# Spectra of Dirichlet Laplacian in 3-dimensional polyhedral layers

Bakharev F. L.†<EMAIL_ADDRESS>and  Matveenko S. G.†<EMAIL_ADDRESS>
###### Abstract.

The structure of the spectrum of the three-dimensional Dirichlet Laplacian in
the 3D polyhedral layer of fixed width is studied. It appears that the
essential spectrum is defined by the smallest dihedral angle that forms the
boundary of the layer while the discrete spectrum is always finite. An example
of a layer with the empty discrete spectrum is constructed. The spectrum is
proved to be nonempty in regular polyhedral layer.

###### Key words and phrases:

Laplace operator, Dirichlet layers, discrete spectrum, continuous spectrum

†Chebyshev Laboratory, St. Petersburg State University, 14th Line V.O., 29,
Saint Petersburg 199178 Russia

E-mail addresses<EMAIL_ADDRESS><EMAIL_ADDRESS>
The work is supported by the Russian Science Foundation grant 19-71-30002

##  1\. Introduction

The study of the Dirichlet spectral problem for the Laplace operator in
various unbounded domains has attracted the attention of researchers during
last decades. In recent years, many papers on spectra of quantum waveguides
(regions with cylindrical outlets to infinity and Dirichlet conditions at the
boundary) have appeared. A slightly smaller number of works discuss the so-
called Dirichlet layers and their junctions, which structurally may be much
more complicated than waveguides. Nevertheless, spectra of problems in the
layers are similar to spectra of problems in waveguides. This is due to the
fact that in layers, as well as in waveguides, on the one hand, a countable
number of identical non-intersecting cubes can be inscribed, and on the other
hand, one cannot inscribe an arbitrarily large cube. This leads to the fact
that the corresponding problems usually have an essential spectrum
$\sigma_{ess}$ occupying the ray on the real axis with a positive cut-off
point. Below its threshold there may exist a discrete spectrum $\sigma_{d}$
corresponding to trapped waves. The appearance of trapped waves is often a
consequence of the structure of the domain contained in a sufficiently large
ball (the shape of the junction), while the characteristics of the essential
spectrum are determined by the structure of the waveguide or layer at
infinity.

Several works focused on Dirichlet layers are worth mentioning. In [12] the
authors study the spectra of Dirichlet layers with constant width built around
a special type of two-dimensional smooth surfaces in $\mathbb{R}^{3}$,
assuming that the surface ‘‘flattens out’’ at infinity, meaning its curvature
approaches zero. The authors provide sufficient conditions for trapped wave
emergence in such layers. In [7] the authors expand on the ideas from [12],
dropping some assumptions, and consider more complex topological structures
and a wider class of layers. Nonetheless, the flattening out of the generating
surface at infinity remains a key assumption. Note also [19] and [20] which
develop and generalize the results from the aforementioned two articles, in
particular, to the multi-dimensional case. In [17], the authors allow for a
curvature explosion at one of the surface points, considering a conical layer.
This leads to a surprising result: an infinite number of eigenvalues below the
threshold appears.

This work studies the spectral properties of the Dirichlet problems in
constant width layers constructed from polyhedral three-dimensional angles.
The analog in two-dimensional plane are the ‘‘${\sf L}$-shaped’’ waveguides,
the spectra of which have been well studied (see, for example, works [16, 21,
11, 22] and monograph [14]). In the first known work [8] the question of the
structure of the essential spectrum of a Fichera layer (a layer constructed
from a trihedral angle with all flat angles equal to $90^{\circ}$) is studied
and the finiteness of the number of eigenvalues below the threshold is
established. The result is refined in work [4] where the existence of trapped
waves is proven, and the same result is established in higher dimensions. The
uniqueness of the eigenvalue below the threshold of the continuous spectrum in
this situation remains an open question, which has only been verified
numerically.

This work extends the research in [4] by considering the Dirichlet problem for
the Laplacian operator in a layer $\Pi$ built on a three-dimensional
polyhedral angle. The assumption that a sphere can be placed in the angle to
touch all its faces is imposed. The structure of the essential spectrum is
established, namely, it is shown that it covers the ray with a cut-off point
which coincides with the first eigenvalue $\Lambda_{\dagger}$ of the Dirichlet
problem in an L-shaped waveguide with an opening angle equal to the minimum of
the dihedral angles of $\Pi$. As the essential spectrum consists of non-
isolated points, it coincides with the continuous spectrum of the problem. It
should be noted that the polyhedral layer may have a rich symmetry group and
hence it may occur eigenvalues embedded in continuous spectrum (see, for
instance, [13]). Thus the fine structure of the essential spectrum remains an
open question. It is then established that for regular polyhedral angles the
corresponding layers necessarily have a non-empty discrete spectrum. However,
a general statement of this sort cannot be made — an example of a trihedral
layer with an empty discrete spectrum is provided. Note that in [19] it is
proven that for sufficiently thin constant thickness layers built on the
graphs of smooth strictly convex functions in $\mathbb{R}^{3}$ the discrete
spectrum is non-empty when the layers are flattened at infinity.

The work has the following structure. Section 2 contains a description of the
domain, the statement of the Dirichlet problem and preliminary information
about the spectrum. In Section 3 known results on L-shaped waveguides are
collected. In Section 4 we study the structure of the spectrum, specifically,
we establish the finiteness of the discrete spectrum and prove that the
essential spectrum is the ray $[\Lambda_{\dagger},+\infty)$. Note that this
proof is slightly different from the ones proposed in the works [8] and [4]
and, in our opinion, is more elementary.

In Section 5 an example of a trihedral layer is given, for which the discrete
spectrum is empty. In Section 6 we provide a proof of the non-emptiness of the
discrete spectrum for regular polyhedral layers. This proof generally repeats
the proof from the work [4] and is given for the sake of completeness. Section
7 contains possible generalizations and questions that remain unanswered.

##  2\. Problem statement

###  2.1. Domain geometry

We consider a convex $n$-gon $P_{1}^{\prime}P_{2}^{\prime}\ldots
P_{n}^{\prime}$ ($n\geq 3$) in $\mathbb{R}^{3}$, assuming that its plain
doesn’t contain an origin $O$. A convex hull of rays
$\ell^{\prime}_{j}=O^{\prime}P_{j}^{\prime}$ forms a polyhedral angle with $n$
faces

$\Upsilon^{\prime}=\mathop{\rm conv}\nolimits\\{\ell^{\prime}_{j}\colon 1\leq
j\leq n\\}.$

Its boundary $\Gamma^{\prime}=\partial\Upsilon^{\prime}$ consists of vertex
angles $\Gamma_{j}^{\prime}=\mathop{\rm
conv}\nolimits\\{\ell^{\prime}_{j},\ell^{\prime}_{j+1}\\}$, which are equal to
$\alpha_{j}=\angle P_{j}^{\prime}O^{\prime}P_{j+1}^{\prime}$. The value of the
inner dihedral angle with the edge $\ell^{\prime}_{j}$ we define by
$\beta_{j}$. Here and below we assume that indices are equivalent to each
other with respect to congruence modulo $n$, i.e. $n+1=1$, $n+2=2$, ….

