Nagaland | 681
Odisha | 41
Punjab | 195
Rajasthan | 186
Telangana | 235
Tripura | 365
West Bengal | 125
National | 598
Other Books | 1442
Total | 7554
Table 16: State-wise statistics of School textbooks collected Language | Number of Instances
---|---
Assamese | 2
Bengali | 2619
English | 1178
Hindi | 2808
Kannada | 7
Malayalam | 7571
Oriya | 3
Sindhi | 30
Tamil | 223
Telugu | 20
Urdu | 129
Total | 14590
Table 17: Language wise statistics of subtitles collected from OpenSubtitles Language | Number of Courses
---|---
Assamese | 1
Bengali | 91
English | 523
Gujarati | 106
Hindi | 184
Kannada | 89
Malayalam | 108
Marathi | 85
Punjabi | 1
Tamil | 150
Telugu | 98
Total | 1436
Table 18: Language wise statistics of the course transcripts collected from NPTEL Language | Number of Instances
---|---
Assamese | 63
Bengali | 91
English | 410
Gujarati | 92
Hindi | 89
Kannada | 78
Malayalam | 89
Marathi | 90
Manipuri | 65
Oriya | 82
Punjabi | 81
Tamil | 85
Telugu | 89
Urdu | 64
Total | 1468
Table 19: Language-wise Mann Ki Baat transcripts collected Languages | Average Page Count
---|---
asm | 4.52
ben | 3.44
guj | 3.06
hin | 2.39
kan | 2.46
mal | 2.39
mar | 3.16
nep | 2.44
ori | 3.1
pan | 2.85
san | 2.68
tam | 2.55
tel | 2.38
urd | 2
Table 20: Showing average page count of PDF documents after merge operation Language | No.of Questions
---|---
Assamese | 2.73M
Bengali | 4.54M
Bodo | 2.66M
Gujarati | 6.41M
Hindi | 10.54M
Kannada | 6.16M
Kashmiri | 2.21M
Malayalam | 3.98M
Marati | 4.36M
Meitei | 2.02M
Nepali | 1.89M
Odia | 5.35M
Punjabi | 5.23M
Sanskrit | 5.73M
Tamil | 3.59M
Telugu | 3.72M
Urdu | 3.21M
Total | 74M
Table 21: Number of instruction-answer pairs for each language in the IndoWordNet split of IndicAlign - Instruct Code | Wikimedia | IndicCorp V1 | IndicCorp V2 | Sangraha Verified
---|---|---|---|---
asm | 10 | 59 | 132 | 292
ben | 129 | 1795 | 2330 | 10604
brx | - | - | 4 | 1.5
doi | - | - | 0.1 | 0.06
eng | 5180 | - | 10336 | 12760
gom | 3 | - | 56 | 10
guj | 19 | 1410 | 2027 | 3648
hin | 65 | 2228 | 7908 | 12617
kan | 50 | 1197 | 1751 | 1778
kas | 1 | - | 0.12 | 0.45
mai | 2 | - | 23 | 15
mal | 11 | 1425 | 2205 | 2731
mar | 23 | 777 | 1290 | 2827
mni | 1 | - | 1 | 7.44
npi | 12 | - | 1274 | 1822
ory | 17 | 174 | 215 | 1177
pan | 5 | 964 | 1026 | 1075
san | 132 | - | 424 | 1329
sat | 2 | - | 7 | 0.33
snd | 6 | - | 19 | 258
tam | 74 | 989 | 980 | 3985
tel | 76 | 1149 | 1478 | 3707
urd | 48 | - | 872 | 3658
Total | 5869 | 12168 | 24023 | 64306
Table 22: Detailed Language-wise comparison of number of tokens (in Millions)
in Wikimedia, IndicCorp-V1, IndicCorp-V2 and Sangraha Verified Figure 26:
Prompt template used for generating toxic prompta data using Mistral-7B Chat.
Figure 27: Prompt template used for classifying whether a prompt is toxic or
not. Figure 28: Prompt template used for generating conversations for the
Wiki-Conv data.

(a) Intent LLM (b) Init User LLM

(c) Next User LLM (d) Assistant LLM

Figure 29: Prompt templates used for generating Wiki-Chat data

(a) English

(b) Tamil

Figure 30: Example Conversation - IndoWordNet

(a) English

(b) Hindi

Figure 31: Example Conversation - Anudesh

(a) English

(b) Tamil

Figure 32: Example Conversation - Wiki-Conv

(a) English

(b) Malayalam

Figure 33: Example Conversation - Indic Share llama

(a) English

(b) Gujarati

Figure 34: Example Conversation - Dolly

(a) English

(b) Telugu

Figure 35: Example Conversation - Wiki-Chat

(a) English

(b) Bengali

Figure 36: Example Conversation - OpenAssitant

(a) English

(b) Hindi

Figure 37: Example Conversation - HH-RLHF

(a) English

(b) Hindi

Figure 38: Example Conversation - Wiki How

(a) English

(b) Hindi

Figure 39: Example Conversation - Toxic Matrix


# Conditions for realizing one-point interactions from a multi-layer structure
model

Alexander V. Zolotaryuk and Yaroslav Zolotaryuk  Bogolyubov Institute for
Theoretical Physics, National Academy of Sciences of Ukraine, Kyiv 03143,
Ukraine

###### Abstract

A heterostructure composed of $N$ parallel homogeneous layers is studied in
the limit as their widths $l_{1},\ldots,l_{N}$ shrink to zero. The problem is
investigated in one dimension and the piecewise constant potential in the
Schrödinger equation is given by the strengths $V_{1},\ldots,V_{N}$ as
functions of $l_{1},\ldots,l_{N}$, respectively. The key point is the
derivation of the conditions on the functions
$V_{1}(l_{1}),\ldots,V_{N}(l_{N})$ for realizing a family of one-point
interactions as $l_{1},\ldots,l_{N}$ tend to zero along available paths in the
$N$-dimensional space. The existence of equations for a squeezed structure,
the solution of which determines the system parameter values, under which the
non-zero tunneling of quantum particles through a multi-layer structure
occurs, is shown to exist and depend on the paths. This tunneling appears as a
result of an appropriate cancellation of divergences.

###### keywords:

one-dimensional quantum systems, multi-layer heterostructures, point
interactions, resonance sets.

##  1 Introduction

The differential operators with singular zero-range potentials are frequently
used to model different physical systems. The particular class of such models
is represented by Schrödinger operators in one dimension with regular
realistic potentials, which can be approximated by point (called also contact)
interactions (see books [1, 2] for details and references) defined on the sets
consisting of isolated points, curves or surfaces. In this asymptotic
approach, for a general class of potentials $V$, under certain conditions the
functions $\varepsilon^{-1}V(x/\varepsilon)$ and
$\varepsilon^{-2}V(x/\varepsilon)$ converge in the sense of distributions as
$\varepsilon\to 0$ to the Dirac delta-function $\delta(x)$ and its first
derivative $\delta^{\prime}(x)$, respectively, realizing one-point
interactions. Thus, the one-dimensional Schrödinger operator with the
$\delta^{\prime}(x)$ distribution has rigorously been treated in works [3, 4,
5, 6] and afterwards these studies were extended to the quantum graph theory
[7, 8, 9]. Some particular cases of the regularization of the
$\delta^{\prime}(x)$ function for realizing one-point interactions have been
used in [10]. Using a special class of piecewise constant potentials $V$, it
was possible [11, 12, 13] to construct explicitly the matrices, which connect
the two-sided boundary conditions at the point of singularity.

In a similar way, the Schrödinger operator with the combined potential
$a\delta^{\prime}(x)+b\delta(x)$ was investigated [14, 15, 16], including the
studies of the discrete spectrum [17, 18]. The one-dimensional quantum
Hamiltonians with potentials having a regular unperturbed part of different
shape and a singular perturbation of the form $a\delta^{\prime}(x)+b\delta(x)$
have been examined in publications [19, 20, 21].

Various aspects have been studied in numerous works (see, e.g., publications
[3, 22, 23, 24, 25, 26, 27, 28], a few to mention) concerning the
$\delta^{\prime}$-interaction, which should not be confused with the
$\delta^{\prime}$-potential. It is worth mentioning the works on the three
delta approximation of the $\delta^{\prime}$-interaction [29, 30] as well as
publications [21, 31, 32], where the $\delta^{\prime}$-interaction was
considered as a perturbation of regular potentials.

The most simple example of regularizing the $\delta^{\prime}$-potential is the
sequence of piecewise constant functions, which from the physical point of
view can be related to planar heterostructures. In particular, for the double-
layer case, using a three-scale parametrization of the layer parameters such
as their strengths, thickness and the distance between the layers, a whole
family of one-point interactions has been constructed, where the scattering
data [13, 33, 34] as well as the existence of a bound state [35, 36] have been
examined. In general, here it is not required that a piecewise constant
potential must have a well-defined limit (for instance, in the sense of
distributions) as the thickness of layers and the distance between them shrink
to zero. As a result, within the three-scaled squeezing procedure, it was
possible to describe both the resonant point interactions [4, 5, 6, 10, 11,
14, 15, 16] and the point interaction produced by the potential
$a\delta^{\prime}(x)$ potential, that comes from the theory developed by
Kurasov and coworkers [1, 37, 38], for which the elements of the diagonal
connection matrix are $\theta=(2+a)/(2-a)$ and $\theta^{-1}$ [17], within an
unique scheme [34].

The present paper is devoted to the investigation of a planar heterostructure
composed of an arbitrary number of layers. The layers may be separated by
distances and these spaces (if any) are considered as a particular case of
layers with zero strength potential. The goal of the paper is to study this
multi-layer structure in the limit as both the layer thickness and the
distance between the layers shrink to zero. The electron motion in this system
is supposed to be confined in the longitudinal direction (say, along the
$x$-axis) being perpendicular to the transverse planes where electronic motion
is free. Then the three-dimensional Schrödinger equation of such a structure
can be separated into longitudinal and transverse parts, resulting in the
reduced stationary one-dimensional Schrödinger equation

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

with respect to the longitudinal component of the wave function $\psi(x)$ and
the electron energy $E$. Here, $V(x)$ is the potential for quantum particles
in the form of a piecewise constant function and the prime stands for the
differentiation over $x$.

##  2 Transmission matrix for a multilayer structure

Consider one-dimensional stationary Schrödinger equation (1), where the
potential $V(x)$ is a piecewise constant function defined on the interval
$(x_{0},x_{N})$ with $N$ subsets $(x_{i-1},x_{i})$, where $i=1,\ldots,N$ and
$N=1,2,\ldots$ . Each strength $V_{i}\in{\mathbb{R}}$ defined on this interval
is supposed to depend on the $i$th layer width $l_{i}:=x_{i}-x_{i-1}$, so that
we have the set of functions: $V_{1}(l_{1}),\ldots,V_{N}(l_{N})$.

The transmission matrix $\Lambda_{i}(x_{i-1},x_{i})$ for the $i$th layer
connects the values of the wave function $\psi_{i}(x)$ and its derivative
$\psi^{\prime}_{i}(x)$ at the boundaries $x=x_{i-1}$ and $x=x_{i}$ according
to the matrix equation

$\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left(\begin{array}[]{cc}\\!\psi_{i}(x_{i})\\\
\psi^{\prime}_{i}(x_{i})\end{array}\\!\right)=\Lambda_{i}(x_{i-1},x_{i})\left(\\!\begin{array}[]{cc}\psi_{i}(x_{i-1})\\\
\psi^{\prime}_{i}(x_{i-1})\end{array}\\!\right),~{}\Lambda_{i}(x_{i-1},x_{i})=\left(\\!\begin{array}[]{cc}\lambda_{i,11}~{}~{}\lambda_{i,12}\\\
\lambda_{i,21}~{}~{}\lambda_{i,22}\end{array}\\!\right).$ (2)

The transmission matrix for each layer defined on the interval
$(x_{i-1},x_{i})$ can be computed through the solutions $u_{i}(x)$ and
$v_{i}(x)$ and their derivatives taken at the boundaries $x=x_{i-1}$ and
$x=x_{i}$. Let $u_{i}(x)$ and $v_{i}(x)$ be linearly independent solutions on
the interval $(x_{i-1},x_{i})$ obeying the initial conditions

$u_{i}(x_{i-1})=1,~{}~{}u^{\prime}_{i}(x_{i-1})=0,~{}~{}v_{i}(x_{i-1})=0,~{}~{}v^{\prime}_{i}(x_{i-1})=1.$
(3)

Then the representation of the $\Lambda_{i}$-matrix reads

$\Lambda_{i}(x_{i-1},x_{i})=\left(\begin{array}[]{cc}u_{i}(x_{i})~{}~{}v_{i}(x_{i})\\\
u^{\prime}_{i}(x_{i})~{}~{}v^{\prime}_{i}(x_{i})\end{array}\right),~{}~{}~{}\det\Lambda_{i}(x_{i-1},x_{i})=1.$
(4)

Particularly, for the piecewise constant function $V_{i}(x)$, the solutions
$u_{i}(x)$ and $v_{i}(x)$, $x_{i-1}\leq x\leq x_{i}$, obeying conditions (3),
read

$u_{i}(x)=\cos[q_{i}(x-x_{i-1})],~{}~{}v_{i}(x)=q_{i}^{-1}\sin[q_{i}(x-x_{i-1})],$
(5)

where

$q_{i}=\sqrt{E-V_{i}}\,.$ (6)

Therefore the $\Lambda_{i}$-matrix in this particular case is

$\Lambda_{i}(x_{i-1},x_{i})=\left(\,\begin{array}[]{cc}\cos(q_{i}l_{i}){}{}&q_{i}^{-1}\sin(q_{i}l_{i})\\\
-q_{i}\sin(q_{i}l_{i}){}{}&\cos(q_{i}l_{i})\end{array}\right),~{}~{}i=\overline{1,N},$
(7)

and the full transmission matrix, connecting the boundary conditions at
$x=x_{0}$ and $x=x_{N}$, is the product

$\Lambda(x_{0},x_{N})=\Lambda_{N}(x_{N-1},x_{N})\ldots\Lambda_{1}(x_{0},x_{1})=\left(\begin{array}[]{cc}\lambda_{11}~{}~{}\lambda_{12}\\\
\lambda_{21}~{}~{}\lambda_{22}\end{array}\right).$ (8)

Clearly, due to the form of matrices (7), $\det\Lambda(x_{0},x_{N})=1.$

##  3 Structure of the full transmission matrix $\Lambda(x_{0},x_{N})$

Using matrix representation (7) in product (8), by induction one can derive
some properties of the $\Lambda(x_{0},x_{N})$-matrix elements. First of all
note that each element of this matrix consists of $2^{N-1}$ terms, each of
which is the $N$-multiple product of sines and cosines involved in matrices
(7). By lengthy and straightforward computation one can derive by induction
the structure of these terms:

$\left\\{\begin{array}[]{llll}\lambda_{11}\\\ \lambda_{12}\\\ \lambda_{21}\\\
\lambda_{22}\end{array}\right\\}=\prod_{n=1}^{N}\cos(q_{n}l_{n})\times\left\\{\begin{array}[]{llll}~{}~{}\,1+{\cal
Q}_{11}\,,\\\ ~{}~{}\sum_{i=1}^{N}q_{i}^{-1}t_{i}+{\cal Q}_{12}\,,\\\
-\sum_{i=1}^{N}q_{i}t_{i}-{\cal Q}_{21}\,,\\\ ~{}~{}\,1+{\cal
Q}_{22}\,,\end{array}\right.$ (9)

where $t_{i}:=\tan(q_{i}l_{i})$ and

${\cal
Q}_{11}=\sum_{n=1}^{[N/2]}(-1)^{n}\sum_{1=i_{1}<j_{1}<i_{2}<j_{2}<\ldots<i_{n}<j_{n}=N}D_{i_{1}j_{1}}\ldots
D_{i_{n}j_{n}},$ (10) $\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\cal
Q}_{12}=\sum_{n=1}^{[(N-1)/2]}(-1)^{n}\sum_{1=i_{1}<j_{1}<i_{2}<j_{2}<\ldots<i_{n}<j_{n}<k_{n}=N}D_{j_{1}i_{1}}\ldots
D_{j_{n}i_{n}}q_{k_{n}}^{-1}t_{k_{n}},~{}~{}$ (11)
$\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\cal
Q}_{21}=\sum_{n=1}^{[(N-1)/2]}(-1)^{n}\sum_{1=i_{1}<j_{1}<i_{2}<j_{2}<\ldots<i_{n}<j_{n}<k_{n}=N}D_{i_{1}j_{1}}\ldots
D_{i_{n}j_{n}}q_{k_{n}}t_{k_{n}},~{}~{}$ (12) ${\cal
Q}_{22}=\sum_{n=1}^{[N/2]}(-1)^{n}\sum_{1=i_{1}<j_{1}<i_{2}<j_{2}<\ldots<i_{n}<j_{n}=N}D_{j_{1}i_{1}}\ldots
D_{j_{n}i_{n}},$ (13)

with

$D_{ij}:=(q_{i}/q_{j})t_{i}t_{j},~{}~{}i,j=\overline{1,N}~{}(i\neq j).$ (14)

Here, $[N/2]$ and $[(N-1)/2]$ stand for the integer parts of $N/2$ and
$(N-1)/2$, respectively.

