Resolvent expansions on hybrid manifolds

TL;DR

This paper investigates the spectral properties of Laplace-type operators on hybrid manifolds composed of 2D manifolds and 1D segments, deriving resolvent expansions and exploring inverse spectral questions.

Contribution

It introduces a method to analyze resolvent expansions for Laplace operators on hybrid manifolds and addresses inverse spectral problems in this context.

Findings

Derived large spectral parameter expansion of the trace of the second resolvent power

Provided insights into inverse spectral theory for hybrid manifolds

Established a framework for spectral analysis on complex geometric configurations

Abstract

We study Laplace-type operators on hybrid manifolds, i.e. on configurations consisting of closed two-dimensional manifolds and one-dimensional segments. Such an operator can be constructed by using the Laplace-Beltrami operators on each component with some boundary conditions at the points of gluing. The large spectral parameter expansion of the trace of the second power of the resolvent is obtained. Some questions of the inverse spectral theory are adressed.

Equations

$$\sum _{i=1}^m{\mu }_i=2n.$$

$$N=2n.$$

$$G^2(x,x;z)=\sum _{k=0}^K\frac{a_k(x)\Gamma (k+1)}{4\pi z^{2k+2}}+O(z^{-2(K+2)}),$$

$$\mathrm{Tr}{R}^2(z)=\sum _{k=0}^K\frac{a_k\Gamma (k+1)}{4\pi z^{2k+2}}+O(z^{-2(K+2)}),$$

$$a_k={\int }_M{a}_k(x)dx$$

$$\mathrm{Tr}{R}^2\sim \frac{\mathrm{vol}M}{4\pi z^2}+\frac{\chi (M)}{6z^4}+\cdots ,$$

$$\operatorname{Tr}R^{2}(z)=\frac{\sum_{i}\operatorname{vol}M_{i}}{4\pi z^{2}}+\frac{\sum_{j}l_{j}}{4z^{3}}+\sum_{j=4}^{k}\frac{c_{j}(\ln z^{2})}{z^{k}}+O\Big{(}\frac{1}{z^{k+\alpha}}\Big{)},$$

$$⟨x,S^{\ast }y⟩-⟨S^{\ast }x,y⟩=⟨{\Gamma }_{1}x,{\Gamma }_{2}y⟩-⟨{\Gamma }_{2}x,{\Gamma }_{1}y⟩,$$

$$A{B}^{\ast }=B{A}^{\ast }\text{ and }\ker \left(\begin{array}{cc}A& -B\\ B& A\end{array}\right)=0,$$

$$(H^{A,B}-z)^{-1}=(H_{0}-z)^{-1}-\gamma(z)\big{(}BQ(z)-A\big{)}^{-1}B\gamma(\bar{z})^{*}.$$

$$\mathcal{G}={\mathbb{C}}^2,{\Gamma }_{1}f=\left(\begin{array}{c}-f^{\prime }(0)\\ f^{\prime }(l)\end{array}\right),{\Gamma }_{2}f=\left(\begin{array}{c}f(0)\\ f(l)\end{array}\right).$$

$$G_{0}(x,y;z)=\begin{cases}\dfrac{\operatorname{cosh}\big{(}\sqrt{-z}(x-l)\big{)}\operatorname{cosh}\big{(}\sqrt{-z}y\big{)}}{\sqrt{-z}\operatorname{sinh}\big{(}\sqrt{-z}l\big{)}},&\quad x\geq y,\\[12.0pt] \dfrac{\operatorname{cosh}\big{(}\sqrt{-z}(y-l)\big{)}\operatorname{cosh}\big{(}\sqrt{-z}x\big{)}}{\sqrt{-z}\operatorname{sinh}\big{(}\sqrt{-z}l\big{)}},&\quad x\leq y,\end{cases}$$

$$\gamma (z)\left(\begin{array}{c}{\xi }_1\\ {\xi }_2\end{array}\right)={\xi }_1{G}_0(\cdot ,0;z)+{\xi }_2{G}_0(\cdot ,l;z)$$

$$\begin{gathered} Q(z)=\left(\begin{array}{cc}{G}_0(0,0;z)& G_0(l,0;z)\\ G_0(0,l;z)& G_0(l,l;z)\end{array}\right)= \\ \frac{1}{\sqrt{-z}\sinh l\sqrt{-z}}\left(\begin{array}{cc}\cosh l\sqrt{-z}& 1\\ 1& \cosh l\sqrt{-z}\end{array}\right). \end{gathered}$$