Figure 1. Layers built on regular trihedral and tetrahedral angles

The polyhedral layer of unit width is defined by the formula

$\Pi=\\{x_{\dagger}\in\Upsilon^{\prime}\colon\mathop{\rm
dist}\nolimits(x,\Gamma^{\prime})\in(0,1)\\}.$

The boundary $\partial\Pi$ consists of two polyhedral surfaces: the outer one
coincides with the $\Gamma^{\prime}$ and the inner one we denote by $\Gamma$.
If a ball can be inscribed into the angle $\Upsilon^{\prime}$, then
$\Gamma^{\prime}$ and $\Gamma$ can be matched by shifting. Otherwise, the
inner polyhedral surface $\Gamma$ may be more complex, but in this work we
consider only the first of these situations, which occurs, for instance, in
the case of any trihedral angle, as well as in the case of a regular
polyhedral angle (an angle with equal vertex angles and equal dihedral
angles). For elements of $\Gamma$, corresponding to elements
$\Gamma^{\prime}$, we use the same symbols, but removing the stroke.

We associate a rectangular Cartesian coordinate system $(x_{j},y_{j},z_{j})$
with each dihedral angle of the polyhedral angle $\Upsilon$ defined by its
origin at point $O$ and an orthonormal basis $({\bf e}_{j}^{1},{\bf
e}_{j}^{2},{\bf e}_{j}^{3})$, where the latter vector of the triple is
directed along the ray $\ell_{j}$, the first of the triple lies within one of
the faces, and the second is directed into the half-space containing the
angle.

Below we need one natural decomposition of $\Pi$. For each vertex angle
$\Gamma_{j}$, we will draw a bisector and construct a plane $\pi_{j}$ passing
through this bisector, perpendicular to the plane $\Gamma_{j}$. The region cut
out from the layer $\Pi$ by two such planes $\pi_{j-1}$ and $\pi_{j}$ is
denoted by $\varpi_{j}$. So, the closure of layer $\Pi$ is represented as a
union

$\overline{\Pi}=\bigcup\limits_{j=1}^{n}\overline{\varpi_{j}}.$ (1)

Figure 2. The set $\varpi_{3}$ for the trihedral layer $\Pi$ on Fig. 1

Let us note that the sections of $\varpi_{j}$, which are orthogonal to the
axes $Oz_{j}$ for $z_{j}>0$, are truncated broken waveguides of unit width
with inner angle magnitude equals $\beta_{j}$. The coordinate system
$z_{j}^{\perp}=(x_{j},y_{j})$, whose origin coincides with the inner vertex,
corresponds to this section.

###  2.2. Spectral problem and preliminary information about the spectrum

In this work we study the spectral problem of the Laplace operator

$-\Delta u(x)=\Lambda u(x),\quad x\in\Pi;\quad u(x)=0,\quad x\in\partial\Pi$
(2)

in various domains $\Pi$. The problem (2) admits a variational formulation

$\mathfrak{a}_{\Pi}[u,v]:=(\nabla u,\nabla
v)_{\Pi}=\Lambda(u,v)_{\Pi}\quad\forall v\in\hbox
to0.0pt{$\displaystyle\overset{\circ}{\hphantom{H^{\prime}}\vphantom{\prime}}$\hss}H\vphantom{H}^{1}(\Pi),$

where $(u,v)_{\Pi}$ denotes the standard scalar product in $L_{2}(\Pi)$. The
closed positive definite sesquilinear form $\mathfrak{a}_{\Pi}$ on the Sobolev
space of functions with zero trace on the boundary $\partial\Pi$ in $\hbox
to0.0pt{$\displaystyle\overset{\circ}{\hphantom{H^{\prime}}\vphantom{\prime}}$\hss}H\vphantom{H}^{1}(\Pi)$
generates a positive self-adjoint operator $\mathfrak{A}_{\Pi}$. The structure
of the spectrum of this operator in the domains from Sec. 2.1 is the main
subject of study of this work.

As will be shown in Theorem 1, the essential spectrum of the operator
$\sigma_{ess}(\mathfrak{A}_{\Pi})$ occupies the ray
$[\Lambda_{\dagger},+\infty)$ with a cut-off
$\Lambda_{\dagger}=\Lambda_{\dagger}(\Pi)$ which coincides with the first
eigenvalue in the ${\sf L}$-shaped waveguide with the inner vertex angle
equals to the smallest of the dihedral angles $\beta_{j}$.

The eigenvalues of the operator $\mathfrak{A}_{\Pi}$ that are located below
the threshold $\Lambda_{\dagger}$ are elements of the discrete spectrum
$\sigma_{d}(\mathfrak{A}_{\Pi})$ and can be found using the max-min principle
(see [6, Theorem 10.2.2]).

$\Lambda_{j}(\Pi)\ \ =\sup\limits_{\mathop{\rm codim}\nolimits E=j-1}\
\inf\limits_{u\in E\setminus{\\{\vec{0}\\}}}\ \frac{\|\nabla
u;L_{2}(\Pi)\|^{2}}{\|u;L_{2}(\Pi)\|^{2}},$

where the supremum is taken over all subspaces $E$ of codimension $j-1$ in the
space $\hbox
to0.0pt{$\displaystyle\overset{\circ}{\hphantom{H^{\prime}}\vphantom{\prime}}$\hss}H\vphantom{H}^{1}(\Pi)$.
In particular, the existence of a function $w\in\hbox
to0.0pt{$\displaystyle\overset{\circ}{\hphantom{H^{\prime}}\vphantom{\prime}}$\hss}H\vphantom{H}^{1}(\Pi)$
such that

$\|\nabla w;L_{2}(\Pi)\|^{2}<\Lambda_{\dagger}\|w;L_{2}(\Pi)\|^{2},$

is equivalent to the existence of points of the discrete spectrum below the
threshold. As for the first eigenvalue $\Lambda_{1}$ of the Dirichlet problem
if it exists, it is known to be simple, and the corresponding eigenfunction
can be chosen positive.

Theorem 1 states that the discrete spectrum, if exists, is finite and lies
below the threshold. For a regular polyhedral angular layer the discrete
spectrum is always non-empty (see theorem 3). Theorem 2 gives an example of an
polyhedral angular layer with an empty discrete spectrum.