It is remarkable that all the $Q$-series appear in the form of pairwise terms
(14), called ‘dyads’ from now on. The $n$th group of terms in (10) or (13)
consists of ${N\choose\,2n\,}=\frac{N!}{(2n)!(N-2n)!}$ summands. Thus, for
$n=1$ there are $N(N-1)/2$ dyads (14), where $i=1,\ldots,N-1$ and
$j=2,\dots,N$ with the ordering relation $i<j$. Next, the terms in the group
with $n=2$ are of the form $D_{i_{1}j_{1}}D_{i_{2}j_{2}}$ in $\lambda_{11}$
and $D_{j_{1}i_{1}}D_{j_{2}i_{2}}$ in $\lambda_{22}$, where
$i_{1}=1,\ldots,N-3$, $j_{1}=2,\ldots,N-2$, $i_{2}=3,\ldots,N-1$ and
$j_{2}=4,\ldots,N$ with the ordering relation $i_{1}<j_{1}<i_{2}<j_{2}$. In a
similar way, one can present the terms in the group with $n=3$ and so on.

Similarly, the $n$th group of terms in (11) and (12) consists of
${N\choose\,2n+1\,}=\frac{N!}{(2n+1)!(N-2n-1)!}$ summands. Thus, for $n=1$ we
have the sum $N(N-1)(N-2)/6$ terms in real-valued functions $V_{i}(l_{i})$ the
form of ‘triads’ $D_{ji}q_{k}^{-1}t_{k}=(q_{j}/q_{i}q_{k})t_{i}t_{j}t_{k}$ in
$\lambda_{12}$ and $D_{ij}q_{k}t_{k}=(q_{i}q_{k}/q_{j})t_{i}t_{j}t_{k}$ in
$\lambda_{21}$, where $i=1,\ldots,N-2$, $j=2,\ldots,N-1$ and $k=3,\ldots,N$
($i<j<k$). In a similar way, one can present the terms in the group with $n=2$
and so on.

##  4 Conditions for realizing one-point interactions with finite diagonal
elements in the connection matrix

A family of one-point interactions can be realized from equation (1) with a
piecewise constant potential $V(x)$ defined on the interval $x_{0}\leq x\leq
x_{N}$ if all the $N$ layer widths $l_{1},\ldots,l_{N}$ shrink to zero,
setting then $x_{0}\to-\,0$ and $x_{N}\to+\,0$. Any finite configuration of
the layer widths can be associated with a point (vector)
$l:=\mbox{col}(l_{1},\ldots,l_{N})$ in the orthogonal angle of dimension $N$
with the vertex at the origin $l=0$, so that the squeezing of the full layer
thickness $x_{N}-x_{0}$ to zero can be described as a path approaching the
origin $l=0$. At the same time, for the realization of a non-trivial point
interaction, at least one of the potential strengths $V_{1},\ldots,V_{N}$,
$i=1,\ldots,N$, must tend to infinity with some rate. Otherwise, the
transmission through the squeezed structure will be trivial, i.e., the limit
transmission (connection) matrix will be just the identity. In general, under
the squeezing procedure, the behavior of the functions
$V_{1}(l_{1}),\ldots,V_{N}(l_{N})$ versus the paths approaching the origin
$l=0$ are to be analyzed below.

While the elements $\lambda_{11}$ and $\lambda_{22}$ [see equations (10) and
(13)] are composed of only dyads (14), the summands in the off-diagonal
elements contain additional factors: $q_{k}^{-1}t_{k}$ in $\lambda_{12}$ and
$q_{k}t_{k}$ in $\lambda_{21}$. Therefore, for the existence of finite
squeezed limits of the elements $\lambda_{11}$ and $\lambda_{22}$, it is
sufficient to estimate only the dyads $D_{ij}$ and $D_{ji}$ ($i<j$). Next,
according to (11) and (12), these estimates are to be used for the analysis of
the elements $\lambda_{12}$ and $\lambda_{21}$. Hence, the squeezing analysis
of the full $\Lambda$-matrix is reduced to studying the conditions on all the
pairs of functions $V_{i}(l_{i})$ and $V_{j}(l_{j})$ versus the paths
projected onto the $(N-1)N/2$ $(l_{i},l_{j})$-faces of the $N$-dimensional
orthogonal angle.

###  4.1 A $\delta$-like squeezing limit

Consider first the case with the asymptotic behavior $V_{i}(l_{i})l_{i}\to
c\in{\mathbb{R}}$ for all the functions $V_{i}(l_{i})$ as $l_{i}\to 0$. Here
and in the following $c$ stands for an arbitrary constant. Assume that the
functions $V_{i}(l_{i})$ belong to the sets

${\cal G}_{i}^{0}:=\left\\{V_{i}\in{\mathbb{R}}\,\Big{|}\,\lim_{l_{i}\to
0}V_{i}(l_{i})\,l_{i}=\alpha_{i}\in{\mathbb{R}}\right\\}\\!,~{}~{}~{}i=\overline{1,N}.$
(15)

Note that the free space $(x_{i-1},x_{i})$ with $V_{i}\equiv 0$, is also
treated as a ‘layer’ with strength zero.

For any two variables $\beta_{1}$ and $\beta_{2}$, we write
$\beta_{1}\sim\beta_{2}$ if they are of the same order. Let us now estimate
series (10)–(13) in the case as all the functions $V_{i}$ belong to ${\cal
G}_{i}^{0}$ ($i=1,\dots,N$). In the limit as $l\to 0$, we have
$D_{ij}\sim\alpha_{i}l_{j}\to 0$ and $D_{ji}\sim\alpha_{j}l_{i}\to 0$ as well
as $D_{ji}q_{k}^{-1}t_{k}\sim\alpha_{j}l_{i}l_{k}\to 0$ and
$D_{ij}q_{k}t_{k}\sim\alpha_{i}l_{j}\alpha_{k}\to 0$. Moreover, the shrinking
of all the widths $l_{i}$’s to zero is available to be arranged in a repeated
manner (not simultaneously). For instance, the shrinking may pass along some
edges of the orthogonal angle or be found in its ($l_{i},l_{j}$)-faces. Hence,
all the paths with $l\to 0$ constitute in this case a pencil that coincides
with the $N$-dimensional orthogonal angle. Thus, as follows from equations
(10)–(13), the $Q$-series in equations (9) vanish, resulting in the limit
transmission (connection) matrix of the form

$\Lambda_{0}:=\lim_{l\to
0}\Lambda(x_{0},x_{N})=\left(\begin{array}[]{lr}1~{}~{}0\\\
\alpha~{}~{}1\end{array}\right),~{}~{}~{}~{}\alpha:=\sum_{i=1}^{N}\alpha_{i}\,.$
(16)

In the case if $\alpha>0$, we have the point interaction specified by a
$\delta$-potential barrier, while for $\alpha<0$, we are dealing with an
effective $\delta$-well potential. Using the general equation for bound state
levels $\kappa$ ($k={\rm i}\kappa$, $\kappa>0$), written in terms of the
$\Lambda$-matrix elements as (see, e.g., [36, 39])

$\lambda_{11}(\kappa)+\lambda_{22}(\kappa)+\kappa\lambda_{12}(\kappa)+\kappa^{-1}\lambda_{21}(\kappa)=0,$
(17)

we obtain $\kappa=-\,\alpha/2$. Thus, one can formulate

Conclusion 1 : Let all the functions $V_{i}(l_{i})$, $i=1,\ldots,N$, belong to
the ${\cal G}_{i}^{0}$-sets defined by condition (15) and the vector
$l=\mbox{col}(l_{1},\ldots,l_{N})$ tends to the origin $l=0$ along any path
within the pencil formed by the $N$-dimensional orthogonal angle with the
vertex at the origin $l=0$. Then, in the squeezing limit $l\to 0$, the point
interactions are realized as the family of $\delta$-potentials with the
connection matrix given by equation (16).

###  4.2 A singular squeezing limit

The set of functions $V_{i}(l_{i})\in{\cal G}_{i}^{0}$, used for materializing
$\delta$-potentials, can be extended to include the functions with a more
singular behavior like $|V_{i}(l_{i})|\,l_{i}\to\infty$ as $l_{i}\to 0$. In
the case if at least one of the functions $V_{i}(l_{i})$, $i=1,\ldots,N$,
possesses such an infinite behavior, the shrinking of the full system to zero
dimension will be referred to as a singular squeezing limit from here on.

The sufficient requirement for realizing a point interaction is that the
arguments $q_{i}l_{i}$ of the trigonometric functions in matrix expressions
(7) must be finite in the limit as $l_{i}\to 0$ even if the product
$V_{i}(l_{i})\,l_{i}$ diverges as $l_{i}\to 0$. In order to derive the
conditions under which the $l_{i}\to 0$ limit is finite, we define the sets of
real-valued functions $V_{i}(l_{i})$, $i=1,\ldots,N$, by the conditions

${\cal G}_{i}:=\left\\{V_{i}(l_{i})\in{\mathbb{R}}~{}\big{|}~{}\lim_{l_{i}\to
0}|V_{i}(l_{i})|^{1/2}l_{i}=0\right\\},$ (18)

with the corresponding complements

${\cal
G}^{\prime}_{i}:=\left\\{V_{i}(l_{i})\in{\mathbb{R}}~{}\big{|}~{}\lim_{l_{i}\to
0}|V_{i}(l_{i})|^{1/2}l_{i}=c>0\right\\}.$ (19)

Hence, the trigonometric functions in expressions (9)–(14) make sense if each
function $V_{i}(l_{i})$ belongs to the union $\bar{\cal G}_{i}:={\cal
G}_{i}\cup{\cal G}^{\prime}_{i}$ .

The condition $V_{i}(l_{i})\in\bar{\cal G}_{i}$, $i=1,\ldots,N$, is not
sufficient for the existence of finite squeezed limits of the diagonal
elements $\lambda_{11}$ and $\lambda_{22}$ because of the presence of the
factors $q_{i}$ with $|q_{i}|\to\infty$ in series (10) and (13). However, this
divergence can be suppressed due to the presence of the ‘neighboring’ factors
$q_{j}^{-1}$ ($j\neq i$), leading to the ambiguity of type $\infty/\infty$.
Contrary to the $\delta$-potential case, here the finite limits are possible
only if all the thickness parameters $l_{i}$, $i=1,\ldots,N$, converge to the
origin simultaneously. Thus, for the existence of finite squeezed limits of
the elements $\lambda_{11}$ and $\lambda_{22}$, it is sufficient to estimate
both the dyads $D_{ij}$ and $D_{ji}$ ($i<j$). Due to definition (14), the
asymptotic expressions for these dyads, written in terms of the functions
$V_{i}(l_{i})$ and $V_{j}(l_{j})$ as well as the parameters $l_{i}$ and
$l_{j}$, read

$|D_{ij}|\sim\left\\{\begin{array}[]{llll}|V_{i}(l_{i})/V_{j}(l_{j})|^{1/2}\\\
|V_{i}(l_{i})|\,l_{i}\,|V_{j}(l_{j})|^{-1/2}\\\ |V_{i}(l_{i})|^{1/2}\,l_{j}\\\
|V_{i}(l_{i})|\,l_{i}\,l_{j}\end{array}\right.\mbox{for}~{}\left\\{\begin{array}[]{llll}V_{i}\in{\cal
G}^{\prime}_{i},~{}V_{j}\in{\cal G}^{\prime}_{j},\\\ V_{i}\in{\cal
G}_{i},~{}V_{j}\in{\cal G}^{\prime}_{j},\\\ V_{i}\in{\cal
G}^{\prime}_{i},~{}V_{j}\in{\cal G}_{j},\\\ V_{i}\in{\cal
G}_{i},~{}V_{j}\in{\cal G}_{j},\end{array}\right.$ (20)

in series (10) for $\lambda_{11}$ and

$|D_{ji}|\sim\left\\{\begin{array}[]{llll}|V_{j}(l_{j})/V_{i}(l_{i})|^{1/2}\\\
|V_{j}(l_{j})|^{1/2}\,l_{i}\\\ |V_{j}(l_{j})|\,l_{j}\,|V_{i}(l_{i})|^{-1/2}\\\
|V_{j}(l_{j})|\,l_{i}\,l_{j}\end{array}\right.\mbox{for}~{}\left\\{\begin{array}[]{llll}V_{i}\in{\cal
G}^{\prime}_{i},~{}V_{j}\in{\cal G}^{\prime}_{j},\\\ V_{i}\in{\cal
G}_{i},~{}V_{j}\in{\cal G}^{\prime}_{j},\\\ V_{i}\in{\cal
G}^{\prime}_{i},~{}V_{j}\in{\cal G}_{j},\\\ V_{i}\in{\cal
G}_{i},~{}V_{j}\in{\cal G}_{j},\end{array}\right.$ (21)

in series (13) for $\lambda_{22}$. Both the terms $D_{ij}$ and $D_{ji}$
($i<j$) will be finite in the limit as $l_{i}\to 0$ and $l_{j}\to 0$ if