$$\mathcal{D}(S^{*})=\big{\{}f=f_{0}+\sum_{j=1}^{\mu}\varphi_{j}G(\cdot,q_{j};z):\,f_{0}\in\mathcal{D}(\Delta),\quad\Phi=(\varphi_{1},\dots,\varphi_{\mu})\in\mathbb{C}^{\mu}\big{\}}.$$

$$G(x,y;z)=\frac{1}{2\pi }\ln \frac{1}{d(x,y)}+F(x,y;z),$$

$$f(x)=\frac{1}{2\pi }\ln \frac{1}{d(x,y)}{a}_j(f)+b_j(f)+o(1)\text{ for }x\to q_j,$$

$$\Gamma_{1}f=\big{(}a_{j}(f)\big{)}_{j=1,\dots,\mu},\quad\Gamma_{2}f=\big{(}b_{j}(f)\big{)}_{j=1,\dots,\mu}.$$

$$\gamma (z)({\xi }_j)=\sum _j{\xi }_{j}G(\cdot ,q_j;z),Q_{jk}(z)=\{\begin{array}{ll}G(q_j,q_k;z),& j\ne k,\\ F(q_j,q_j;z),& j=k.\end{array}$$

$$F(x,x,z)=\frac{1}{4\pi }\left(-\ln z^2-2\gamma +\sum _{n=1}^M\frac{\Gamma (n)a_n(x,x)}{z^{2n}}\right)+O(z^{-2(M+1)}).$$

$$\mathcal{D}(D_i)=\{f\in \mathcal{D}({\Delta }_{M_i}):f(q_{is})=0,s=1,\dots ,{\mu }_i\}.$$

$$S=D_1\oplus \cdots \oplus D_m\oplus D_1^s\oplus \cdots \oplus D_n^s$$

$$A=\left(\begin{array}{cc}(a_{i,i})& (a_{i,i+N})\\ (a_{i+N,i})& (a_{i+N,i+N})\end{array}\right),B=\left(\begin{array}{cc}(b_{i,i})& (b_{i,i+N})\\ (b_{i+N,i})& (b_{i+N,i+N})\end{array}\right).$$

$$A_j=\left(\begin{array}{cc}{a}_{i,i}& a_{i,i+N}\\ a_{i+N,i}& a_{i+N,i+N}\end{array}\right),B_j=\left(\begin{array}{cc}{b}_{i,i}& b_{i,i+N}\\ b_{i+N,i}& b_{i+N,i+N}\end{array}\right),$$

$$A_j=1-{\Phi }_j,B_j=i(1+{\Phi }_j)$$

$${\Phi }_i=\left(\begin{array}{cc}{\phi }_{i,i}& {\phi }_{i,i+N}\\ {\phi }_{i+N,i}& {\phi }_{i+N,i+N}\end{array}\right)\in U(2)$$

$$\Phi =\left(\begin{array}{cc}({\phi }_{i,i})& ({\phi }_{i,i+N})\\ ({\phi }_{i+N,i})& ({\phi }_{i+N,i+N})\end{array}\right)\in U(2N).$$

$${\int }_{M}G(x,u,z)G(u,y,z)du=-\frac{1}{2z}{G}_z^{\prime }(x,y,z)$$

$$T(z)=\gamma (z)[BQ(z)-A{]}^{-1}B(\gamma (\bar{z}){)}^{\ast },$$

$$\mathrm{Tr}{R}^2(z)=\mathrm{Tr}{R}_0^2(z)-\mathrm{Tr}(R_0(z)T(z))-\mathrm{Tr}(T(z)R_0(z))+\mathrm{Tr}{T}^2(z).$$

Full text

1 Introduction

The spectral theory on compact Riemannian manifolds has been studied for a long time and takes its roots in physical problems. A great number of important results has been obtained and this subject has a lot of ramifications. The main objects of investigation in spectral theory are Laplace type operators on a compact manifold, constructed from a Riemannian metric. A related subject which is developing very actively is the spectral theory on manifolds with singularities, see e.g. [5, 8]. Generalizing in another related direction, one can be interested in the spectral theory on metric graphs, or quantum graphs [10],  or more general multistructures [2]. In the present paper we are studying some spectral properties of the Laplacian on spaces composed of one- and two-dimensional pieces.