##  3\. Spectra of ${\sf L}$-shaped waveguides

Below we will need some properties of planar waveguides formed by a broken at
an angle $\theta\in(0,\pi)$ strip of unit width (see Fig. 3). Let us define
them in a similar way to the section 2.1. Consider two rays
$\ell_{1}^{\prime}$ and $\ell_{2}^{\prime}$ on a plane with the vertex
$O^{\prime}$ and angle $\theta$ between them. Their union
$\gamma^{\prime}=\ell_{1}^{\prime}\cup\ell_{2}^{\prime}$ forms an exterior
part of the waveguide boundary. Define $\omega(\theta)$ as a set of all points
inside the angle located at a distance less than one from $\gamma^{\prime}$.
The boundary of such waveguide is a disjunctive union
$\partial\omega(\theta)=\gamma(\theta)\cup\gamma^{\prime}(\theta)$ of the
‘‘interior’’ and ‘‘exterior’’ parts. The vertex of the interior angle we
denote by $O$.

Figure 3. L-shaped waveguide $\omega(\theta)$, truncated waveguides
$\omega^{R-1}(\theta)$ и $\omega^{R}(\theta)$, the support of the function
$\chi^{R}$

Moreover, let us define the truncated waveguide $\omega^{R}(\theta)$ as a
bounded part of $\omega(\theta)$ with the outlets of length $R$ (see. Fig. 3).
The lengths of the outlets of the truncated waveguide $\omega^{R}$ are
measured from the vertex $O$ of the interior angle $\gamma(\theta)$ to the
cross-sections perpendicular to the outlets axis.

The following properties of the Dirichlet spectral problem for the Laplace
operator in the domains $\omega(\theta)$ are valid.

######  Proposition 1.

  * (i)

For all $\theta\in(0,\pi)$ the essential spectrum coincides with a ray
$[\pi^{2},+\infty)$.

  * (ii)

For all $\theta\in(0,\pi)$ the discrete spectrum is nonempty.

  * (iii)

For all $\theta\in(2\arctan(\sqrt{3}/4),\pi)\supset(\pi/2,\pi)$ there is
unique eigenvalue $\lambda_{1}(\omega(\theta))$ below the threshold of the
continuous spectrum.

  * (iv)

For all $\theta\in(0,\pi)$ there exists such $C_{\theta}>0$ that for all $R>1$
for the first eigenvalue $\lambda_{1}(\omega^{R}(\theta))$ of the Laplacian in
the truncated waveguide $\omega^{R}(\theta)$ with the Dirichlet conditions on
the boundary $\partial\omega^{R}(\theta)\cap\partial\omega(\theta)$ and the
Neumann conditions on the cross-sections
$\partial\omega^{R}(\theta)\setminus\partial\omega(\theta)$ the following
estimate holds

$\lambda_{1}(\omega(\theta))-C_{\theta}e^{-2\nu
R}\leq\lambda_{1}(\omega^{R}(\theta))\leq\lambda_{1}(\omega(\theta)),$

with some positive $\nu$. Moreover, the corresponding eigenfunction
$v_{1}(\omega(\theta))$ decays exponentially along the outlet axes and the
following estimate holds

$\|v_{1}(\omega(\theta));\,H^{1}(\omega(\theta)\setminus\omega^{R}(\theta))\|\leq
C_{\theta}e^{-\nu R/2}.$

  * (v)

For any integer $k$ there exists a number $\theta_{k}>0$ such that when
$\theta<\theta_{k}$, there are at least $k$ eigenvalues below the threshold.

  * (vi)

The first eigenvalue $\lambda_{1}(\omega(\theta))$ is a monotonically
increasing function on $(0,\pi)$, while
$\lambda_{1}(\omega(\theta))\to\pi^{2}$ as $\theta\to\pi$ and
$\lambda_{1}(\omega(\theta))\to\pi^{2}/4$ for $\theta\to 0$.

The monograph [14] contains a proof of items (i)-(iii) without exact estimates
for $\theta$. Estimates of the angle at which the eigenvalue is unique are
obtained in [23]. The proof of item (iv) for rectangular waveguides is
contained in [3] and can be easily adapted to the case of an arbitrary angular
waveguide, see the note below. Item (v) of Proposition 1 and the first
statement in item (vi) are proven in [1]. The proof of the second statement of
the last item of the proposition is given in [11].

######  Remark 1.

_Proof of item (iv) for an arbitrary angular waveguide._ The main difference
from the proof presented in [3] is that below the threshold the discrete
spectrum of an ${\sf L}$-shaped waveguide with a sufficiently sharp angle may
consist of several eigenvalues (see item (v) of Proposition 1):

$\lambda_{1}(\omega(\theta))<\lambda_{2}(\omega(\theta))\leq\ldots\leq\lambda_{\kappa}(\omega(\theta))\leq\pi^{2}=\lambda_{\dagger}.$

For the first eigenvalues the identity holds

$\lambda_{1}(\omega(\theta))-\lambda_{1}^{R}(\omega(\theta))=\frac{(v_{1}^{R},[\Delta,\chi^{R}]v_{1})_{\omega^{R}(\theta)}}{(v_{1}^{R},\chi^{R}v_{1})_{\omega^{R}(\theta)}},$

where $v_{1}$, $v_{1}^{R}$ are the normed eigenfunctions corresponding to
$\lambda_{1}(\omega(\theta))$ and $\lambda_{1}^{R}(\omega(\theta))$,
respectively, and $[\Delta,\chi^{R}]$ is the commutator of the Laplacian and
the smooth cut-off function, which equals to 1 in $\omega^{R-1}(\theta)$ and
equals to 0 in $\omega(\theta)\setminus\omega^{R}(\theta)$ (see Fig. 3). The
method of separating variables guarantees exponential decay (with an exponent
of $\sqrt{\pi^{2}-\lambda_{1}(\omega(\theta))}$) of the eigenfunction $v_{1}$
together with its gradient in the outlet of the infinite waveguide, so the
numerator of the fraction in the right part is exponentially small, and it
remains to show that the denominator is separated away from zero.

The denominator $A_{R}=(v_{1}^{R},\chi^{R}v_{1})_{\omega^{R}(\theta)}$ equal
to the coefficient in the decomposition of $\chi^{R}v_{1}$ into the orthogonal
sum

$\chi^{R}v_{1}=A_{R}v_{1}^{R}+B_{R}w^{R},$

where $\|w^{R};L_{2}(\omega^{R}(\theta))\|=1$ and
$(v_{1}^{R},w^{R})_{\omega^{R}(\theta)}=0$. The exponential decay of the
eigenfunction and its gradient in the infinite waveguide implies the estimates

$|A_{R}|^{2}+|B_{R}|^{2}=\|\chi_{R}v_{1};L_{2}(\omega^{R}(\theta)\|^{2}=1-O(e^{-2\sqrt{\pi^{2}-\lambda_{1}(\omega(\theta))}R})$
(3)

and

$\|\nabla
v_{1};L_{2}(\omega^{R}(\theta))\|^{2}=\lambda_{1}(\omega(\theta))+O(e^{-2\sqrt{\pi^{2}-\lambda_{1}(\omega(\theta))}R}).$
(4)