$\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left\\{\begin{array}[]{llllllll}|V_{i}(l_{i})/V_{j}(l_{j})|\to
c>0\\\ |V_{i}(l_{i})|\,l_{i}\,l_{j}\to c\geq 0,~{}l_{i}/l_{j}\to c\geq 0\\\
|V_{j}(l_{j})|\,l_{i}\,l_{j}\to c\geq 0,~{}l_{j}/l_{i}\to c\geq 0\\\
\,l_{i}/l_{j}\to c\geq 0\\\ \,l_{j}/l_{i}\to c\geq 0\\\
|V_{i}(l_{i})|\,l_{i}\,l_{j}\to c\geq 0,~{}|V_{j}(l_{j})|\,l_{i}\,l_{j}\to
c\geq 0\\\ |V_{j}(l_{j})|\,l_{i}\,l_{j}\to c\geq 0\\\
|V_{i}(l_{i})|\,l_{i}\,l_{j}\to c\geq
0\end{array}\right.\mbox{for}~{}\left\\{\begin{array}[]{llllllll}V_{i}\in{\cal
G}^{\prime}_{i}\,,~{}V_{j}\in{\cal G}^{\prime}_{j}\,,\\\ V_{i}\in{\cal
G}_{i}\setminus{\cal G}_{i}^{0}\,,~{}V_{j}\in{\cal G}^{\prime}_{j}\,,\\\
V_{i}\in{\cal G}^{\prime}_{i}\,,~{}V_{j}\in{\cal G}_{j}\setminus{\cal
G}_{j}^{0}\\\ V_{i}\in{\cal G}_{i}^{0}\,,~{}V_{j}\in{\cal
G}^{\prime}_{j}\,,\\\ V_{i}\in{\cal G}^{\prime}_{i}\,,~{}~{}V_{j}\in{\cal
G}_{j}^{0}\,,\\\ V_{i}\in{\cal G}_{i}\setminus{\cal
G}_{i}^{0}\,,~{}V_{j}\in{\cal G}_{j}\setminus{\cal G}_{j}^{0}\,,\\\
V_{i}\in{\cal G}_{i}^{0}\,,~{}V_{j}\in{\cal G}_{j}\setminus{\cal
G}_{j}^{0}\,,\\\ V_{i}\in{\cal G}_{i}\setminus{\cal
G}_{i}^{0}\,,~{}~{}V_{j}\in{\cal G}_{j}^{0}\,.\end{array}\right.$ (22)

These finite limits can be fulfilled if $l_{i}$ and $l_{j}$ tend to zero
simultaneously, forming a pencil of paths in the $N$-dimensional orthogonal
angle with the vertex at $l=0$. Therefore we need to describe these paths
projected onto each $(l_{i},l_{j})$-face. Thus, let us introduce the following
pencil projections onto a given $(l_{i},l_{j})$-face:

$\begin{array}[]{lll}\Gamma_{ij}\\!\\!&:=&\\!\\!\left\\{(l_{i},l_{j})\to
0~{}|~{}l_{i}/l_{j}\to 0\right\\},\\\
\Gamma^{\prime}_{ij}\\!\\!&:=&\\!\\!\left\\{(l_{i},l_{j})\to
0~{}|~{}l_{i}/l_{j}\to c>0\right\\}=\Gamma^{\prime}_{ji}\,,\\\
\Gamma_{V_{i}j}\\!\\!&:=&\\!\\!\left\\{V_{i}(l_{i})\in{\cal
G}_{i}\setminus{\cal
G}_{i}^{0}\,,~{}~{}(l_{i},l_{j})\in\Gamma_{ij}~{}\big{|}~{}|V_{i}(l_{i})|\,l_{i}\,l_{j}\to
0\right\\},\\\
\Gamma^{\prime}_{V_{i}j}\\!\\!&:=&\\!\\!\left\\{V_{i}(l_{i})\in{\cal
G}_{i}\setminus{\cal
G}_{i}^{0}\,,~{}(l_{i},l_{j})\in\Gamma_{ij}~{}\big{|}~{}|V_{i}(l_{i})|\,l_{i}\,l_{j}\to
c>0\right\\},\end{array}$ (23)

where $i,j=1,\ldots,N$ and $i\neq j$. Here, the linear
$(l_{i},l_{j})$-projective paths $\Gamma^{\prime}_{ij}=\Gamma^{\prime}_{ji}$
determine the pencil composed of all the straight lines filling in the
interior of the $N$-dimensional orthogonal angle. They are boundary (limit)
cluster sets of both the pencils $\Gamma_{ij}$ and $\Gamma_{ji}$, which
consist of curved lines in this interior. The projective nonlinear paths
$\Gamma_{V_{i}j}$ and $\Gamma^{\prime}_{V_{i}j}$ may be termed as ‘adjoint’
($l_{i},l_{j}$)-curves of the strength $V_{i}(l_{i})$. More precisely, the
paths $\Gamma_{V_{i}j}$ mean that $l_{i}/l_{j}$ tends to zero more slowly than
$|V_{i}(l_{i})|\,l_{i}^{2}$ (notice that $V_{i}\in{\cal G}_{i}\setminus{\cal
G}_{i}^{0}$), i.e., $|V_{i}(l_{i})|\,l_{i}^{2}/(l_{i}/l_{j})\to 0$, while for
the limit sets $\Gamma^{\prime}_{V_{i}j}$, this ratio is non-zero:
$l_{i}/l_{j}\sim|V_{i}(l_{i})|\,l_{i}^{2}\to 0$. The unions

$\begin{array}[]{ll}\bar{\Gamma}_{ij}:=\Gamma_{ij}\cup\Gamma^{\prime}_{ij}=\left\\{(l_{i},l_{j})\to
0~{}|~{}l_{i}/l_{j}\to c\geq 0\right\\},\\\
\bar{\Gamma}_{V_{i}j}:=\Gamma_{V_{i}j}\cup\Gamma^{\prime}_{V_{i}j}=\left\\{(l_{i},l_{j})\to
0~{}\big{|}~{}l_{i}/l_{j}\to 0,\,|V_{i}(l_{i})|\,l_{i}\,l_{j}\to c\geq
0\right\\},\end{array}$ (24)

where $\bar{\Gamma}_{V_{i}j}\subset\Gamma_{ij}$ , will also be used from now
on. The full family of the $(l_{i},l_{j})$-projective paths, under which all
conditions (22) hold true, can be collected together as follows

$\left\\{\begin{array}[]{lllllllll}\Gamma^{\prime}_{ij}\\\
\bar{\Gamma}_{V_{i}j}\cup\Gamma^{\prime}_{ij}\\\ \bar{\Gamma}_{ij}\\\
\bar{\Gamma}_{V_{j}i}\cup\Gamma^{\prime}_{ji}\\\ \bar{\Gamma}_{ji}\\\
\bar{\Gamma}_{V_{i}j}\cup\bar{\Gamma}_{V_{j}i}\cup\Gamma^{\prime}_{ij}\\\
\bar{\Gamma}_{V_{j}i}\cup\bar{\Gamma}_{ij}\\\
\bar{\Gamma}_{V_{i}j}\cup\bar{\Gamma}_{ji}\\\
\bar{\Gamma}_{ij}\cup\bar{\Gamma}_{ji}\end{array}\right.\mbox{if}~{}~{}\left\\{\begin{array}[]{lllllllll}V_{i}\in{\cal
G}^{\prime}_{i}\,,~{}V_{j}\in{\cal G}^{\prime}_{j}\,,\\\ V_{i}\in{\cal
G}_{i}\setminus{\cal G}_{i}^{0}\,,~{}V_{j}\in{\cal G}^{\prime}_{j}\,,\\\
V_{i}\in{\cal G}^{0}_{i}\,,~{}V_{j}\in{\cal G}^{\prime}_{j}\,,\\\
V_{i}\in{\cal G}^{\prime}_{i}\,,~{}V_{j}\in{\cal G}_{j}\setminus{\cal
G}_{j}^{0}\,,\\\ V_{i}\in{\cal G}^{\prime}_{i}\,,~{}V_{j}\in{\cal
G}^{0}_{j}\,,\\\ V_{i}\in{\cal G}_{i}\setminus{\cal
G}_{i}^{0}\,,~{}V_{j}\in{\cal G}_{j}\setminus{\cal G}_{j}^{0}\,,\\\
V_{i}\in{\cal G}_{i}^{0}\,,~{}V_{j}\in{\cal G}_{j}\setminus{\cal
G}_{j}^{0}\,,\\\ V_{i}\in{\cal G}_{i}\setminus{\cal
G}_{i}^{0}\,,~{}~{}V_{j}\in{\cal G}_{j}^{0}\,,\\\ V_{i}\in{\cal
G}_{i}^{0}\,,~{}V_{j}\in{\cal G}_{j}^{0}\,,\end{array}\right.~{}~{}(i<j).$
(25)

Thus, under any of nine conditions (25) fulfilled on all the
$(l_{i},l_{j})$-faces, the $l\to 0$ limit of the $\Lambda$-matrix elements
$\lambda_{11}$ and $\lambda_{22}$ is finite. Next, under these conditions, we
also have $\lambda_{12}\to 0$ and this immediately follows from formulas (9)
and (11) because $q_{i}^{-1}t_{i}$’s and therefore all the terms in the series
$Q_{12}$ tend to zero. Contrary, in expressions (9) and (12), at least one of
the summands $q_{i}t_{i}$ and possibly some terms in the series $Q_{21}$ in
general diverge as $l\to 0$, except for some cases as all appearing
divergences will mutually canceled out and this situation will be analyzed
below. Hence, under the conditions ensuring the finite squeezed limits of
$\lambda_{11}$ and $\lambda_{22}$, the resulting point interactions are
separated. Thus, the above results can be summarized in the form of

Conclusion 2 : Let all the functions $V_{i}(l_{i})$, $i=1,\ldots,N$, belong to
any of the subsets ${\cal G}_{i}^{0}$, ${\cal G}_{i}\setminus{\cal G}_{i}^{0}$
or ${\cal G}^{\prime}_{i}$ of the $\bar{\cal G}_{i}$-set defined by conditions
(15), (18) and (19). Assume that for at least one layer, the function
$V_{i}(l_{i})$ belongs to ${\cal G}_{i}\setminus{\cal G}_{i}^{0}$ or ${\cal
G}^{\prime}_{i}$ . Then the element $\lambda_{21}$ diverges in the limit as
$l\to 0$ and under any of conditions (25) fulfilled on all the
$(l_{i},l_{j})$-faces, $i,j=1,\ldots,N$ ($i<j$), the family of one-point
interactions is realized in the squeezing limit with the two-sided boundary
conditions on the wave function $\psi(x)$ being of the Dirichlet type:
$\psi(\pm 0)=0$.

##  5 Resonant tunneling one-point interactions

We have derived above the conditions, under which the $l\to 0$ limit of the
matrix elements $\lambda_{11}$ and $\lambda_{22}$ is finite as well as the
limit relation $\lambda_{12}\to 0$. Here, on the basis of expressions (9) and
(12), we will analyze the element $\lambda_{21}$, which turns out to be the
most singular $\Lambda$-matrix element as $l\to 0$. The behavior of each term
in this element depends on the paths, along which the squeezing limit is
implemented.

As shown above, the squeezed $\delta$-potentials have been materialized for
all the strengths $V_{i}(l_{i})$ obeying the condition
$|V_{i}(l_{i})|\,l_{i}\to c\geq 0$ and the corresponding $l\to 0$ paths were
arbitrary, including various repeated limits $l_{i}\to 0$, $i=1,\ldots,N$. In
this (regular) case, the connection matrix is of form (16). In the following
for this particular case, the summands in $\sum_{i=1}^{N}q_{i}t_{i}$ will be
referred to as regular terms. Since no additional restrictions on this
behavior were imposed, the transmission through this point potential can be
referred to as a non-resonant tunneling.

The situation crucially changes if at least one of the terms $q_{i}$, for
which $V_{i}(l_{i})\in\bar{\cal G}_{i}\setminus{\cal G}_{i}^{0}$, is present
in series (9) and (12). In this (singular) case,
$|V_{i}(l_{i})|\,l_{i}\to\infty$ as $l\to 0$ and in spite of the validity of
conditions (25), the divergent terms in series (9) and (12) in general will be
present. We refer these terms to as singular ones. One can write
$\lambda_{21}=\lambda_{21}^{c}+\lambda_{21}^{\infty}$, where
$\lambda_{21}^{c}\to\alpha\in{\mathbb{R}}$ and each term from
$\lambda_{21}^{\infty}$ diverges as $l\to 0$. Under some conditions on the
functions $V_{i}(l_{i})$ and the paths converging to the origin $l=0$, the
cancellation of divergent terms in the group $\lambda_{21}^{\infty}$ can be
materialized, resulting in a finite limit value of $\lambda_{21}$. However,
this is possible if all the divergent terms are of the same order on certain
paths. Therefore it is necessary to ‘measure’ the divergence rate of these
terms by the multiplication them by an appropriate factor $L\to 0$, chosen in
such a way that all the resulting products are finite non-zero quantities in
the limit as $l\to 0$. In other words, a one-to-one correspondence between
these quantities and the divergent terms must be provided in the squeezed
limit. As a result, the equation for the cancellation of divergent terms is of
the form $\lim_{l\to 0}(\lambda_{21}^{\infty}L)=0$, being a constraint on some
limit parameters that describe the configuration $V_{1},\ldots,V_{N}$ and the
pencil sets. In the following, the conditions on these limit parameters will
be referred to as resonance equations and their solutions resonance sets (the
notion introduced by Golovaty and Man’ko [4]).

It is important to stress that the divergence rate of terms in series (9) and
(12) depends on the paths, on which the resonance equations are derived.
Consider, for instance, the convergence of a given term $q_{i}t_{i}$ on a
$(l_{i},l_{j})$-face, for which $V_{i}\in{\cal G}_{i}\setminus{\cal
G}^{0}_{i}$ . Approaching the origin along the paths from the pencil
$\Gamma^{\prime}_{ij}$ ($l_{i}/l_{j}\to c>0$), we have
$|q_{i}t_{i}|l_{j}\sim|q_{i}^{2}|l_{i}l_{j}\to 0$, while on the pencil
$\Gamma^{\prime}_{V_{i}j}$ ($l_{i}/l_{j}\sim|V_{i}|\,l_{i}^{2}\to 0$), this
limit is non-zero: $|q_{i}^{2}|l_{i}l_{j}\to c>0$. This means that the
divergence of the term $q_{i}^{2}l_{i}$ on the pencil $\Gamma^{\prime}_{ij}$
is lower than on $\Gamma^{\prime}_{V_{i}j}$ . Therefore the factor $L$ must be
chosen in such a way that all the divergent terms, participating in the
cancellation of divergences, have to be of the same order if this possible.