We consider the so-called “hybrid manifolds”, which are unions of smoth manifolds connected by line segments. More specifically, let us describe it from the topological point of view. Consider a set of $m$ compact Riemannian manifolds $M_{1},...,M_{m}$ and a set of $n$ segments $L_{1},...,L_{n}$. On each manifold $M_{i}$ we fix some points $q_{is}$, $s=1,\dots,\mu_{i}$, $\mu_{i}>0,\,i=1,\dots,m$. First of all we consider the disjoint union of all initial elements: $M_{1}\sqcup\dots\sqcup M_{m}\sqcup L_{1}\sqcup\dots\sqcup L_{n}$. Then we construct a one-to-one correspondence between the set of end points of all segments and the set of points $q_{is},s=1,\dots,\mu_{i},i=1,\dots,M$. The following natural condition on the number of elements must be satisfied

[TABLE]

Finally, according to this correspondence, we glue each end of each segment to the corresponding point on one of the manifolds, which results in a certain topological space. We assume it to be path connected which immediately implies that $n\geqslant m-1$. The topological space obtained by gluing the initial manifolds and segments as described is called a hybrid manifold. If the manifolds are zero-dimensional, then we have a quantum graph. However, from now on, we will assume that our manifolds are 2-dimensional. We will also often relabel the gluing points as $q_{1},\ldots q_{N}$, where

[TABLE]

Our first task will be to define Laplace type operators on a hybrid space. This construction involves the Laplacians on each smooth part as well as boundary conditions at the points of gluing, and will be explained in Section 3. The spectral properties of the operators obtained in this way can be studied using their resolvents or some function of it. In our approach we consider the trace of the squared resolvent. This is a standard procedure in spectral theory. Indeed, if we consider the usual (positive) Laplacian $\Delta$ on a Riemannian surface $M$, then it is known (see e.g. [19]) that $R^{2}(z)=(\Delta+z^{2})^{-2}$ is trace class. If $G^{2}(x,y;z)$ is the integral kernel of this operator, then we have the following asymptotic expansion as $z\rightarrow\infty$ : for all $K\geqslant 0$,

[TABLE]

where the coefficients $a_{k}$ are the local heat kernel invariants of $M$ and can be expressed by some universal expressions containing the metric tensor and its derivates. Furthermore,

[TABLE]

where the coefficients

[TABLE]

are the global heat kernel invariants of $M$. In particular, the first terms of this expansion reveal important geometric and topological characteristics of the surface:

[TABLE]

where $\operatorname{vol}M$ and $\chi(M)$ are the volume and the Euler characteristic of $M$, respectively; this is closely related to the Weyl asymptotics for the eigenvalues, see e.g. [5]. Note that traces of some functions of the Laplacian are also of use in the inverse spectral theory of quantum graphs, see e.g. [4, 12, 15, 18].

Our aim is to obtain similar results for a Laplace type operator on a hybrid manifold. In fact, due to the singular structure of the hybrid space, we won’t get an asymptotic expansion in the usual way but in a certain extended form. More precisely, we show (subsection 3.3)

Theorem 1**.**

Let $R(z)$ denote the resolvent of a Laplace type operator on a hybrid manifold. Then $\operatorname{Tr}R^{2}(z)$ has an expansion of the following form: for any $k\geq 4$ and $\alpha\in(0,1)$ one has

[TABLE]

where $\operatorname{vol}M_{i}$ (resp. $l_{j}$) is the volume of the manifold $M_{i}$ (resp. of the segment $L_{j}$) used to construct the hybrif manifold, and the coefficients $c_{j}$ are rational functions.

Each coefficient $c_{j}$ is actually a rational function of nonpositive degree. Therefore, each $c_{j}(\ln z^{2})$ has an aymptotic expansion in powers of $1/\ln z^{2}$ for large $z$. For a rather large class of boundary conditions (see the beginning of Section 5 for more details), we are able to compute the first terms of the expansion of $c_{k}$. This allows us to address some quesions in inverse spectral theory : given the expansion of $\operatorname{Tr}R^{2}(z)$ in Theorem 1, can we recover some topological and metric parameters of the hybrid manifold from the coefficients? Under some generic assumptions we prove the following result (subsection 4.2):

Theorem 2**.**

Consider the expansion of the trace of the square of the resolvent as in Theorem 1. The knowledge of $TrR^{2}$ determines:

• •

whether this manifold is hybrid or smooth;

• •

the sum of the volumes of all manifolds taking part in the construction;

• •

the sum of the Euler characteristics of all manifolds;

• •

the number of segments used in this hybrid manifold;

• •

the sum of the lengths of these segments;

• •

the Euler characteristic of the hybrid manifold.

We mention that a rather similar result for quantum graphs was obtained in [14].