Applying the max-min principle (see [6, Theorem 10.2.2]) to the function
$w^{R}$ we obtain

$\|\nabla(\chi^{R}v_{1});L_{2}(\omega^{R}(\theta))\|^{2}\geq|A_{R}|^{2}\lambda_{1}(\omega^{R}(\theta))+|B_{R}|^{2}\lambda_{2}(\omega^{R}(\theta)).$
(5)

According to [5, Lemma 2.1], the functions $\lambda_{1}(\omega^{R}(\theta))$
and $\lambda_{2}(\omega^{R}(\theta))$ monotonically increase with $R$. From
the proof in [5, Proposition 2] it follows that
$\lambda_{1}(\omega^{R}(\theta))\to\lambda_{1}(\omega(\theta))$ as
$R\to+\infty$, and the limit of $\lambda_{2}(\omega^{R}(\theta))$ is either
equals $\lambda_{2}(\omega(\theta))$, or coincides with the threshold
$\lambda_{\dagger}$, or lies above the threshold. The first eigenvalue
$\lambda_{1}(\omega(\theta))$ of the Dirichlet problem is simple, so in any of
the above cases the limits of the first two eigenvalues of the truncated
waveguides cannot coincide. Thus, from the equalities (3) and (4) and
inequality (5) it follows that $|A_{R}|$ is equal to 1 up to an exponentially
small correction, which proves that the denominator does not vanish for
sufficiently large values of $R$. See [3] for details.

##  4\. Structure of the spectrum

######  Proposition 2.

The following statement holds

$\sigma_{ess}(\mathfrak{A}_{\Pi})\supset[\Lambda_{\dagger},+\infty),$

where $\Lambda_{\dagger}$ coincides with the first eigenvalue
$\lambda_{1}(\omega(\beta_{\dagger}))$ of the Dirichlet spectral problem in a
planar broken waveguide $\omega(\beta_{\dagger})$ with the angle
$\beta_{\dagger}=\min\limits_{1\leq j\leq n}\beta_{j}$.

###### Proof.

For all $\kappa\geq 0$ we construct the Weil sequence for the operator
$\mathfrak{A}$ at the point $\lambda_{1}(\omega(\beta_{\dagger}))+\kappa^{2}$.
Denote $v_{\dagger}$ the non-negative eigenfunction in the waveguide
$\omega(\beta_{\dagger})$, corresponding to the eigenvalue
$\lambda_{1}(\omega(\beta_{\dagger}))$ and normed in the space
$L_{2}(\omega(\beta_{\dagger}))$. The layer $\Pi$ for all $R>0$ contains
arbitrarily long cylinders with sections $\omega^{R}(\beta_{\dagger})$. We
define the elements of the Weyl sequence in such cylinders by multiplying the
eigenfunction $v_{\dagger}$ in their cross section by an appropriate imaginary
exponent depending on the longitudinal coordinate, as well as by the necessary
cutting functions. Let us describe the construction in more details.

Suppose $\beta_{j}=\beta_{\dagger}$. Consider a part of the layer
$\varpi_{j}=\varpi_{\dagger}$ from the partition (1). Associate with
$\varpi_{\dagger}$ the Cartesian system
$(x_{\dagger},y_{\dagger},z_{\dagger})=(x_{j},y_{j},z_{j})$. Its sections
orthogonal to the $Oz_{\dagger}$ axis at $z_{\dagger}>0$ are truncated
waveguides of unit width broken at an angle $\beta_{\dagger}$. By analogy, we
denote the corresponding coordinate system in these sections by
$z_{\dagger}^{\perp}=(x_{\dagger},y_{\dagger})$.

Denote by $\mathcal{X}=\mathcal{X}(z)$ a smooth function that vanishes at
$z\leq 0$, equals one at $z\geq 1$, and satisfies the inequality
$0\leq\mathcal{X}(z)\leq 1$. With its help we define cut-off functions along
the $z$ axis: $\mathcal{X}_{n}(z)=\mathcal{X}(z-2^{n})\mathcal{X}(2^{n+1}-z)$.
In each section orthogonal to the $Oz_{\dagger}$ axis we define, as before,
the cut-off function $\chi^{R(n)}=\chi^{R(n)}(x_{\dagger},y_{\dagger})$, where
$R(n)$ is the length of the smallest of the outlets in a cross-section
corresponding to the plane $z_{\dagger}=2^{n}$. Note that $R(n)$ tends to
infinity as $n$ grows. Finally, we define the sequence of cut-off functions
$\eta_{n}(x_{\dagger},y_{\dagger},z_{\dagger})=\mathcal{X}_{n}(z_{\dagger})\chi^{R(n)}(x_{\dagger},y_{\dagger})$.
The sequence of functions

$\Psi_{n}(x)=\begin{cases}2^{-n/2}v_{\dagger}(x_{\dagger},y_{\dagger})e^{i\kappa
z_{\dagger}}\eta_{n}(x_{\dagger},y_{\dagger},z_{\dagger}),\qquad&(x_{\dagger},y_{\dagger},z_{\dagger})\in\varpi_{\dagger},\\\
0,&(x_{\dagger},y_{\dagger},z_{\dagger})\in\Pi\setminus\varpi_{\dagger}\end{cases}$

converges weakly to zero in $L_{2}(\varpi_{\dagger})$, since their
$L_{2}(\varpi_{\dagger})$-norms do not exceed one and their supports do not
intersect. Moreover, the Proposition 1 (iv) entails the estimate

$\|\Psi_{n};\,L_{2}(\varpi_{\dagger})\|^{2}\geq
1-C_{\dagger}e^{-c_{\dagger}n},\qquad C_{\dagger},\,c_{\dagger}>0,$

from which it follows that the norms of $\Psi_{n}$ are separated away from
zero. The norm of discrepancy

$\|-\Delta\Psi_{n}-(\lambda_{1}(\omega(\beta_{\dagger}))+\kappa^{2})\Psi_{n};\,L_{2}(\varpi_{\dagger})\|$

can be estimated by the sum

$2^{-n/2}\left(\|\Delta_{z_{\dagger}^{\perp}}\eta_{n}v_{\dagger};\,L_{2}(\varpi_{\dagger})\|+2\|\nabla_{z_{\dagger}^{\perp}}\eta_{n}\nabla_{z_{\dagger}^{\perp}}v_{\dagger};\,L_{2}(\varpi_{\dagger})\|\right)+\\\
+2^{-n/2}\left(\|\partial^{2}_{z_{\dagger}}\eta_{n}v_{\dagger};\,L_{2}(\varpi_{\dagger})\|+2\kappa\|\partial_{z_{\dagger}}\eta_{n}v_{\dagger};\,L_{2}(\varpi_{\dagger})\|\right).$