The limit parameters

$s_{i}:=\lim_{l_{i}\to\,0}(q_{i}l_{i}),~{}~{}\tau_{i}:=\lim_{l_{i}\to
0}(q_{i}l_{i}t_{i})=s_{i}\tan s_{i}\,,~{}~{}i=\overline{1,N},$ (26)

will be used for each strength $V_{i}\in{\cal G}^{\prime}_{i}$ . Next, on
paths (23), we also define

$\chi_{ij}:=\lim_{(l_{i},\,l_{j})\in\Gamma^{\prime}_{ij}}\\!(l_{i}/l_{j})=\chi_{ji}^{-1}~{}~{}~{}\mbox{and}~{}~{}~{}\eta_{ij}:=\lim_{(l_{i},\,l_{j})\in\Gamma^{\prime}_{V_{i}j}}\\!(q_{i}^{2}l_{i}l_{j})$
(27)

for $V_{i}\in{\cal G}_{i}$ , where $i,j=1,\ldots,N$ ($i\neq j)$. Clearly, if
$V_{i}\in{\cal G}_{i}^{0}$, then $\eta_{ij}\equiv 0$. Thus, the cancellation
of divergences imposes some constraints on the configurations
$V_{1}(l_{1}),\ldots,V_{N}(l_{N})$ and the path pencils, which have to be
defined properly in the limit as $l\to 0$.

Let us consider two simple examples illustrating how the cancellation
procedure does work, namely the pairwise configurations (i) $V_{i}\in{\cal
G}^{\prime}_{i}$ and $V_{j}\in{\cal G}^{\prime}_{j}$ , and (ii) $V_{i}\in{\cal
G}_{i}\setminus{\cal G}_{i}^{0}$ and $V_{j}\in{\cal G}^{\prime}_{j}$, whereas
the rest of strengths are regular terms. In case (i), the cancellation
equation reads $\lambda_{21}^{\infty}=q_{i}t_{i}+q_{j}t_{j}=0$ and both the
divergent terms in this equation are of the same rate in the squeezing limit
if the $(l_{i},l_{j})$-paths belong to the pencil $\Gamma^{\prime}_{ij}$ .
Therefore this equation can be multiplied by either $l_{i}$ or $l_{j}$. Thus,
multiplying it by $l_{i}$ , in the limit as $l_{i}\to 0$ and $l_{j}\to 0$, we
arrive at the resonance equation

$\tau_{i}+\chi_{ij}\tau_{j}=0,$ (28)

written in terms of limit parameters (26) and (27), which in their turn depend
only on the asymptotic behavior of the layer parameters $V_{i}(l_{i})$,
$V_{j}(l_{j})$, $l_{i}$ and $l_{j}$. In particular, on the paths with
$l_{i}=l_{j}$ and for the configuration $V_{i}(l_{i})=-V_{j}(l_{j})$, equation
(28) reduces to [11]

$\tan s_{i}=\tanh s_{i},~{}~{}~{}s_{i}=\lim_{l_{i}\to
0}|V_{i}(l_{i})|^{1/2}l_{i}\,.$ (29)

In the multi-layer case described above, the solutions
$\\{s_{i,n}\\}_{n=-\infty}^{\infty}$ of this equation form the discrete
resonance set of the point interaction that corresponds to the potential
$V(x)=a\delta^{\prime}(x)+b\delta(x)$. At the values $a=s_{i,n}$ , this
interaction is partially transparent, whereas beyond these values the
transmission is zero.

Similarly, in the other case (ii), it would be possible to multiply the
equation $\lambda_{21}^{\infty}=q_{i}^{2}l_{i}+q_{j}t_{j}=0$, where both the
summands diverge as $(l_{i},l_{j})\to 0$ ($V_{i}\in{\cal G}_{i}\setminus{\cal
G}^{0}_{i}$ and $V_{j}\in{\cal G}^{\prime}_{j}$), by $l_{i}$ . However, in
this case, $|q_{i}^{2}|\,l_{i}^{2}\to 0$, while the other term
$\chi_{ij}\tau_{j}$ is non-zero. This means that the divergence of the terms
$q_{i}^{2}l_{i}$ and $q_{j}t_{j}$ is of different order on the pencil
$\Gamma^{\prime}_{ij}$ , where $l_{i}$ and $l_{j}$ approach the origin
linearly. Therefore this pencil is not appropriate for the cancellation of
divergences. Instead, let us consider the cancellation of divergences on the
pencil $\Gamma^{\prime}_{V_{i}j}$ of curved paths, for which
$l_{i}/l_{j}\sim|V_{i}(l_{i})|\,l_{i}^{2}\to 0$, where the divergence rate of
both terms is of the same order:
$|q_{i}^{2}|\,l_{i}\sim|V_{i}(l_{i})|\,l_{i}\sim l_{j}^{-1}$ and
$|q_{j}t_{j}|\sim|V_{j}(l_{j})|\sim l_{j}^{-1}$. In this case, the parameter
$L=l_{j}$ can be used as a multiplier for the comparison of the divergence of
$q_{i}^{2}l_{i}$ and $q_{j}t_{j}$ . Then the resonance equation is well-
defined on the $\Gamma^{\prime}_{V_{i}j}$-paths and it can be written in terms
of finite quantities as follows

$\lim_{(l_{i},l_{j})\in\Gamma^{\prime}_{V_{i}j}}\\!\\!(q_{i}^{2}l_{i}+q_{j}t_{j})\,l_{j}=\eta_{ij}+\tau_{j}=0.$
(30)

For a negative strength $V_{j}$ ($s_{j}>0$), this equation (with respect to
the variable $s_{j}$) admits a countable set of solutions. The configuration
with some $V_{i}\in{\cal G}_{i}^{0}$ can also be considered and then in
equation (30) we have $\eta_{ij}=0$. This situation is the limiting case as
the divergence vanishes due to the solution of the equation $\tau_{j}=0$,
forming the resonance set consisting of points $s_{j}=n_{j}\pi$ with integers
$n_{j}$.

Since, in the limit as $l\to 0$, the dyads $D_{ij}$ are finite under
conditions (25), for the systems with $N\geq 3$, the divergence rate of the
terms in series (12) does not exceed that of $q_{i}t_{i}$’s. Moreover, due to
the presence of the triads

$T_{ijk}:=D_{ij}q_{k}t_{k}=(q_{i}q_{k}/q_{j})t_{i}t_{j}t_{k}~{}~{}~{}(1\leq
i<j<k\leq N)$ (31)

in the $Q_{21}$-series, rewritten in the form

$\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\cal
Q}_{21}=\sum_{n=1}^{[(N-1)/2]}(-1)^{n}\\!\\!\sum_{1=i_{1}<j_{1}<\ldots<i_{n}<j_{n}<k_{n}=N}D_{i_{1}j_{1}}\ldots\stackrel{{\scriptstyle\surd}}{{D}}_{i_{m}j_{m}}\ldots
D_{i_{n}j_{n}}T_{i_{m}j_{m}k_{n}},~{}~{}$ (32)

where the products $D_{i_{m}j_{m}}q_{k_{n}}t_{k_{n}}$ have been replaced by
the triads $T_{i_{m}j_{m}k_{n}}$ and the symbol $\surd$ stands for the absence
of an indicated term (here the factor ${D}_{i_{m}j_{m}}$), it is possible that
the $l\to 0$ limit of these products will be of type $0\cdot\infty\sim c\geq
0$ if $D_{ij}\to 0$. Hence, depending on the paths, there are two
possibilities: (i) the triads $T_{ijk}$ are divergent and the order of their
divergence is the same as that of the terms $q_{i}t_{i}$’s and (ii)
$|T_{ijk}|\to c\geq 0$, leading to the cancellation of divergences only in the
sum $\sum_{i=1}^{N}q_{i}t_{i}$. Clearly, some terms in the latter sum as well
as in the series $Q_{21}$ may be finite belonging to the group
$\lambda_{21}^{c}$.

Let us analyze the asymptotic behavior of a $T_{ijk}$-triad, where all the
strengths $V_{i}$ , $V_{j}$ and $V_{k}$ belong to the corresponding ${\cal
G}^{\prime}$-sets. In this situation, we have the following asymptotic
relation:

$T_{ijk}\sim(l_{j}/l_{i}l_{k})\,\tau_{i}\tau_{j}\tau_{k}/s_{j}^{2}\,\sim\,l_{j}^{-1}\chi_{ji}\,\chi_{jk}\,\tau_{i}\tau_{j}\tau_{k}/s_{j}^{2}.$
(33)

Hence, in the linear squeezing limit ($l_{i}\sim l_{j}\sim l_{k}$), $T_{ijk}$
diverges as $l_{i}^{-1}\sim l_{j}^{-1}\sim l_{k}^{-1}$. Therefore, if one
would like to have the limit $|T_{ijk}|\to c\geq 0$, it is necessary that at
least $l_{j}/l_{i}\to 0$ or $l_{j}/l_{k}\to 0$ sufficiently fast. This
nonlinear behavior could be realized on the adjoint paths $\Gamma_{V_{j}i}$ or
$\Gamma_{V_{j}k}$ , but the strength $V_{j}$ in this case must belong to
${\cal G}_{j}\setminus{\cal G}_{j}^{0}$, that contradicts with the assumption
$V_{j}\in{\cal G}^{\prime}_{j}$ . Therefore the only possibility for
materializing a finite limit of $T_{ijk}$ is a nonlinear extension of the
linear squeezing. To this end, we introduce a parameter $\sigma\in[1,\infty)$
and extend the notions of the boundary pencils $\Gamma^{\prime}_{ij}$ and
$\Gamma^{\prime}_{V_{i}j}$ given in equations (23) to

$\begin{array}[]{ll}\Gamma^{\sigma}_{ij}~{}\,:=\left\\{(l_{i},\,l_{j})\to
0~{}\big{|}~{}l_{i}^{1/\sigma}/l_{j}\to c>0\right\\},\\\
\Gamma^{\sigma}_{V_{i}j}:=\left\\{V_{i}(l_{i})\in{\cal G}_{i}\setminus{\cal
G}_{i}^{0}\,,~{}(l_{i},\,l_{j})\in\Gamma_{ij}~{}\big{|}~{}|V_{i}(l_{i})|\,l_{i}\,l_{j}^{1/\sigma}\to
c>0\right\\},\end{array}$ (34)

so that
$\Gamma^{\sigma}_{ij}|_{\sigma=1}\equiv\Gamma^{\prime}_{ij}=\Gamma^{\prime}_{ji}$
and $\Gamma^{\sigma}_{V_{i}j}|_{\sigma=1}\equiv\Gamma^{\prime}_{V_{i}j}$ .
Since $\sigma\geq 1$, we have $\Gamma_{ij}^{\sigma}\subset\bar{\Gamma}_{ij}$
and $\Gamma_{V_{i}j}^{\sigma}\subset\bar{\Gamma}_{V_{i}j}$ . Therefore
conditions (25) are not violated on paths (34). Note that for the paths
$\Gamma_{V_{i}j}^{\sigma}$, as follows from definition (34), we have the
relation $l_{i}/l_{j}^{1/\sigma}\sim|V_{i}(l_{i})|\,l_{i}^{2}\to 0$.
Accordingly, for each $\sigma$ we introduce a new set of the functions
$V_{i}(l_{i})$ as

${\cal G}^{\sigma}_{i}:=\left\\{V_{i}(l_{i})\in\bar{\cal G}_{i}\setminus{\cal
G}^{0}_{i}~{}\big{|}~{}\lim_{l_{i}\to
0}|V_{i}(l_{i})|\,l_{i}^{1+1/\sigma}=c>0\right\\},~{}1\leq\sigma<\infty,$ (35)

where ${\cal G}^{\sigma}_{i}\big{|}_{\sigma=1}\equiv{\cal G}^{\prime}_{i}$ .
Next, for $\sigma\in(1,\infty)$, we have ${\cal G}^{\sigma}_{i}\subset{\cal
G}_{i}\setminus{\cal G}^{0}_{i}$ and $\lim_{\sigma\to\infty}{\cal
G}^{\sigma}_{i}={\cal G}^{0}_{i}$, so that the pencil ${\cal G}^{\sigma}_{i}$
is found between the sets ${\cal G}^{0}_{i}$ and ${\cal G}^{\prime}_{i}$ .

Next, we have to extend the definition of the limit quantities $\chi_{ij}$ and
$\eta_{ij}$ given by equations (27) for all values $\sigma\in[1,\infty)$.
Thus, we define ($i,j=1,\ldots,N$)

$\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\chi_{ij}^{\sigma}=\lim_{(l_{i},\,l_{j})\in\Gamma^{\sigma}_{ij}}\\!(l_{i}^{1/\sigma}/l_{j}),~{}i\neq
j,~{}~{}\mbox{and}~{}~{}\eta_{ij}^{\sigma}:=\lim_{(l_{i},\,l_{j})\in\Gamma^{\sigma}_{V_{i}j}}\\!\\!(q_{i}^{2}l_{i}l_{j}^{1/\sigma})~{}~{}\mbox{if}~{}~{}V_{i}\in{\cal
G}_{i},~{}~{}$ (36)

where the definition of the pencils $\Gamma^{\sigma}_{ij}$ and
$\Gamma^{\sigma}_{V_{i}j}$ is given in equations (34). According to notations
(27), $\chi_{ij}^{\sigma}|_{\sigma=1}\equiv\chi_{ij}=\chi_{ji}^{-1}$,
$\eta_{ij}^{\sigma}|_{\sigma=1}\equiv\eta_{ij}$ and $\eta_{ij}^{\sigma}\equiv
0$ if $V_{i}\in{\cal G}^{0}_{i}$ . Contrary to the case $\sigma=1$, for
$\sigma>1$ it is possible to consider the particular case $j=i$. From
definition (35), we have $|V_{i}|\sim l_{i}^{-(1+1/\sigma)}$ for
$V_{i}\in{\cal G}^{\sigma}_{i}$ , $i=1,\ldots,N$.

Thus, asymptotic relation (33) can be generalized to

$T_{ijk}\sim(l_{j}^{2/\sigma}/l_{i}l_{k})\,l_{j}^{1-2/\sigma}\tau_{i}\tau_{j}\tau_{k}/s_{j}^{2}\,\sim\,l_{j}^{1-2/\sigma}\chi_{ji}^{\sigma}\,\chi_{jk}^{\sigma}\,\tau_{i}\tau_{j}\tau_{k}/s_{j}^{2}\,,$
(37)

being valid on the paths, where $l_{j}^{1/\sigma}\sim l_{i}$ and
$l_{j}^{1/\sigma}\sim l_{k}$ , i.e., on the pencil projections
$\Gamma^{\prime}_{ik}$ , $\Gamma_{ji}^{\sigma}$ and $\Gamma_{jk}^{\sigma}$ .
Hence, representation (37) coincides with (33) at $\sigma=1$ and at the same
time it also holds true on the whole interval $2\leq\sigma<\infty$. Exactly,
at $\sigma=2$, it was possible to prove the existence of a bound state for the
pure $\delta^{\prime}$-potential realized on the resonance set [36].

