting point for further studies on regular approximations of point interactions and understanding the resonance mechanism.

Acknowledgments

The author acknowledges the financial support from the Department of Physics and Astronomy of the National Academy of Sciences of Ukraine under project No. 0117U000240. He is indebted to one of Referees for criticisms and recommendations, resulting in the significant improvement of the paper. He would like to express gratitude to Yaroslav Zolotaryuk for stimulating discussions and valuable suggestions.

References