Since in each section the supports of the functions
$\Delta_{z_{\dagger}^{\perp}}\eta_{n}$ and
$\nabla_{z_{\dagger}^{\perp}}\eta_{n}$ are enclosed inside
$\omega^{R(n)}(\theta_{\dagger})\setminus\omega^{R(n)-1}(\theta_{\dagger})$,
while the $v_{\dagger}$ eigenfunction and its gradient decay exponentially
along each outlet (see Proposition 1 (iv)), the first term decreases
exponentially as $n$ increases. The supports of the functions
$\partial_{z_{\dagger}}\mathcal{X}_{n}$ and
$\partial^{2}_{z_{\dagger}}\mathcal{X}_{n}$ contained in the union of two
cylinders of the unit height:

$\omega^{R(n)}(\theta_{\dagger})\times\Big{(}[2^{n},2^{n}+1]\cup[2^{n+1}-1,2^{n+1}]\Big{)}.$

The function $v_{\dagger}$ is normalized in the space
$L_{2}(\omega(\theta_{\dagger}))$, therefore, due to the exponential smallness
of the normalizing factor $2^{-n/2}$, the second term in the discrepancy
estimate also decreases exponentially with increasing of $n$. As a result, the
$L_{2}(\varpi_{\dagger})$-norm of the discrepancy tends to zero as $n$
increases.

Thus, the functions $\Psi_{n}$ form the Weil sequence for the number
$\lambda_{1}(\omega(\beta_{\dagger}))+\kappa^{2}$, which according to [6,
Theorem 9.1.2] belongs to the essential spectrum. ∎

To prove the finiteness of the discrete spectrum, we need an auxiliary Hardy-
type inequality.

######  Lemma 1.

For all functions $v\in H^{1}(1,+\infty)$ the inequality holds

$\|\rho v;L_{2}(2,+\infty)\|^{2}\leq
4\|v^{\prime};L_{2}(2,+\infty)\|^{2}+2\|v^{\prime};L_{2}(1,2)\|^{2}+2\|v,L_{2}(1,2)\|^{2}$

for $\rho(z)=z^{-1}$.

###### Proof.

Denote the left-hand side of the inequality by $A^{2}$ and, using the
Newton–Leibniz formula, estimate it as follows

$A^{2}=\int_{2}^{+\infty}z^{-2}\left|2\int_{2}^{z}v(t)v^{\prime}(t)dt+v^{2}(2)\right|dz\leq\\\
\leq
2\int_{2}^{+\infty}z^{-2}\int_{2}^{z}|v(t)||v^{\prime}(t)|dt\,dz+\frac{1}{2}|v(2)|^{2}.$

Changing the order of integration using the Tonelli theorem in the last
integral and then applying the Cauchy–Swartz inequality, we conclude that

$A^{2}\leq 2AB+\frac{1}{2}C^{2},\quad\mbox{где}\quad
B=\left(\int_{2}^{+\infty}|v^{\prime}(t)|^{2}\right)^{\frac{1}{2}},\,C=|v(2)|.$

Applying the arithmetic – geometric mean inequality to the right hand side, we
arrive at

$A^{2}\leq\frac{1}{2}A^{2}+2B^{2}+\frac{1}{2}C^{2},$

and hence

$A^{2}\leq 4B^{2}+C^{2}.$ (6)

To estimate the last term $C^{2}=|v(2)|^{2}$, we write the Newton–Leibniz
formula

$v(t)-v(s)=\int_{s}^{t}v^{\prime}(z)\,dz$

for $t,s\in[1,2]$. Integrating both parts of this equality with respect to the
variable $s$ from 1 to 2, we obtain the relation

$v(t)=\int_{1}^{2}v(t)\,ds\leq\int_{1}^{2}|v(s)|\,ds+\int_{1}^{2}\int_{s}^{t}|v^{\prime}(z)|\,dz\,ds\\\
\leq\int_{1}^{2}|v(s)|\,ds+\int_{1}^{2}|v^{\prime}(z)|\,dz\leq\|v;L_{2}(1,2)\|+\|v^{\prime};L_{2}(1,2)\|.$

In the last estimate, we used the Cauchy–Swartz inequality. As a result, we
get that

$|v(t)|^{2}\leq 2\|v^{\prime};L_{2}(1,2)\|^{2}+2\|v;L_{2}(1,2)\|^{2}$ (7)

for $t\in[1,2]$. Combining together the inequalities (6) and (7), we arrive at
the required assertion. ∎

We need a slightly more rough estimate from the following corollary.

######  Corollary 1.

For any $v\in H^{1}(1,+\infty)$ and any $R_{0}\geq 2$ the inequality

$\|\rho v;L_{2}(R_{0},+\infty)\|^{2}\leq
4\|v^{\prime};L_{2}(1,+\infty)\|^{2}+2\|v,L_{2}(1,R_{0})\|^{2}$

for $\rho(z)=z^{-1}$.

######  Theorem 1.

The discrete spectrum of the operator $\mathfrak{A}_{\Pi}$ in the angle layer
$\Pi$ is finite and lies below the threshold $\Lambda_{\dagger}$. The
essential spectrum of the operator $\mathfrak{A}_{\Pi}$ occupies the ray
$[\Lambda_{\dagger},+\infty)$.

###### Proof.

From the Proposition 2 it follows that the discrete spectrum lies below the
threshold. To prove its finiteness, we will use an analogue of the
Dirichlet–Neumann bracketing (see [24, XIII.15]), namely, cut the Dirichlet
layer $\Pi$ into domains $\varpi_{j}$ (partition (1)), preserving the
Dirichlet condition on the boundary $\partial\Pi\cap\partial\varpi_{j}$ and
assigning the Neumann condition on the flat boundaries of the cuts
$\pi_{j}\cap\Pi$. This partition corresponds to the operator
$\widetilde{\mathfrak{A}}_{\Pi}$ with quadratic form

$\widetilde{\mathfrak{a}}_{\Pi}(u,u)=\sum_{j=1}^{n}\|\nabla
u;L_{2}(\varpi_{j})\|^{2},$

defined on the space

$\mathcal{H}(\Pi)=\\{u\in L^{2}(\Pi)\colon u|_{\varpi_{j}}\in
H^{1}(\varpi_{j})\mbox{ for }1\leq j\leq n,u|_{\partial\Pi}=0\\}.$

According to the max-min principle (see [24, Theorem XIII.2] or [6, Theorem
10.2.2]), the eigenvalues of the operator $\widetilde{\mathfrak{A}}_{\Pi}$ do
not exceed the corresponding eigenvalues of the operator $\mathfrak{A}_{\Pi}$,
and hence the finiteness of the discrete spectrum of the mixed boundary value
problem entails the finiteness of the discrete spectrum of the operator
$\mathfrak{A}_{\Pi}$.