In addition to (37), where $V_{j}\in{\cal G}^{\prime}_{j}$ , one can write the
asymptotic representation for the other ‘lateral’ configurations
$V_{i}\in{\cal G}_{i}$ and $V_{k}\in{\cal G}_{k}$ as follows

$T_{ijk}\sim l_{j}^{1-2/\sigma}\left\\{\begin{array}[]{lll}\vskip 3.0pt plus
1.0pt minus
1.0pt\chi_{jk}^{\sigma}\,\eta_{ij}^{\sigma}\,\tau_{j}\tau_{k}/s_{j}^{2}\\\
\chi_{ji}^{\sigma}\,\eta_{kj}^{\sigma}\,\tau_{i}\tau_{j}/s_{j}^{2}\par\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\eta_{ij}^{\sigma}\,\eta_{kj}^{\sigma}\,\tau_{j}/s_{j}^{2}\end{array}\right.~{}\mbox{for}~{}\left\\{\begin{array}[]{lll}\vskip
3.0pt plus 1.0pt minus 1.0ptV_{i}\in{\cal G}_{i}\setminus{\cal
G}^{0}_{i}\,,~{}V_{k}\in{\cal G}^{\prime}_{k}\,,\\\ V_{i}\in{\cal
G}^{\prime}_{i}\,,~{}V_{k}\in{\cal G}_{k}\setminus{\cal
G}_{k}^{0}\,,\par\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ V_{i}\in{\cal
G}_{i}\setminus{\cal G}^{0}_{i}\,,~{}V_{k}\in{\cal G}_{k}\setminus{\cal
G}_{k}^{0}\end{array}\right.$ (38)

and

$T_{ijk}\sim l_{j}^{1-1/\sigma}\left\\{\begin{array}[]{lllll}\vskip 3.0pt plus
1.0pt minus 1.0pt\chi_{jk}^{\sigma}\,\tau_{j}\tau_{k}/s_{j}^{2}\\\
\eta_{kj}^{\sigma}\,\tau_{j}/s_{j}^{2}\par\vskip 3.0pt plus 1.0pt minus
1.0pt\\\ \chi_{ji}^{\sigma}\,\tau_{i}\tau_{j}/s_{j}^{2}\par\vskip 3.0pt plus
1.0pt minus 1.0pt\\\ \eta_{ij}^{\sigma}\,\tau_{j}/s_{j}^{2}\par\vskip 3.0pt
plus 1.0pt minus 1.0pt\\\
\,0\end{array}\right.~{}\mbox{for}~{}\left\\{\begin{array}[]{lllll}\vskip
3.0pt plus 1.0pt minus 1.0ptV_{i}\in{\cal G}^{0}_{i}\,,~{}V_{k}\in{\cal
G}^{\prime}_{k}\,,\\\ V_{i}\in{\cal G}^{0}_{i}\,,~{}V_{k}\in{\cal
G}_{k}\setminus{\cal G}_{k}^{0}\,,\par\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
V_{i}\in{\cal G}^{\prime}_{i}\,,~{}V_{k}\in{\cal G}_{k}^{0}\,,\par\vskip 3.0pt
plus 1.0pt minus 1.0pt\\\ V_{i}\in{\cal G}_{i}\setminus{\cal
G}^{0}_{i}\,,~{}V_{k}\in{\cal G}_{k}^{0}\,,\par\vskip 3.0pt plus 1.0pt minus
1.0pt\\\ V_{i}\in{\cal G}^{0}_{i}\,,~{}V_{k}\in{\cal
G}_{k}^{0}\,.\end{array}\right.$ (39)

Equations (33) and (37)–(39) can also be used for the case as $V_{j}\in{\cal
G}_{j}$, just setting in these formulas $\tau_{j}/s^{2}_{j}=1$. The asymptotic
representation given by relations (38) holds true for $\sigma=1$ and
$\sigma\in[2,\infty)$, whereas representation (39) is valid on the whole
interval $1\leq\sigma<\infty$. Below the cases $\sigma=1$ and
$1<\sigma<\infty$ will be considered separately.

###  5.1 Conditions for the existence of resonance sets on the paths with
$\sigma=1$

The cancellation procedure can particularly be performed on the paths with
$\sigma=1$, which are defined by the pencil projections $\Gamma^{\prime}_{ij}$
and $\Gamma^{\prime}_{V_{i}j}$ [see relations (23)]. Consider first the case
$N=3$, where series (32) reduces to the single summand $Q_{21}=T_{ijk}$
($i=1$, $j=2$, $k=3$). Assume that all the strengths belong to either ${\cal
G}^{\prime}$\- or ${\cal G}\setminus{\cal G}^{0}$-sets. Then, according to the
asymptotic representation given by relations (37) and (38) with $\sigma=1$,
the $T_{ijk}$-triad diverges as $l_{j}^{-1}\sim l_{i}^{-1}\sim l_{k}^{-1}$.
Hence, all the terms in $\lambda_{21}$ are divergent and the condition for the
cancellation of divergences yields the equation

$q_{i}t_{i}+q_{j}t_{j}+q_{k}t_{k}=T_{ijk}\,.$ (40)

For the comparison of the divergence rate of all the terms on the paths with
$\sigma=1$, this equation can be multiplied by one of the widths $l_{i}$,
$l_{j}$ or $l_{k}$ , but $V_{i}$, $V_{j}$ or $V_{k}$ must belong to the
corresponding ${\cal G}^{\prime}$-sets. More precisely, if for instance, the
strength $V_{j}$ belongs to ${\cal G}_{j}\setminus{\cal G}_{j}^{0}$, resulting
in the behavior $|q_{j}t_{j}|\sim|q_{j}^{2}|\,l_{j}\to\infty$, the
multiplication of this term by $l_{j}$ leads to its disappearance as $l_{j}\to
0$ because $|q_{j}|\,l_{j}\to 0$. Therefore the width $l_{j}$ cannot be used
here for ‘measuring’ the rate of the divergence of $|q_{j}^{2}|\,l_{j}$ and in
this case another parameter, say $l_{i}$ or $l_{k}$ should be chosen as a
multiplier if $V_{i}\in{\cal G}^{\prime}_{i}$ or $V_{k}\in{\cal
G}^{\prime}_{k}$, respectively. Thus, at least the strength in one of the
summands of the left-hand side of equation (40), where all the terms are
singular, has to belong to the corresponding ${\cal G}^{\prime}$-set.

Consider now the situation as $V_{i}\in{\cal G}^{\prime}_{i}$ and
$V_{j}\in{\cal G}_{j}\setminus{\cal G}_{j}^{0}$. According to the above
arguments, in this case, equation (40) can be multiplied by $l_{i}$ . For
$V_{k}\in{\cal G}^{\prime}_{k}$ , the resulting products make sense on the
paths, where $l_{j}/l_{i}\sim|V_{j}|\,l_{j}^{2}\to 0$, $l_{i}\sim l_{k}$ and
$l_{j}\sim l_{k}$ [see representation (33)]. However, the last two relations
contradict with the first one. Similarly, for $V_{k}\in{\cal
G}_{k}\setminus{\cal G}_{k}^{0}$ , we have
$l_{j}/l_{i}\sim|V_{j}|\,l_{j}^{2}\to 0$,
$l_{k}/l_{i}\sim|V_{k}|\,l_{k}^{2}\to 0$ and
$l_{k}/l_{j}\sim|V_{k}|\,l_{k}^{2}\to 0$. From the last two relations, one
obtains that $l_{i}\sim l_{j}$ and this again contradicts with the first
relation. Thus, the configuration of the singular strengths $V_{i}$, $V_{j}$
and $V_{k}$ in equation (40), where the ‘middle’ strength $V_{j}$ belongs to
${\cal G}_{j}\setminus{\cal G}_{j}^{0}$ cannot be used for deriving resonance
equations in the squeezing limit. In other words, for the strength
configurations with ${\cal G}_{j}\setminus{\cal G}_{j}^{0}$, it is impossible
to find appropriate paths, on which the divergent terms are of the same order.
However, if $V_{j}\in{\cal G}^{\prime}_{j}$ , multiplying then equation (40)
by $l_{j}$ and using asymptotic relations (33) and (38) with $\sigma=1$, one
obtains the correct resonance equations, which are collected in table 1.

Table 1:  Resonance equations for a three-layer structure defined on the corresponding paths in the case as $V_{j}\in{\cal G}^{\prime}_{j}$ .  $V_{i}(l_{i}),~{}V_{k}(l_{k})$ | Resonance equations | Paths
---|---|---
$V_{i}\in{\cal G}^{\prime}_{i}\,,~{}V_{k}\in{\cal G}^{\prime}_{k}$ | $\chi_{ji}\tau_{i}+\tau_{j}+\chi_{jk}\tau_{k}=\chi_{ji}\chi_{jk}\tau_{i}\tau_{j}\tau_{k}/s^{2}_{j}$ | $\Gamma^{\prime}_{ij}\,,\,\Gamma^{\prime}_{jk}$
$V_{i}\in{\cal G}_{i}\setminus{\cal G}^{0}_{i}\,,~{}V_{k}\in{\cal G}^{\prime}_{k}$ | $\eta_{ij}+\tau_{j}+\chi_{jk}\tau_{k}=\eta_{ij}\chi_{jk}\tau_{j}\tau_{k}/s_{j}^{2}$ | $\Gamma^{\prime}_{V_{i}j}\,,\,\Gamma^{\prime}_{jk}$
$V_{i}\in{\cal G}^{\prime}_{i}\,,~{}V_{k}\in{\cal G}_{k}\setminus{\cal G}_{k}^{0}$ | $\chi_{ji}\tau_{i}+\tau_{j}+\eta_{kj}=\eta_{kj}\chi_{ji}\tau_{i}\tau_{j}/s_{j}^{2}$ | $\Gamma^{\prime}_{ij}\,,\,\Gamma^{\prime}_{V_{k}j}$
$V_{i}\in{\cal G}_{i}\setminus{\cal G}^{0}_{i}\,,~{}V_{k}\in{\cal G}_{k}\setminus{\cal G}_{k}^{0}$ | $\eta_{ij}+\tau_{j}+\eta_{kj}=\eta_{ij}\,\eta_{kj}\tau_{j}/s_{j}^{2}$ | $\Gamma^{\prime}_{V_{i}j}\,,\,\Gamma^{\prime}_{V_{k}j}$

Here, all the four terms in equation (40) are divergent being of the same
order, so that all the terms in the resonance equations are finite and non-
zero. Note that for the configurations pointed out in the second, the third
and the fourth lines in table 1, it is impossible to derive resonance
equations on the pencils $\\{\Gamma^{\prime}_{ij}\,,\,\Gamma^{\prime}_{jk}\\}$
because the divergent terms of equation (40) in this situation are of
different order.

Next, we have to consider the configurations $V_{i}$, $V_{j}$ and $V_{k}$
($N=3$), where one or two strengths belongs to ${\cal G}^{0}$-sets. Thus,
setting in table 1 $\tau_{j}=0$ and $\tau_{j}/s^{2}_{j}=1$, one obtains the
resonance equations for the configuration with $V_{j}\in{\cal G}^{0}_{j}$ .
The general form of these equations reads

$A_{ij}+A_{kj}=A_{ij}A_{kj}\,,$ (41)

where ($i\neq j$)

$A_{ij}:=\left\\{\begin{array}[]{ll}\lim_{(l_{i},\,l_{j})\in\Gamma^{\prime}_{ij}}(q_{i}t_{i}l_{j})=\chi_{ji}\tau_{i}&\mbox{if}~{}~{}V_{i}\in{\cal
G}^{\prime}_{i}\,,\\\
\lim_{(l_{i},\,l_{j})\in\Gamma^{\prime}_{V_{i}j}}(q_{i}^{2}l_{i}l_{j})=\eta_{ij}&\mbox{if}~{}~{}V_{i}\in{\cal
G}_{i}\setminus{\cal G}_{i}^{0}\,.\end{array}\right.$ (42)

The solution of equations (41) are curves on the $\\{A_{ij},A_{kj}\\}$-plane.
The corresponding resonance sets have been analyzed in detail for various
situations earlier in works [34, 36]. Thus, we conclude that equation (40)
with the singular lateral terms $q_{i}t_{i}$ and $q_{k}t_{k}$ multiplied by
$l_{j}$ makes sense only if the middle strength $V_{j}\in{\cal
G}^{\prime}_{j}\cup{\cal G}_{j}^{0}$.

Finally, for the case $N=3$, where one of the lateral strengths $V_{i}$ or
$V_{k}$ (with the $\delta$-like shrinking) belongs to ${\cal G}^{0}_{i}$ or
${\cal G}^{0}_{k}$, respectively, we have to use asymptotic representation
(39), where $|T_{ijk}|\to c\geq 0$. For instance, for the structure with
$V_{i}\in{\cal G}^{\prime}_{i}$ , $V_{j}\in{\cal G}^{\prime}_{j}$ and
$V_{k}\in{\cal G}^{0}_{k}$ , multiplying equation (40) by $l_{i}$ , we arrive
at equation (28). Similarly, for $V_{i}\in{\cal G}^{\prime}_{i}$ ,
$V_{j}\in{\cal G}_{j}\setminus{\cal G}_{j}^{0}$ and $V_{k}\in{\cal G}^{0}_{k}$
, one obtains $\tau_{i}+\eta_{ji}=0$, in fact, equation (30). In both these
situations, the cancellation of divergences occurs only between two terms in
equation (40) as in a double-layer system.

The above arguments can be extended to the general case with $N>3$. Consider
the configuration $V_{1}(l_{1}),\ldots,V_{N}(l_{N})$, for which at least three
terms $q_{i}t_{i}$, $q_{j}t_{j}$ and $q_{k}t_{k}$ ($1\leq i<j<k\leq N$)
diverge in the squeezing limit. Then, according to asymptotic relations (37)
and (38), the corresponding triad $T_{ijk}$ will also be a divergent term at
$\sigma=1$. Let the lateral strengths $V_{i}$ and $V_{k}$ belong to the
corresponding ${\cal G}^{\prime}$\- or ${\cal G}\setminus{\cal G}^{0}$-sets,
while $V_{j}\in{\cal G}_{j}\setminus{\cal G}_{j}^{0}$. For the comparison of
the divergence rate of these four terms, one can multiply them by some $l_{m}$
, $m=1,\ldots,N$ ($m\neq i,j,k$). Thus, multiplying the sum
$q_{i}t_{i}+q_{j}^{2}l_{j}+q_{k}t_{k}$ ($V_{i}\in{\cal G}^{\prime}_{i}$ and
$V_{k}\in{\cal G}^{\prime}_{k}$) and $T_{ijk}=q_{i}t_{i}\,l_{j}\,q_{k}t_{k}$
by $l_{m}$, one obtains that the resulting products will be finite and non-
zero quantities on the paths characterized by the relations $l_{i}\sim l_{m}$,
$l_{j}/l_{m}\sim|V_{j}|\,l_{j}^{2}\to 0$, $l_{k}\sim l_{m}$ and $l_{j}\sim
l_{k}$ . From the last two relations we have $l_{j}\sim l_{m}$, but this
contradicts with $l_{j}/l_{m}\to 0$. Next, for instance, in the case with
$V_{i}\in{\cal G}_{i}\setminus{\cal G}_{i}^{0}$ and $V_{k}\in{\cal
G}_{k}\setminus{\cal G}_{k}^{0}$, one obtains the relations
$l_{i}/l_{m}\sim|V_{i}|\,l_{i}^{2}\to 0$,
$l_{j}/l_{m}\sim|V_{j}|\,l_{j}^{2}\to 0$,
$l_{k}/l_{m}\sim|V_{k}|\,l_{k}^{2}\to 0$ and
$l_{k}/l_{j}\sim|V_{k}|\,l_{k}^{2}\to 0$. Again, from the last two relations
we obtain $l_{j}\sim l_{m}$ that contradicts with $l_{j}/l_{m}\to 0$. This
zero convergence is replaced by $l_{j}\sim l_{m}$ if $V_{j}\in{\cal
G}^{\prime}_{j}$ , so that the contradiction vanishes. In the case as
$V_{j}\in{\cal G}_{j}^{0}$ , the restriction $l_{j}/l_{m}\to 0$ is absent.
Hence, the ‘internal’ strengths $V_{j}\in{\cal G}_{j}\setminus{\cal
G}_{j}^{0}$ with $j=2,\ldots,N-1$ have to be excluded in the procedure of
deriving resonance equations and, as a result, each strength $V_{j}$ in the
sum $\sum_{i=2}^{N-1}q_{i}t_{i}$ must belong to either ${\cal G}^{\prime}_{j}$
or ${\cal G}^{0}_{j}$. Note that the lateral terms in the sum
$\sum_{i=1}^{N}q_{i}t_{i}$ (from one to several ones) may belong to ${\cal
G}^{0}$-sets, so that the next neighboring strengths appear to be singular
ones. Then, the applying of relations (39) results in the same result: for the
existence of resonance equations, the internal strengths must belong to ${\cal
G}^{\prime}\cup{\cal G}^{0}$-sets.

Thus, multiplying the element $\lambda_{21}$ given by series (9) and (12) by
one of $l_{j}$ with $j=2,\ldots,N-1$, from the condition $\lim_{l\to
0}(\lambda_{21}^{\infty}\,l_{j})=\lim_{l\to 0}(\lambda_{21}l_{j})=0$, we
obtain the limit equation

$\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{i=1}^{N}A_{ij}\,+\\!\\!\sum_{n=1}^{[(N-1)/2]}(-1)^{n}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{1=i_{1}<j_{1}<i_{2}<j_{2}<\ldots<i_{n}<j_{n}<k_{n}=N}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(A_{i_{1}j_{1}}\tau_{j_{1}}/s^{2}_{j_{1}})\ldots(A_{i_{n}j_{n}}\tau_{j_{n}}/s^{2}_{j_{n}})A_{k_{n}j}=0,~{}~{}$
(43)

where we have used the limits

$\left\\{\begin{array}[]{llll}\lim_{(l_{i},\,l_{j})\in\Gamma^{\prime}_{ij}}D_{ij}\\\
\lim_{(l_{i},\,l_{j})\in\Gamma^{\prime}_{ij}}D_{ij}\\\
\lim_{(l_{i},\,l_{j})\in\Gamma^{\prime}_{V_{i}j}}D_{ij}\\\
\lim_{(l_{i},\,l_{j})\in\Gamma^{\prime}_{V_{i}j}}D_{ij}\end{array}\right\\}=A_{ij}\tau_{j}/s_{j}^{2}~{}~{}\mbox{if}~{}~{}\left\\{\begin{array}[]{llll}V_{i}\in{\cal
G}^{\prime}_{i}\,,~{}V_{j}\in{\cal G}^{\prime}_{j}\,,\\\ V_{i}\in{\cal
G}^{\prime}_{i}\,,~{}V_{j}\in{\cal G}_{j}\,,\\\ V_{i}\in{\cal
G}_{i}\setminus{\cal G}_{i}^{0}\,,\,V_{j}\in{\cal G}^{\prime}_{j}\,,\\\
V_{i}\in{\cal G}_{i}\setminus{\cal G}_{i}^{0}\,,\,V_{j}\in{\cal
G}_{j}\,.\end{array}\right.$ (44)

Here, the limit quantities $A_{ij}$ are defined by equations (42). In addition
to these equations, notice that $A_{ij}=0$ if $V_{i}\in{\cal G}^{0}_{i}$
($i\neq j$). Next, in equation (43), $A_{jj}=\tau_{j}$ if $V_{j}\in{\cal
G}^{\prime}_{j}$ , while $A_{jj}=0$ and $\tau_{j}/s^{2}_{j}=1$ if
$V_{j}\in{\cal G}_{j}^{0}$. Clearly, all the terms from the group
$\lambda^{c}_{21}$ disappear in equation (43). All the equations obtained from
the condition $\lim_{l\to 0}(\lambda_{21}l_{j})=0$, where $j=2,\ldots,N-1$,
are equivalent, so that the resonance sets do not depend on the subscript $j$
in equation (43).

Thus, the solutions of resonance equation (43) determine the conditions on the
limit parameters $\chi_{ij}$, $s_{i}$ and $\eta_{ij}$ under which a non-zero
(resonant) transmission through the squeezed structure is possible. These
conditions form one or several hypersurfaces in the
$\\{\ldots,\chi_{ji},\ldots,s_{i},\ldots,\eta_{ij},\ldots\\}$-space. The
number and structure of resonance sets (hypersurfaces) depends on the
configuration $V_{1}(l_{1}),\ldots,V_{N}(l_{N})$ and the corresponding paths
converging to the origin $l=0$. For a fixed number of layers $N$, several
resonance sets can exist and its number increases with $N$.

Finally, consider the particular case as all the functions $V_{i}(l_{i})$ are
non-negative. For realizing a non-trivial point interaction in the singular
squeezing limit, assume that some of these functions $V_{i}(l_{i})\in\bar{\cal
G}_{i}\setminus{\cal G}_{i}^{0}$ are positive (barriers). Then
$s_{i}\in{\mathbb{I}}$, so that $\tau_{i}<0$ ($s^{2}_{i}<0$) and
$\eta_{ij}<0$. As a result, all the terms in equation (43) become negative,
and consequently, the total left-hand expression in this equation will also be
negative. Hence, in the case of the absence of well-like potential strengths
$V_{i}$ , there are no cancellations of divergences resulting in a finite
limit of $\lambda_{21}$ .

However, in the case if at least one well-like potential strength ($V_{i}<0$)
is present, equation (43) admits a countable set of non-trivial solutions due
to the functions $\tan s_{i}$ with $s_{i}\in{\mathbb{R}}$. This leads to the
realization of non-separated point interactions in the squeezing limit. For
instance, one of the particular cases illustrating the existence of solutions
to equation (43) is the configuration, where each divergent potential strength
$V_{i}$ belongs to the set ${\cal G}^{\prime}_{i}$ and all these potential
strengths are defined on the paths $\Gamma^{\prime}_{ij}$. In this case, all
the divergent terms in (43) are of the same order. Hence, in the presence of
at least one well-like potential strength, the cancellation of divergences
provides the appearance of non-separated point interactions in the squeezing
limit. Thus, from the above arguments we single out the results, which can be
formulated as

Conclusion 3 : (a) The resonance sets for $\sigma=1$ are obtained as solutions
to limit equation (43) defined for the configurations
$V_{1}(l_{1}),\ldots,V_{N}(l_{N})$, $i=1,\ldots,N$, where the lateral
strengths $V_{i}$ ($i=1,N$) are singular, i.e., they belong to either the
${\cal G}^{\prime}$\- or ${\cal G}\setminus{\cal G}^{0}$-sets, whereas the
internal strengths $V_{i}$ ($i=2,\ldots,N-1$) are from the ${\cal
G}^{\prime}_{i}\cup{\cal G}_{i}^{0}$-sets. (b) In the case as some lateral
strengths (from one to several) belong to the corresponding ${\cal
G}^{0}$-sets, more precisely, $V_{1}\in{\cal G}^{0}_{1}\,,\ldots,V_{p}\in{\cal
G}^{0}_{p}$ (left layers) and $V_{N-r}\in{\cal
G}^{0}_{N-r}\,,\ldots,V_{N}\in{\cal G}^{0}_{N}$ (right layers), so that the
strengths $V_{p+1}$ and $V_{N-r-1}$ are singular, now the strengths
$V_{p+2}\,,\ldots,V_{N-r-2}$ must belong to either the ${\cal G}^{\prime}$\-
or ${\cal G}^{0}$-sets. If one of the lateral strengths or both these are
regular, i.e., $V_{1}\in{\cal G}_{1}^{0}$ and $V_{N}\in{\cal G}_{N}^{0}$, then
only the $\delta$-like squeezing of these strengths makes sense. (c) The
presence in the configuration $V_{1}\,,\ldots,V_{N}$ at least one strength of
a well-like form, say, $V_{i}<0$, implies the existence of a countable set of
solutions to the resonance equations.

###  5.2 Conditions for the existence of resonance sets in the case $\sigma>1$

Similarly to the case with $\sigma=1$, consider first the situation as $N=3$
and analyze in detail all possible cancellations of divergences in
$\lambda_{21}$ on the paths with $\sigma>1$. Here, the cancellation can also
be possible within equation (40), where $i=1$, $j=2$ and $k=3$. Assume that
all the terms $q_{i}t_{i}$, $q_{j}t_{j}$ and $q_{k}t_{k}$ diverge in the
squeezing limit. The configuration $V_{i}\in{\cal G}^{\prime}_{i}$,
$V_{j}\in{\cal G}^{\prime}_{j}$ and $V_{k}\in{\cal G}^{\prime}_{k}$ is the
most singular case. The divergence rate of all the terms in this case could be
compared by multiplying equation (40) by one of the factors
$l^{1/\sigma}_{i}$, $l^{1/\sigma}_{j}$ or $l^{1/\sigma}_{k}$ with $\sigma>1$.
However, the multiplication by any of these factors does not lead to finite
values. For instance, the resulting term $l_{i}^{1/\sigma}q_{i}t_{i}\sim
l_{i}^{1/\sigma-1}\tau_{i}$ diverges for $\sigma>1$ as $l_{i}\to 0$. Hence, in
the three-layer structure, the cancellation of divergences on the paths with
$\sigma>1$ is impossible if all the three strengths $V_{i}$, $V_{j}$ and
$V_{k}$ belong to the corresponding ${\cal G}^{\prime}$-sets. Therefore at
least one of the strengths must belong to the ${\cal G}$-set, which can be
either lateral ($V_{i}$ or $V_{k}$) or middle ($V_{j}$).

In addition to relations (37) and (38), for the case $V_{j}\in{\cal
G}^{\prime}_{j}$ , we also need the asymptotic representation

$T_{ijk}\sim\left\\{\begin{array}[]{lll}\vskip 3.0pt plus 1.0pt minus
1.0ptl_{i}^{-1/\sigma}\chi_{jk}\,\eta_{ii}^{\sigma}\,\tau_{j}\tau_{k}/s_{j}^{2}\\\
l_{k}^{-1/\sigma}\chi_{ji}\,\eta_{kk}^{\sigma}\,\tau_{i}\tau_{j}/s_{j}^{2}\par\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
l_{i}^{-1/\sigma}\eta_{ii}^{\sigma}\,\eta_{kj}\,\tau_{j}/s_{j}^{2}\par\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
l_{k}^{-1/\sigma}\eta_{ij}\,\eta_{kk}^{\sigma}\,\tau_{j}/s_{j}^{2}\end{array}\right.~{}~{}\mbox{for}~{}\left\\{\begin{array}[]{lll}\vskip
3.0pt plus 1.0pt minus 1.0ptV_{i}\in{\cal G}_{i}^{\sigma}\,,\,V_{k}\in{\cal
G}^{\prime}_{k}\,,\\\ V_{i}\in{\cal G}^{\prime}_{i}\,,\,V_{k}\in{\cal
G}_{k}^{\sigma}\,,\par\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ V_{i}\in{\cal
G}_{i}^{\sigma}\,,\,V_{k}\in{\cal G}_{k}\,,\par\vskip 3.0pt plus 1.0pt minus
1.0pt\\\ V_{i}\in{\cal G}_{i}\,,\,V_{k}\in{\cal
G}_{k}^{\sigma}\,.\end{array}\right.$ (45)

Using next these relations after the multiplication of equation (40) by
$l_{i}^{1/\sigma}$ and $l_{i}^{1/\sigma}$, respectively, we get the four
resonance equations for $V_{j}\in{\cal G}^{\prime}_{j}$ and $\sigma>1$, which
are collected in table 2. Here, the cancellation of divergences occurs between
all the four terms if $V_{i}\in{\cal G}_{i}\setminus{\cal G}^{0}_{i}$ or
$V_{k}\in{\cal G}_{k}\setminus{\cal G}^{0}_{k}$ . In the case as
$V_{i}\in{\cal G}^{0}_{i}$ and $V_{k}\in{\cal G}^{0}_{k}$, we have accordingly
$\eta^{\sigma}_{ii}=0$ and $\eta^{\sigma}_{kk}=0$ in table 2 and therefore
only two terms take place in the cancellation.

Table 2:  Resonance equations and available paths for a three-layer structure, resulting from the multiplication of equation (40) by $l_{i}^{1/\sigma}$ or $l_{k}^{1/\sigma}$, where $V_{j}\in{\cal G}^{\prime}_{j}$ and $\sigma>1$.  $V_{i}(l_{i})$, $V_{k}(l_{k})$ | Resonance equations | Paths
---|---|---
$V_{i}\in{\cal G}_{i}^{\sigma}$ , $V_{k}\in{\cal G}^{\prime}_{k}$ | $\eta^{\sigma}_{ii}+\chi^{\sigma}_{ij}\tau_{j}+\chi^{\sigma}_{ik}\tau_{k}=\chi_{jk}\,\eta^{\sigma}_{ii}\,\tau_{j}\tau_{k}/s_{j}^{2}$ |  $\Gamma^{\prime}_{jk}$ , $\Gamma^{\sigma}_{ij}$ , $\Gamma^{\sigma}_{ik}$
$V_{i}\in{\cal G}^{\prime}_{i}$ , $V_{k}\in{\cal G}_{k}^{\sigma}$ | $\chi^{\sigma}_{ki}\tau_{i}+\chi_{kj}^{\sigma}\tau_{j}+\eta_{kk}^{\sigma}=\chi_{ji}\,\eta^{\sigma}_{kk}\,\tau_{i}\tau_{j}/s_{j}^{2}$ |  $\Gamma^{\prime}_{ij}$ , $\Gamma^{\sigma}_{ki}$ , $\Gamma^{\sigma}_{kj}$
$V_{i}\in{\cal G}_{i}^{\sigma}$ , $V_{k}\in{\cal G}_{k}$ | $\eta_{ii}^{\sigma}+\chi_{ij}^{\sigma}\tau_{j}+\eta^{\sigma}_{ki}=\eta^{\sigma}_{ii}\,\eta_{kj}\,\tau_{j}/s_{j}^{2}$ |  $\Gamma^{\prime}_{V_{k}j}$ , $\Gamma^{\sigma}_{ij}$ , $\Gamma^{\sigma}_{V_{k}i}$
$V_{i}\in{\cal G}_{i}$ , $V_{k}\in{\cal G}_{k}^{\sigma}$ | $\eta_{ik}^{\sigma}+\chi_{kj}^{\sigma}\tau_{j}+\eta^{\sigma}_{kk}=\eta_{ij}\,\eta_{kk}^{\sigma}\,\tau_{j}/s_{j}^{2}$ |  $\Gamma^{\prime}_{V_{i}j}$ , $\Gamma^{\sigma}_{kj}$ , $\Gamma^{\sigma}_{V_{i}k}$

In the ‘middle’ case as $V_{j}\in{\cal G}_{j}$, we have to apply the
asymptotic representation given by equations (37) and (38), where $\sigma\geq
2$ resulting in the limit $|T_{ijk}|\to c\geq 0$. Multiplying equation (40) by
$l_{j}^{1/\sigma}$, we obtain the four resonance equations, which are present
in table 3.