The operator $\widetilde{\mathfrak{A}}_{\Pi}$ is a direct sum of Laplacians
with mixed boundary conditions on the partition elements, so it suffices to
prove that, given on $\varpi_{j}$, the Laplace operator with the Neumann
conditions on the cuts
$\partial\varpi_{j}\setminus(\Gamma\cup\Gamma^{\prime})$ and the Dirichlet
conditions on the rest of the boundary has a finite number of eigenvalues
below $\Lambda_{\dagger}$.

In what follows we will not write the index $j$. The assertion that the
discrete spectrum is finite follows from the existence of such finite-
dimensional space $E\subset L_{2}(\Pi)$, that for all $u$ orthogonal $E$ the
following inequality holds

$\frac{\|\nabla u;\ L_{2}(\varpi)\|^{2}}{\|u;\
L_{2}(\varpi)\|^{2}}\geq\Lambda_{\dagger}.$ (8)

Figure 4. The set $\varpi_{\bullet}$ and the part of the cylinder $Q$ that
fell into it in the case of a symmetric domain $\varpi$

Let us introduce some notation. We denote by $\ell(R)=cR$ the length of the
smaller of the two arms in the cross-section $\varpi$ by the plane $z=R$.
Denote by $Q$ the cylinder $\omega^{\ell(1)}\times(1,+\infty)$ inscribed in
$\varpi$. The set $\varpi$ is divided into two parts: the compact
$\varpi_{\bullet}=\varpi\cap\\{z\leq R_{0}\\}$ and the unbounded
$\widetilde{\varpi}=\varpi\cap\\{z\geq R_{0}\\}$ (parameter $R_{0}$ will be
chosen later). For the function $u\in\mathcal{H}(\Pi)$ denote by
$u_{z}(x,y)=u(x,y,z)$.

For $z\geq 0$ the inequality holds

$\|\nabla
u_{z};L_{2}(\omega^{\ell(z)})\|^{2}\geq\lambda_{1}(\omega^{\ell(z)})\|u_{z};L_{2}(\omega^{\ell(z)})\|^{2},$
(9)

and for $z\geq 1$ outside the cylinder $Q$, due to the Friedrichs inequality,
we can write that

$\|\nabla u_{z};L_{2}(\omega^{\ell(z)})\|^{2}\geq\|\nabla
u_{z};L_{2}(\omega^{\ell(z)}\setminus\omega^{\ell(1)})\|^{2}\geq\pi^{2}\|u_{z};L_{2}(\omega^{\ell(z)}\setminus\omega^{\ell(1)})\|^{2}.$
(10)

Multiplying the inequality (9) by $(1-(4z\pi)^{-2})$

$(1-(4z\pi)^{-2})\|\nabla
u_{z};L_{2}(\omega^{\ell(z)})\|^{2}\geq(1-(4z\pi)^{-2})\lambda_{1}(\omega^{\ell(z)})\|u_{z};L_{2}(\omega^{\ell(z)})\|^{2},$

and writing the inequality (10) as

$(4z\pi)^{-2}\|\nabla
u_{z};L_{2}(\omega^{\ell(z)}\setminus\omega^{\ell(1)})\|^{2}+(4z)^{-2}\|u_{z};L_{2}(\omega^{\ell(1)})\|^{2}\geq(4z)^{-2}\|u_{z};L_{2}(\omega^{\ell(z)})\|^{2},$

add the resulting estimates. As a result, we have

$\|\nabla
u_{z};L_{2}(\omega^{\ell(z)})\|^{2}+(4z)^{-2}\|u_{z};L_{2}(\omega^{\ell(1)})\|^{2}\geq\\\
\geq\Big{(}(1-(4z\pi)^{-2})\lambda_{1}(\omega^{\ell(z)})+(4z)^{-2}\Big{)}\|u_{z};L_{2}(\omega^{\ell(z)})\|^{2}.$
(11)

By virtue of Proposition 1 (iv) the coefficient on the right-hand side of (11)
satisfies the inequality

$(1-(4z\pi)^{-2})\lambda_{1}(\omega^{\ell(z)})+(4z)^{-2}\geq\lambda_{1}(\omega)+(4z)^{-2}(1-\lambda_{1}(\omega)\pi^{-2})-Ce^{-\nu\ell(z)},$

and hence there exists $R_{0}\geq 2$ such that for $z\geq R_{0}$ the
inequality holds

$(1-(4z\pi)^{-2})\lambda_{1}(\omega^{\ell(z)})+(4z)^{-2}>\lambda_{1}(\omega),$

and in the end

$\|\nabla
u_{z};L_{2}(\omega^{\ell(z)})\|^{2}+(4z)^{-2}\|u_{z};L_{2}(\omega^{\ell(1)})\|^{2}\geq\lambda_{1}(\omega)\|u_{z};L_{2}(\omega^{\ell(z)})\|^{2}.$

Integrating the last estimate over $z\in[R_{0},+\infty)$, we derive the
inequality

$\|\nabla_{z^{\perp}}u;L_{2}(\widetilde{\varpi})\|^{2}+\frac{1}{16}\|\rho
u;L_{2}(\widetilde{\varpi}\cap
Q)\|^{2}\geq\lambda_{1}(\omega)\|u;L_{2}(\widetilde{\varpi})\|^{2}.$

Recall that, as before, $\rho(z)=z^{-1}$. Applying the Corollary 1 to the
second term, we obtain the relation

$\|\nabla_{z^{\perp}}u;L_{2}(\widetilde{\varpi})\|^{2}+\frac{1}{4}\|\partial_{z}u;L_{2}(Q)\|^{2}\geq\\\
\geq\lambda_{1}(\omega)\|u;L_{2}(\widetilde{\varpi})\|^{2}-\frac{1}{8}\|u;L_{2}(\varpi_{\bullet}\cap
Q)\|^{2}.$ (12)

Note that by imposing the orthogonality conditions to the first eigenfunctions
of the Laplace operator with the Dirichlet conditions on
$\partial\varpi_{\bullet}\cap\partial\Pi$ and the Neumann conditions on the
rest of the boundary on the function $u$, for any $M>0$ one can achieve the
inequality

$\|\nabla u;L_{2}(\varpi_{\bullet})\|^{2}>M\|u;L_{2}(\varpi_{\bullet})\|^{2}.$

Adding this inequality, divided by 2, with the inequality (12), we obtain the
estimate

$\|\nabla_{z^{\perp}}u;L_{2}(\widetilde{\varpi})\|^{2}+\frac{1}{4}\|\partial_{z}u;L_{2}(\widetilde{\varpi}\cap
Q)\|^{2}+\\\ \frac{1}{4}\|\partial_{z}u;L_{2}(\varpi_{\bullet}\cap
Q)\|^{2}+\frac{1}{2}\|\nabla u;L_{2}(\varpi_{\bullet})\|^{2}\geq\\\
\lambda_{1}(\omega)\|u;L_{2}(\widetilde{\varpi})\|^{2}-\frac{1}{8}\|u;L_{2}(\varpi_{\bullet}\cap
Q)\|^{2}+\frac{M}{2}\|u;L_{2}(\varpi_{\bullet})\|^{2}.$

If $M\geq 2(\lambda_{1}(\omega)+1)$, then

$\|\nabla
u;L_{2}(\varpi)\|^{2}\geq\lambda_{1}(\omega)\|u;L_{2}(\varpi)\|^{2},$

which gives an estimate for the threshold of the continuous spectrum. One can
choose the linear span of eigenfunctions from the orthogonality condition on
$\varpi_{\bullet}$, extended by zero to $\varpi$, as the space $E$.