Table 3:  Resonance equations and available paths for a three-layer structure, resulting from the multiplication of equation (40) by $l_{j}^{1/\sigma}$, where $V_{j}\in{\cal G}_{j}^{\sigma}$ and $\sigma\geq 2$.  $V_{i}(l_{i})$, $V_{k}(l_{k})$ | Resonance equations | Paths
---|---|---
$V_{i}\in{\cal G}^{\prime}_{i}$ , $V_{k}\in{\cal G}^{\prime}_{k}$ | $\chi_{ji}^{\sigma}\tau_{i}+\eta^{\sigma}_{jj}+\chi^{\sigma}_{jk}\tau_{k}=0$ | $\Gamma^{\sigma}_{ji}\,,\,\Gamma^{\sigma}_{jk}$
$V_{i}\in{\cal G}^{\prime}_{i}$ , $V_{k}\in{\cal G}_{k}\setminus{\cal G}^{0}_{k}$ | $\chi^{\sigma}_{ji}\tau_{i}+\eta_{jj}^{\sigma}+\eta_{kj}^{\sigma}=0$ | $\Gamma^{\sigma}_{ji}\,,\,\Gamma^{\sigma}_{V_{k}j}$
$V_{i}\in{\cal G}_{i}\setminus{\cal G}^{0}_{i}$ , $V_{k}\in{\cal G}^{\prime}_{k}$ | $\eta^{\sigma}_{ij}+\eta^{\sigma}_{jj}+\chi^{\sigma}_{jk}\tau_{k}=0$ | $\Gamma^{\sigma}_{V_{i}j}\,,\,\Gamma^{\sigma}_{jk}$
$V_{i}\in{\cal G}_{i}\setminus{\cal G}^{0}_{i}$ , $V_{k}\in{\cal G}_{k}\setminus{\cal G}^{0}_{k}$ | $\eta_{ij}^{\sigma}+\eta^{\sigma}_{jj}+\eta^{\sigma}_{kj}=0$ | $\Gamma^{\sigma}_{V_{i}j}\,,\,\Gamma^{\sigma}_{V_{k}j}$

Note that the paths given in this table and those derived for the existence of
relations (37) and (38) are the same. The general form of the resonance
equations in table 3 reads

$A^{\sigma}_{ij}+A^{\sigma}_{kj}=0,$ (46)

where

$A^{\sigma}_{ij}:=\left\\{\begin{array}[]{ll}\lim_{(l_{i},\,l_{j})\in\Gamma^{\sigma}_{ji}}(l_{j}^{1/\sigma}\\!q_{i}t_{i})=\chi^{\sigma}_{ji}\tau_{i}&\mbox{if}~{}~{}V_{i}\in{\cal
G}^{\prime}_{i}\,,~{}i\neq j,\\\
\lim_{(l_{i},\,l_{j})\in\Gamma^{\sigma}_{V_{i}j}}(q_{i}^{2}l_{i}l_{j}^{1/\sigma})=\eta^{\sigma}_{ij}&\mbox{if}~{}~{}V_{i}\in{\cal
G}_{i}\setminus{\cal G}_{i}^{0}\,.\end{array}\right.$ (47)

The solution of equations (46) are curves on the
$\\{A_{ij}^{\sigma},A_{kj}^{\sigma}\\}$-plane, which depend on the parameter
$\sigma\geq 2$. The corresponding resonance sets for the structure composed of
two separated layers ($N=3$, $V_{j}\equiv 0$) have been analyzed in works [34,
36].

Thus, we have established in the case $N=3$ that for the existence of
resonance equations, at least one of the strengths among $V_{i}$ , $V_{j}$ and
$V_{k}$ must belong to the corresponding ${\cal G}$-set. For illustration of
this rule for $N>3$, let us write a couple of resonance equations if $N=4$.
For this particular case, series (32) reduces to the sum

$Q_{21}=T_{ijk}+T_{ijm}+T_{ikm}+T_{jkm}~{}~{}(i=1,\,j=2,\,k=3,\,m=4)$ (48)

and the equation for the cancellation of divergences $\lambda_{21}^{\infty}=0$
becomes $\sum_{i=1}^{4}q_{i}t_{i}=Q_{21}$ if all the terms $q_{i}t_{i}$’s are
divergent.

Let us consider first the configuration as one of the lateral strengths, say
$V_{m}$ , belongs to the ${\cal G}_{m}$-set. Assume that $V_{i}\in{\cal
G}^{\prime}_{i}$ , $V_{j}\in{\cal G}^{\prime}_{j}$ , $V_{k}\in{\cal
G}^{\prime}_{k}$ and $V_{m}\in{\cal G}_{m}$ . Then the equation for the
cancellation of divergences takes the form

$\displaystyle q_{i}t_{i}+q_{j}t_{j}+q_{k}t_{k}+q_{m}^{2}l_{m}$ (49)
$\displaystyle=$ $\displaystyle{q_{i}q_{k}\over
q_{j}}t_{i}t_{j}t_{k}+{q_{i}q_{m}\over q_{j}}t_{i}t_{j}t_{m}+{q_{i}q_{m}\over
q_{k}}t_{i}t_{k}t_{m}+{q_{j}q_{m}\over q_{k}}t_{j}t_{k}t_{m}\,.$

The multiplication of this equation by $l_{m}^{1/\sigma}$ yields the resonance
equation

$\displaystyle\chi_{mi}^{\sigma}\tau_{i}+\chi_{mj}^{\sigma}\tau_{j}+\chi_{mk}^{\sigma}\tau_{k}+\eta_{mm}^{\sigma}$
(50) $\displaystyle=$ $\displaystyle\chi_{ji}\tau_{i}{\tau_{j}\over
s_{j}^{2}}\chi_{mk}^{\sigma}\tau_{k}+\left(\\!\chi_{ji}\tau_{i}{\tau_{j}\over
s_{j}^{2}}+\chi_{ki}\tau_{i}{\tau_{k}\over
s_{k}^{2}}+\chi_{kj}\tau_{j}{\tau_{k}\over
s_{k}^{2}}\right)\eta^{\sigma}_{mm}\,.$

Here, in each of the four triads [see equation (48)], the middle strength is
from the corresponding ${\cal G}^{\prime}$-set and therefore representation
(45) has been applied, similarly as for deriving the first equation in table
2. In equation (50), the available paths are characterized by the relations
$l_{i}\sim l_{j}\sim l_{k}\sim l_{m}^{1/\sigma}$ forming the paths
$\Gamma^{\prime}_{ij}\,,\,\Gamma^{\prime}_{ik}\,,\,\Gamma^{\prime}_{jk}\,,\,\Gamma^{\sigma}_{mi}\,,\,\Gamma^{\sigma}_{mj}\,,\,\Gamma^{\sigma}_{mk}$
with $\sigma\in(1,\infty)$. Besides these conditions, an additional
restriction appears on the strength $V_{m}$ , namely, this strength must
belong to ${\cal G}_{m}^{\sigma}\subset{\cal G}_{m}\setminus{\cal G}_{m}^{0}$
. In the case as $V_{m}\in{\cal G}_{m}^{0}$ , in equation (50) we have
$\eta^{\sigma}_{mm}=0$ and therefore only the four terms participate in
removing divergences, instead of all the eight terms.

Consider now a four-layer system with the configuration as one of the
strengths belonging to a ${\cal G}$-set is localized between the strengths
from ${\cal G}^{\prime}$-sets, e.g., $V_{i}\in{\cal G}^{\prime}_{i}$ ,
$V_{j}\in{\cal G}_{j}$ , $V_{i}\in{\cal G}^{\prime}_{k}$ and $V_{m}\in{\cal
G}^{\prime}_{m}$ . Here, due to representation (37), the squeezing limits of
the triads $T_{ijk}=q_{i}t_{i}\,l_{j}\,q_{k}t_{k}$ and
$T_{ijm}=q_{i}t_{i}\,l_{j}\,q_{m}t_{m}$ are finite on the paths, where
$l_{i}\sim l_{k}\sim l_{m}\sim l_{j}^{1/\sigma}$. Therefore these triads do
not participate in the cancellation of divergences and they have to be omitted
because their middle strength $V_{j}$ belongs to the ${\cal G}_{j}$-set. As a
result, the divergent terms are contained in the equation

$q_{i}t_{i}+q_{j}^{2}l_{j}+q_{k}t_{k}+q_{m}t_{m}={q_{i}q_{m}\over
q_{k}}t_{i}t_{k}t_{m}+q_{j}^{2}l_{j}{t_{k}\over q_{k}}q_{m}t_{m}\,.$ (51)

Multiplying this equation by $l_{j}^{1/\sigma}$ and using relations (45), we
arrive at the resonance equation

$\chi_{ji}^{\sigma}\tau_{i}+\eta_{jj}^{\sigma}+\chi_{jk}^{\sigma}\tau_{k}+\chi_{jm}^{\sigma}\tau_{m}=\chi_{ki}\tau_{i}{\tau_{k}\over
s_{k}^{2}}\chi_{jm}^{\sigma}\tau_{m}+\eta^{\sigma}_{jj}{\tau_{k}\over
s_{k}^{2}}\chi_{km}\tau_{m}\,,$ (52)

valid under the conditions $l_{i}\sim l_{k}\sim l_{m}\sim l_{j}^{1/\sigma}$,
resulting in the paths $\Gamma^{\prime}_{ik}$ , $\Gamma^{\prime}_{im}$ ,
$\Gamma^{\prime}_{km}$ , $\Gamma^{\sigma}_{ji}$ , $\Gamma^{\sigma}_{jk}$ ,
$\Gamma^{\sigma}_{jm}$ . Since representation (37) has been used here, we have
to choose $\sigma\in[2,\infty)$, instead of the interval $1<\sigma<\infty$.

The similar situation takes place as $V_{i}\in{\cal G}^{\prime}_{i}$ ,
$V_{j}\in{\cal G}_{j}$ , $V_{k}\in{\cal G}^{\prime}_{k}$ and $V_{m}\in{\cal
G}_{m}$ . Here, again the same two triads $T_{ijk}$ and $T_{ijm}$ have the
middle strength from ${\cal G}_{j}$ and for the other two triads $T_{ikm}$ and
$T_{jkm}$, the middle strength $V_{k}$ belongs to ${\cal G}^{\prime}_{k}$ . As
a result, again the right-hand part of the resonance equation consists of two
summands. Indeed, multiplying the equation

$q_{i}t_{i}+q_{j}^{2}l_{j}+q_{k}t_{k}+q_{m}^{2}l_{m}=q_{i}t_{i}{t_{k}\over
q_{k}}q_{m}^{2}l_{m}+q_{j}^{2}l_{j}{t_{k}\over q_{k}}q_{m}^{2}l_{m}$ (53)

by $l_{j}^{1/\sigma}$ and applying the representation given by equations (37)
and (38) to the triads $T_{ijk}$ and $T_{ijm}$ , and representation (45) to
$T_{ikm}$ and $T_{jkm}$ , we get the resonance equation in the form

$\chi_{ji}^{\sigma}\tau_{i}+\eta_{jj}^{\sigma}+\chi_{jk}^{\sigma}\tau_{k}+\eta_{mj}^{\sigma}=\chi_{ji}^{\sigma}\tau_{i}{\tau_{k}\over
s_{k}^{2}}\eta_{mk}+\eta^{\sigma}_{jj}{\tau_{k}\over s_{k}^{2}}\eta_{mk}\,.$
(54)

In this equation, the relations $l_{i}\sim l_{k}\sim l_{j}^{1/\sigma}$ and
$|V_{m}|l_{m}\sim l_{j}^{-1/\sigma}$ take place, so that the available paths,
on which equation (54) is well-defined, are $\Gamma^{\prime}_{ik}$ ,
$\Gamma^{\prime}_{V_{m}i}$ , $\Gamma^{\prime}_{V_{m}k}$ ,
$\Gamma^{\sigma}_{ji}$ , $\Gamma^{\sigma}_{jk}$ , $\Gamma^{\sigma}_{V_{m}j}$
with $\sigma\in[2,\infty)$. The form of equations (52) and (54) resembles a
‘mixture’ of the equations collected in tables 2 and 3.

Consider now the configuration $V_{i}\in{\cal G}^{\prime}_{i}$ ,
$V_{j}\in{\cal G}_{j}$ , $V_{k}\in{\cal G}_{k}$ and $V_{m}\in{\cal G}_{m}$ .
Here, all the triads contain the middle strengths $V_{j}$ and $V_{k}$ ,
belonging to the ${\cal G}_{j}$\- and ${\cal G}_{k}$-sets. Therefore only
asymptotic formulas (37) and (38) have to be applied on the paths determined
by the relations $l_{i}\sim l_{j}^{1/\sigma}$, $|V_{k}|\,l_{k}\sim
l_{j}^{-1/\sigma}$, $|V_{m}|\,l_{m}\sim l_{j}^{-1/\sigma}$, $l_{i}\sim
l_{k}^{1/\sigma}$, $|V_{m}|\,l_{m}\sim l_{k}^{-1/\sigma}$, $|V_{j}|\,l_{j}\sim
l_{k}^{-1/\sigma}$. There are no contradictions in these relations, because,
e.g., from the comparison of the relations with $|V_{m}|\,l_{m}$, one finds
that $l_{j}\sim l_{k}$, so that $l_{k}/l_{j}^{1/\sigma}$ and
$l_{j}/l_{m}^{1/\sigma}$ tend here to zero as required. Thus, all the triads
in the cancellation equation have to be omitted and the multiplication of the
equation $q_{i}t_{i}+q_{j}^{2}l_{j}+q_{k}^{2}l_{k}+q_{m}^{2}l_{m}=0$ yields
the resonance equation

$\chi_{ji}^{\sigma}\tau_{i}+\eta_{jj}^{\sigma}+\eta_{kj}^{\sigma}+\eta_{mj}^{\sigma}=0\,,$
(55)

which is similar to the second equation in table 3. The multiplication by
$l_{j}^{1/\sigma}$ imposes the restriction on $V_{j}$ , namely $V_{j}\in{\cal
G}_{j}^{\sigma}$. If some strength belongs to the corresponding ${\cal
G}^{0}$-set, say $V_{m}$ , then in equation (55), we have to set
$\eta_{mj}^{\sigma}=0$. Finally, in a similar way, one can treat the situation
as all the strengths are from the ${\cal G}$-sets. The multiplication of the
equation $q_{i}t_{i}+q_{j}^{2}l_{j}+q_{k}^{2}l_{k}+q_{m}^{2}l_{m}=0$, for
instance, by $l_{j}^{1/\sigma}$ results in the resonance equation

$\eta_{ij}^{\sigma}+\eta_{jj}^{\sigma}+\eta_{kj}^{\sigma}+\eta_{mj}^{\sigma}=0\,,$
(56)

which is similar to the last equation in table 3. Thus, even if all the
strengths belong to the ${\cal G}$-sets, the existence of resonance equations
holds true on the corresponding paths with $\sigma\in[2,\infty)$.