It follows from the Proposition 2 that the ray $[\Lambda_{\dagger},+\infty)$
is contained in the essential spectrum. According to the max-min principle,
the inequality (8) holds true for all functions from $\hbox
to0.0pt{$\displaystyle\overset{\circ}{\hphantom{H^{\prime}}\vphantom{\prime}}$\hss}H\vphantom{H}^{1}(\Pi)$
that are orthogonal to the subspace $E$ in $L_{2}(\Pi)$ and hence it
guarantees that $\Lambda_{\dagger}$ is a cutt-off point of the essential
spectrum, and all eigenvalues below this point belong to the discrete
spectrum. ∎

##  5\. An example of the absence of a discrete spectrum

Despite the fact that all the layers constructed by trihedral angles are
structurally similar, it turns out that the discrete spectrum does not appear
in all cases. A corresponding example is given in the following theorem.

######  Theorem 2.

In the layer $\Pi$, constructed from a trihedral angle with two right vertex
angles at the vertex $O^{\prime}$ (see Fig. 5 on the left), for sufficiently
small values of the third vertex angle $\alpha$ the spectral Dirichlet problem
for the Laplace operator has no eigenvalues below the threshold of the
essential spectrum.

Figure 5. A layer constructed from a trihedral angle with two right vertex
angles and a small third vertex angle and its partition

###### Proof.

The essential spectrum in the domain $\Pi$ by the Theorem 1 is determined by
the smallest dihedral angle, which is equal to $\alpha$. Thus, it is
sufficient to check that for any function $u\in\hbox
to0.0pt{$\displaystyle\overset{\circ}{\hphantom{H^{\prime}}\vphantom{\prime}}$\hss}H\vphantom{H}^{1}(\Pi)$
the following inequality holds

$\|\nabla
u;L_{2}(\Pi)\|^{2}\geq\lambda_{1}(\omega(\alpha))\|u;L_{2}(\Pi)\|^{2}.$ (13)

Assume that $\ell_{1}$ is an edge of a smallest dihedral angle equal to
$\alpha$. Let’s use the coordinate system $(x_{1},y_{1},z_{1})$ associated
with this angle, as described in §2. To check the inequality (13) we divide
$\Pi$ into three parts $\Pi=\Pi_{1}\cup\Pi_{2}\cup\Pi_{3}$

$\displaystyle\Pi_{1}=\Pi\cap\\{z_{1}>0\\},$
$\displaystyle\Pi_{2}=\Pi\cap\\{0<y_{1}<\tan(\alpha)x_{1}\\},$
$\displaystyle\Pi_{3}=\Pi\setminus(\Pi_{1}\cup\Pi_{2}).$

The parts $\Pi_{j}$ are shown schematically in Fig. 5 on the right.

In the part $\Pi_{1}$ we integrate over $z_{1}$ the Friedrichs inequality on
the cross-sections perpendicular to the ray $\ell_{1}$ and obtain the estimate

$\|\nabla
u;L_{2}(\Pi_{1})\|^{2}\geq\lambda_{1}(\omega(\alpha))\|u;L_{2}(\Pi_{1})\|^{2}.$

In the part $\Pi_{2}$ the Friedrichs inequality on segments parallel to the
ray $\ell_{1}$ leads to the estimate

$\|\partial_{z_{1}}u;L_{2}(\Pi_{2})\|^{2}\geq\pi^{2}\|u;L_{2}(\Pi_{2})\|^{2}\geq\lambda_{1}(\omega(\alpha))\|u;L_{2}(\Pi_{2})\|^{2}.$

In the part $\Pi_{3}$ for the $z_{1}$-derivative we write the Friedrichs
inequality along segments parallel to the $\ell_{1}$ axis

$\|\partial_{z_{1}}u;L_{2}(\Pi_{3})\|^{2}\geq\frac{\pi^{2}}{4}\|u;L_{2}(\Pi_{3})\|^{2}.$
(14)

To estimate the transverse gradient, we split the part $\Pi_{3}$ by the
vertical plain of symmetry contained $O^{\prime}\ell^{\prime}_{1}$, and write
the Friedrichs inequality along segments, that are orthogonal to the outer
boundary. As the lengths of the segments do not exceed one, we obtain that

$\|\nabla_{z_{1}^{\perp}}u;L_{2}(\Pi_{3})\|^{2}\geq\frac{\pi^{2}}{4}\|u;L_{2}(\Pi_{3})\|^{2}.$
(15)

Combining together the inequalities (14) and (15), we conclude that

$\|\nabla
u;L_{2}(\Pi_{3})\|^{2}\geq(\pi^{2}/4+\pi^{2}/4)\|u;L_{2}(\Pi_{3})\|^{2}\geq\lambda_{1}(\omega(\alpha))\|u;L_{2}(\Pi_{3})\|^{2}.$

The last inequality by point (vi) of Proposition 1 is true for sufficiently
small angles $\alpha$. ∎

##  6\. Existence of a discrete spectrum for regular angles

Recall that we call an angle layer regular if its outer boundary
$\Gamma^{\prime}$ forms a regular polyhedral cone, i.e. all dihedral angles
$\beta_{j}$ are equal and all vertex angles $\alpha_{j}$ at the vertex
$O^{\prime}$ are equal. In what follows we denote the corresponding quantities
by $\beta$ and $\alpha$; the plane waveguide $\omega(\beta)$ in the section
perpendicular to the edge of the angle layer is denoted by $\omega$.

######  Theorem 3.

In a regular angular layer the discrete spectrum is not empty.