Concerning the double-layer structures ($N=2$), where triads (31) are absent,
instead of resonance equation (30), which is valid for $\sigma=1$, one can
write the equation for the whole interval $1<\sigma<\infty$. Indeed,
multiplying the equation $q_{i}^{2}l_{i}+q_{j}t_{j}=0$ by $l_{i}^{1/\sigma}$,
we get the resonance equation

$\lim_{(l_{i},l_{j})\in\Gamma^{\sigma}_{ij}}\\!\\!(q_{i}^{2}l_{i}+q_{j}t_{j})\,l_{i}^{1/\sigma}=\eta^{\sigma}_{ii}+\chi^{\sigma}_{ij}\tau_{j}=0,$
(57)

where $V_{i}\in{\cal G}^{\sigma}_{i}$ and $l_{i}^{1/\sigma}\sim l_{j}$ . Both
equations (30) for $\sigma=1$ and (57) for $\sigma>1$ admit countable sets of
solutions if $V_{j}<0$, that determine the corresponding resonance sets.

Thus, from the above arguments, one can deduce

Conclusion 4 : For the existence of resonance sets, at least one of the
strengths from the configuration $V_{1}(l_{1}),\ldots,V_{N}(l_{N})$ , say
$V_{j}$ , must belong to the ${\cal G}_{j}$-set. In this case, for deriving a
resonance equation, the equation for the cancellation of divergences
$\lambda_{21}^{\infty}=0$ can be multiplied by $l_{j}^{1/\sigma}$. During this
procedure, the asymptotic representation given by equations (37), (38) and
(45) has to be adopted and this application depends on whether the middle
strength $V_{j}$ in the $T_{ijk}$-triads belongs to the ${\cal
G}^{\prime}_{j}$\- or ${\cal G}_{j}$-set. In the former case, equations (45)
are to be used, while in the latter case, due to relations (37) and (38), the
$T_{ijk}$-triads are finite in the squeezing limit if $\sigma\in[2,\infty)$.
In the case as one of the lateral strengths or both these are from the ${\cal
G}$-sets ($V_{1}\in{\cal G}_{1}$ and $V_{N}\in{\cal G}_{N}$) and all the
internal strengths belong to the ${\cal G}^{\prime}$-sets ($V_{i}\in{\cal
G}^{\prime}_{i}$, $i=2,\ldots,N-1$), then the resonance equations can exist on
the paths with the parameter $\sigma$ from the interval $1<\sigma<\infty$. If
one of the lateral strengths or both these are regular, i.e., $V_{1}\in{\cal
G}_{1}^{0}$ and $V_{N}\in{\cal G}_{N}^{0}$, then the cases $V_{1}\equiv 0$ and
$V_{N}\equiv 0$ have to be excluded from the consideration. Next, in a double-
layer structure ($N=2$), no more than one strength belonging to the
corresponding ${\cal G}^{\prime}$-set is available. For a single-layer
structure ($N=1$), for the existence of resonance sets it is necessary that
the potential must be negative with the strength belonging to the ${\cal
G}^{\prime}$-set.

###  5.3 An example

As a particular example of the functions $V_{i}(l_{i})$ and $V_{j}(l_{j})$,
used for the realization of the singular squeezing limit, one can consider the
following asymptotic expressions:

$V_{i}(l_{i})\sim l_{i}^{-\mu},~{}~{}V_{j}(l_{j})\sim
l_{j}^{-\nu/(1-\mu+\nu)},~{}~{}\mu,\,\nu>0,~{}~{}1-\mu+\nu>0,$ (58)

used in several works [12, 13, 14, 34, 36, 39]. By a proper choosing of the
positive parameters $\mu$ and $\nu$ in (58), all the four singular cases: (i)
$V_{i}\in{\cal G}^{\prime}_{i}$ , $V_{j}\in{\cal G}^{\prime}_{j}$ ; (ii)
$V_{i}\in{\cal G}_{i}\setminus{\cal G}_{i}^{0}$, $V_{j}\in{\cal
G}^{\prime}_{j}$ ; (iii) $V_{i}\in{\cal G}^{\prime}_{i}$ , $V_{j}\in{\cal
G}_{j}\setminus{\cal G}_{j}^{0}$ ; (iv) $V_{i}\in{\cal G}_{i}\setminus{\cal
G}_{i}^{0}$, $V_{j}\in{\cal G}_{j}\setminus{\cal G}_{j}^{0}$ ; which were
considered above, can be materialized on the $(\mu,\nu)$-sets

$\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\begin{array}[]{llll}P~{}:=\\{\mu=\nu=2\\},&V_{i}\in{\cal
G}^{\prime}_{i}\,,~{}V_{j}\in{\cal G}^{\prime}_{j}\,,\\\
L_{1}:=\\{1<\mu<2,\,\nu=2(\mu-1)\\},&V_{i}\in{\cal G}_{i}\setminus{\cal
G}_{i}^{0}\,,~{}V_{j}\in{\cal G}^{\prime}_{j}\,,\\\
L_{2}:=\\{\mu=2,\,2<\nu<\infty\\},&V_{i}\in{\cal
G}^{\prime}_{i}\,,~{}~{}~{}V_{j}\in{\cal G}_{j}\setminus{\cal G}_{j}^{0}\,,\\\
S~{}:=\\{1<\mu<2,\,2(\mu-1)<\nu<\infty\\},&V_{i}\in{\cal G}_{i}\setminus{\cal
G}_{i}^{0}\,,~{}V_{j}\in{\cal G}_{j}\setminus{\cal G}_{j}^{0}\,.\end{array}$
(59)

These sets form the triangle on the $(\mu,\,\nu)$-plane with the vertex at the
point $P$, where $L_{1}$ and $L_{2}$ are the edges, and $S$ the interior of
this angle. One can check that limits (22), namely the first, the second, the
third and the sixth inequalities, are fulfilled accordingly on the paths:
$\Gamma^{\prime}_{ij}$ (at point $P$),
$\Gamma^{\prime}_{ij}\cup\bar{\Gamma}_{V_{i}j}$ (on line $L_{1}$),
$\Gamma^{\prime}_{ij}\cup\bar{\Gamma}_{V_{j}i}$ (on line $L_{2}$) and
$\Gamma^{\prime}_{ij}\cup\bar{\Gamma}_{V_{i}j}\cup\bar{\Gamma}_{V_{j}i}$ (on
plane $S$), where

$\begin{array}[]{ll}\bar{\Gamma}_{V_{i}j}=\left\\{(l_{i},l_{j})\to
0~{}\big{|}~{}l_{i}/l_{j}\to 0,~{}l_{i}^{{1-\mu}}l_{j}\to c\geq 0\right\\},\\\
\bar{\Gamma}_{V_{j}i}=\left\\{(l_{i},l_{j})\to 0~{}\big{|}~{}l_{j}/l_{i}\to
0,~{}l_{i}\,l_{j}^{(1-\mu)/(1-\mu+\nu)}\to c\geq 0\right\\}.\end{array}$ (60)

In other words, inequalities (22) are fulfilled on the paths with
$l_{i}/l_{j}\to c>0$ at the point $P$, while on the lines $L_{1}$ and $L_{2}$
as well as on the plane $S$, the available paths are bounded by the limit
pencils defined by the asymptotic relations: $l_{i}/l_{j}\to c>0$ and
$l_{i}/l_{j}\sim l_{i}^{2-\mu}\to 0$ (on $L_{1}$); $l_{j}/l_{i}\sim
l_{j}^{[2(1-\mu)+\nu]/(1-\mu+\nu)}\to 0$ and $l_{i}/l_{j}\to c>0$ (on
$L_{2}$); $l_{j}/l_{i}\sim l_{j}^{[2(1-\mu)+\nu]/(1-\mu+\nu)}\to 0$ and
$l_{i}/l_{j}\sim l_{i}^{2-\mu}\to 0$ (on $S$).

##  6 Final remarks

Thus, we have formulated above the conditions on the strength configuration
$V_{1}(l_{1}),\ldots,V_{N}(l_{N})$ and the paths
$l=\\{l_{1},\ldots,l_{N}\\}\to 0$, under which the squeezed multi-layer
structure can be described as a one-point interaction model. These conditions
are of two types. The first of them is formulated by Conclusion 2, realizing
the finite and non-zero squeezed limits of the diagonal elements
$\lambda_{11}$ and $\lambda_{22}$ of the transmission matrix $\Lambda$.

Under the conditions of the second type described in Conclusions 3 and 4, the
squeezed $\Lambda$-matrix becomes a well-defined object, due to the
cancellation of divergences in the singular element $\lambda_{21}$. This
cancellation imposes the constraints on the system parameters, resulting in
the appearance of resonance sets. More precisely, using the equation
$\det\Lambda(x_{0},x_{N})=1$ and $\lambda_{12}\to 0$, valid for any strength
configuration, one can assert that
$\lambda_{11}\to\theta\in{\mathbb{R}}\setminus\\{0\\}$ and
$\lambda_{22}\to\theta^{-1}$. As a result, the resulting connection matrix is
of the form

$\Lambda_{0}:=\lim_{l\to
0}\Lambda(x_{0},x_{N})=\left(\begin{array}[]{lr}\theta~{}~{}~{}~{}0\\\
\alpha~{}~{}~{}\theta^{-1}\end{array}\right),~{}~{}~{}~{}\alpha\in{\mathbb{R}}.$
(61)

As follows from equation (17), the bound state level is
$\kappa=-\,\alpha/(\theta+\theta^{-1})$.

Acknowledgments

The authors acknowledge a partial support by the National Academy of Sciences
of Ukraine Grant ‘Functional Properties of Materials Prospective for
Nanotechnologies’ (project No. 0120U100858). They are indebted to Y.D.
Golovaty for careful reading of the manuscript, his suggestions and remarks,
resulting in the improvement of the paper. They also thank the anonymous
referees for the careful reading of this paper, their questions and pointing
out several mistakes and inaccuracies.

References

## References

  * [1] Albeverio S, Gesztesy F, Høegh-Krohn R and Holden H 2005 Solvable Models in Quantum Mechanics (With an Appendix by Pavel Exner) 2nd revised edn (Providence: RI: American Mathematical Society: Chelsea Publishing)
  * [2] Albeverio S and Kurasov P 1999 Singular Perturbations of Differential Operators: Solvable Schrödinger-Type Operators (Cambridge: Cambridge University Press)
  * [3] Šeba P 1986 Rep. Math. Phys. 24 111
  * [4] Golovaty Y D and Man’ko S S 2009 Ukr. Math. Bull. 6 169
  * [5] Golovaty Y D and Hryniv R O 2010 J. Phys. A: Math. Theor. 43 155204
Golovaty Y D and Hryniv R O 2010 J. Phys. A: Math. Theor. 2011 44 049802
(corrigendum)

  * [6] Golovaty Y D and Hryniv R O 2013 Proc. R. Soc. Edinb. A 143 791
  * [7] Man’ko S S 2010 J. Phys. A: Math. Theor. 43 445304
  * [8] Man’ko S S 2012 J. Math. Phys. 53 123521
  * [9] Exner P and Manko S S 2013 J. Phys. A: Math. Theor. 46 345202
  * [10] Toyama F M and Nogami Y 2007 J. Phys. A: Math. Theor. 40 F685
  * [11] Christiansen P L, Arnbak N C, Zolotaryuk A V, Ermakov V N and Gaididei Y B 2003 J. Phys. A: Math. Gen. 36 7589
  * [12] Zolotaryuk A V, Christiansen P L and Iermakova S V 2006 J. Phys. A: Math. Gen. 39 9329
  * [13] Zolotaryuk A V 2008 J. Comput. Theor. Nanoscience  1 187
  * [14] Zolotaryuk A V and Zolotaryuk Y 2011 J. Phys. A: Math. Theor. 44 375305
Zolotaryuk A V and Zolotaryuk Y 2012 J. Phys. A: Math. Theor. 45 119501
(corrigendum)

  * [15] Golovaty Y 2012 Methods Funct. Anal. Topology 18 243
  * [16] Golovaty Y 2013 Integr. Equ. Oper. Theor. 75 341
  * [17] Gadella M, Negro J and Nieto L M 2009 Phys. Lett. A 373 1310
  * [18] Gadella M, Mateos-Guilarte J, Muñoz-Castañeda J M and Nieto L M 2016 J. Phys. A: Math. Theor. 49 015204
  * [19] Gadella M, Glasser M L and Nieto L M 2011 Int. J. Theor. Phys. 50 2144
  * [20] Gadella M, García-Ferrero M A, González-Martín S and Maldonado-Villamizar F H 2014 Int. J. Theor. Phys. 53 1614
  * [21] Fassari S, Gadella M, Glasser M L and Nieto L M 2018 Ann. Phys. (NY) 389 48
  * [22] Albeverio S, Da̧browski L and Kurasov P 1998 Lett. Math. Phys. 45 33
  * [23] Albeverio S and Nizhnik L 2007 J. Math. Anal. Appl. 332 884
  * [24] Lange R-J 2012 J. High Energy Phys. JHEP11(2012)032
  * [25] Brasche J F and Nizhnik L P 2013 Methods Funct. Anal. Topology 19 4
  * [26] Albeverio S and Nizhnik L 2013 Methods Funct. Anal. Topology 19 199
  * [27] Lange R-J 2015 J. Math. Phys. 56 122105
  * [28] Golovaty Y 2018 J. Phys. A: Math. Theor. 51 255202
  * [29] Cheon T and Shigehara T 1998 Phys. Lett. A 243 111
  * [30] Exner P, Neidhardt H and Zagrebnov V A 2001 Commun. Math. Phys. 224 593
  * [31] Albeverio S, Fassari S and Rinaldi F 2013 J. Phys. A: Math. Theor. 46 385305
  * [32] Albeverio S, Fassari S and Rinaldi F 2016 J. Phys. A: Math. Theor. 49 025302
  * [33] Zolotaryuk A V 2018 Physica E: Low-dimensional Systems and Nanostructures 103 81
  * [34] Zolotaryuk A V 2018 Ann. Phys. (NY) 396 479
  * [35] Zolotaryuk A V and Zolotaryuk Y 2020 Low Temperature Phys. 2020 46 927
  * [36] Zolotaryuk A V and Zolotaryuk Y 2021 J. Phys. A: Math. Theor. 54 035201
  * [37] Kurasov P 1996 J. Math. Anal. Appl. 201 297
  * [38] Coutinho F A B, Nogami Y and Pérez J F 1997 J. Phys. A: Math. Gen. 30 3937
  * [39] Zolotaryuk A V and Zolotaryuk Y 2014 Int. J. Mod. Phys. B 28 1350203