###### Proof.

According to the maximum-minimum principle to prove the existence of an
eigenvalue below the threshold $\Lambda_{\dagger}=\lambda_{1}(\omega)$, it is
sufficient to construct a function $V\in\hbox
to0.0pt{$\displaystyle\overset{\circ}{\hphantom{H^{\prime}}\vphantom{\prime}}$\hss}H\vphantom{H}^{1}(\Pi)$,
such that the following inequality holds

$\|\nabla V;\,L_{2}(\Pi)\|^{2}-\lambda_{1}(\omega)\|V;\,L_{2}(\Pi)\|^{2}<0.$
(16)

Similarly to the proof of the Theorem 1, we cut the layer into parts
$\varpi_{j}$ (partition (1)). Since they are all the same, we will denote one
of them as $\varpi_{\dagger}$. Define in $\varpi_{\dagger}$ the function
$V^{\varepsilon}(x_{\dagger},y_{\dagger},z_{\dagger})=v_{\dagger}(x_{\dagger},y_{\dagger})e^{-\varepsilon
z_{\dagger}}$, where $\varepsilon>0$, a $v_{\dagger}$ is the first
eigenfunction of the Dirichlet Laplacian in the broken waveguide $\omega$. A
regular angular layer has a rotational symmetry, thus the function
$V^{\varepsilon}$ is continuous on the faces $\pi_{j-1}\cap\varpi_{j}$ and
$\pi_{j}\cap\varpi_{j}$, and hence $V^{\varepsilon}\in\hbox
to0.0pt{$\displaystyle\overset{\circ}{\hphantom{H^{\prime}}\vphantom{\prime}}$\hss}H\vphantom{H}^{1}(\Pi)$.
Substituting $V^{\varepsilon}$ into the left-hand side of the inequality (16),
after integrating by parts we have

$(-\Delta_{z_{\dagger}^{\perp}}V^{\varepsilon},V^{\varepsilon})_{\varpi_{\dagger}}-\lambda_{1}(\omega)\|V^{\varepsilon};\,L_{2}(\varpi_{\dagger})\|^{2}+\\\
+\varepsilon^{2}\|V^{\varepsilon};\,L_{2}(\varpi_{\dagger})\|^{2}+\int_{\partial\varpi_{\dagger}}\vec{n}\cdot\nabla_{z_{\dagger}^{\perp}}v_{\dagger}(z^{\perp}_{\dagger})v_{\dagger}(z^{\perp}_{\dagger})e^{-2\varepsilon
z_{\dagger}}dS,$ (17)

where $\vec{n}=(n_{x},n_{y},n_{z})$ is the outer normal vector to the boundary
$\partial\varpi_{\dagger}\setminus(\Gamma\cup\Gamma^{\prime})$, and $dS$ is
the standard surface Lebesgue measure on this boundary. Since $v_{\dagger}$ is
an eigenfunction corresponding to $\lambda_{1}(\omega)$, the first two terms
cancel. The plane of symmetry passing through the $Oz_{\dagger}$ axis divides
the layer $\varpi_{\dagger}$ into two parts $\varpi_{+}$ and $\varpi_{-}$, we
assume that $\varpi_{+}$ is directed along $Ox_{\dagger}$. Denote the common
end of $\varpi_{\dagger}$ and $\varpi_{+}$ lying in the plane
$z_{\dagger}=\cot(\alpha/2)x_{\dagger}$ by $\pi_{+}$. As a result, the sum
(17) is converted to the form

$\varepsilon^{2}\|V^{\varepsilon};\,L_{2}(\varpi_{\dagger})\|^{2}+2\\!\int_{\pi_{+}}\\!\\!\\!\\!n_{x}\partial_{x_{\dagger}}v_{\dagger}(x_{\dagger},y_{\dagger})v_{\dagger}(x_{\dagger},y_{\dagger})e^{-2\varepsilon\cot(\alpha/2)x_{\dagger}}dS(x_{\dagger},y_{\dagger}).$

Given that
$dS(x_{\dagger},y_{\dagger})=\sin^{-1}(\alpha/2)\,dx_{\dagger}dy_{\dagger}$,
and $n_{x}=\cos(\alpha_{\dagger}/2)$, the last integral can be written as a
double integral over one of the two symmetrical halves of the waveguide
$\omega$. Let’s denote one of these halves directed along the $Ox_{\dagger}$
axis by $\omega_{+}$, and the other by $\omega_{-}$. In this case the
expression takes the form

$\varepsilon^{2}\|v_{\dagger};\,L_{2}(\varpi_{\dagger})\|^{2}+2\cot\left(\frac{\alpha}{2}\right)\int_{\omega_{+}}\partial_{x_{\dagger}}v_{\dagger}(x_{\dagger},y_{\dagger})v_{\dagger}(x_{\dagger},y_{\dagger})e^{-2\varepsilon\cot(\alpha/2)x_{\dagger}}dx_{\dagger}dy_{\dagger}.$

Integrating by parts we obtain

$\varepsilon^{2}\|v_{\dagger};\,L_{2}(\varpi_{\dagger})\|^{2}+2\varepsilon\cot^{2}\left(\frac{\alpha}{2}\right)\int_{\omega_{+}}\\!\\!|v_{\dagger}(x_{\dagger},y_{\dagger})|^{2}\exp(-2\varepsilon\cot(\alpha/2)x_{\dagger})dx_{\dagger}dy_{\dagger}-\\\
-2\cot\left(\frac{\alpha}{2}\right)\sin\left(\frac{\beta}{2}\right)\int_{\gamma_{0}}|v_{\dagger}(x_{\dagger}(\tau),y_{\dagger}(\tau))|^{2}\exp(-2\varepsilon\cot(\alpha/2)x_{\dagger}(\tau))d\tau,$
(18)

where $\gamma_{0}=\partial\omega_{-}\cap\partial\omega_{+}$, and $\tau$ is the
natural parametrization of the boundary along $\gamma_{0}$. Note that for
sufficiently small $\varepsilon$ the values (18) are negative, so for small
$\varepsilon$ the function $V^{\varepsilon}$ satisfies the inequality (16). ∎

##  7\. Possible generalizations

The results of Theorems 1 and 1 can be extended to layers constructed from
angles that are not necessarily circumscribed around balls. To do this, it is
sufficient to construct an analogue of the partition (1), the structure of
which will be violated only in a compact neighborhood of the angle vertex,
which will not affect the essential spectrum of the operator. Note that the
thickness of the walls of the angular layer can also be made different for
different faces. In general, the results will remain similar, and only the
analysis of the spectra of L-shaped waveguides will become more complicated.

For a regular polygonal layer with small planar angles near the vertex (and
dihedral angles close to $\pi$) it is easy to prove that the number of
eigenvalues below the continuous spectrum can be made arbitrarily large. The
proof is absolutely analogous to the results in [11, 2]. We expect that there
is also monotonicity with respect to the opening angle, similar to one proved
in [9, Proposition 3.1.], but at the moment this statement remains unproven.
In addition, most likely, for sufficiently large opening angles, we expect
uniqueness of the eigenvalue below the continuous spectrum.

Increasing the number of faces in a regular angle inscribed in a given cone
should also lead, in our opinion, to an increase in the number of eigenvalues
below the continuous spectrum, since the region visually becomes similar to a
conical layer (cf. works [17, 10]). However, it turns out that it is not true
that a regular polygonal layer can be inscribed in a conical layer, so it is
not possible to prove this statement by simple estimates between operators.

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