Finally, we will show in Theorem 22 that the expansion given in Theorem 1 contains some information about the boundary conditions used to construct our Laplace type operator. We note that there is a certain canonical choice of boundary conditions for quantum graphs resulting from considering shrinking manifolds approching the quantum graph [11]; no similar results are available for hybrid spaces, hence it is reasonable to take into account a possibly large class of couplings. Our paper is based heavily of the the PhD-thesis of the second named author [17] where a class of the so-called disjoint boundary conditions was discussed. Later in [20] another class of boundary conditions was suggested. We are treating here the most general boundary conditions. It comes out that this requires a certain extension of the notion of pseudoasymptotic expansion introduced in [19], but the most essential features of the coefficients and their interpretation are preserved.

Acknowledgments. As already mentioned, the paper is based and extends the PhD-thesis of Svetlana Roganova. She would like to thank her advisor Jochen Brüning for suggesting the problem and for many useful discussions, hints and comments.

2 Definition of the Laplacian on a hybrid manifold

2.1 Basic facts on self-adjoint extensions

We recall here some basic facts on the theory of self-adjoint boundary conditions; for a more detailed presentation see e.g. [6, 7].

Definition 3**.**

Let $S$ be a closed densely defined symmetric linear operator acting on a Hilbert space $\mathcal{H}$. Let $\Gamma_{1},\Gamma_{2}$ be two linear mappings from $\mathcal{D}(S^{*})$ into a Hilbert space $\mathcal{G}$. The triple $(\mathcal{G},\Gamma_{1},\Gamma_{2})$ is called a boundary value space for $S$ if the following three conditions are satisfied:

• •

for all $x,y\in\mathcal{D}(S^{*})$ there holds

[TABLE]

• •

for any $u,v\in\mathcal{G}$ there exists $x\in\mathcal{D}(S^{*})$ such that $\Gamma_{1}x=u$ and $\Gamma_{2}x=v$,

• •

$\ker\Gamma_{1}\cap\ker\Gamma_{2}=\mathcal{D}(S)$.

The existence of a boundary triple in the above sense is equivalent to the fact that the deficiency indices of $S$ are equal, e.g. that $S$ possesses self-adjoint extensions. Boundary triples deliver a useful machinery for describing all self-adjoint extensions of $S$.

In what follows in this subsection let $(\mathcal{G},\Gamma_{1},\Gamma_{2})$ be a boundary triple for $S$.

Proposition 4**.**

(1) Let $A$ and $B$ be bounded linear operators in $\mathcal{G}$ satisfying

[TABLE]

then the restriction $H^{A,B}$ of $S^{*}$ to the vectors $f\in\mathcal{D}(S^{*})$ with $A\Gamma_{1}f=B\Gamma_{2}f$ is a self-adjoint extension of $S$. Moreover, each self-adjoint extension of $S$ can be represented in such a form.

(2) There exists a bijection between unitary operators $U$ in $\mathcal{G}$ and self-adjoint extensions $H^{U}$ fo $S$ given by $U\leftrightarrow H^{U}=H^{A,B}$ with $A=1-U$ and $B=i(1+U)$.

Due to the above proposition the operator $H_{0}$ which is the restriction of $S^{*}$ to $\ker\Gamma_{1}$ is self-adjoint. For $z\notin\operatorname{spec}H_{0}$ the restriction of $\Gamma_{1}$ onto $\ker(S^{*}-z)$ is a bijection; denote the inverse operator by $\gamma(z)$. Clearly, $\gamma(z)$ acts from $\mathcal{G}$ to $\mathcal{H}$, and we will refer to this operator as to $\gamma$-field for the boundary triple, and the operator $Q(z):=\Gamma_{2}\gamma(z)$ will be called the Weyl function.

Proposition 5** (Krein resolvent formula).**

For $z\notin\operatorname{spec}H^{A,B}\cup\operatorname{spec}H_{0}$ there holds

[TABLE]

2.2 Example 1: Laplacian on a segment

In the Hilbert space $\mathcal{H}=L^{2}[0,l]$, $l>0$, consider the operator $S=-d^{2}/dt^{2}$ with the domain $\mathcal{D}(S)=\{f\in H^{2}[0,l]:f(0)=f(l)=f^{\prime}(0)=f^{\prime}(l))\}$. One can easily show that the adjoint operator $S^{*}$ is given by the same expression on the domain $\mathcal{D}(S^{*})=H^{2}[0,l]$. By direct calculation one can verify that as a boundary triple one can take

[TABLE]

The distinguished self-adjoint extension $H_{0}$ of $S$, i.e. the restriction of $S^{*}$ to $\ker\Gamma_{1}$ is exactly the operator $-d^{2}/dx^{2}$ with the Neumann boundary conditions.

To construct the corresponding $\gamma$-field $\gamma(z)$ and the Weyl function $Q(z)$ it is useful to use the Green function $G_{0}(x,y;z)$, i.e. the integral kernel of the resolvent $(H_{0}-z)^{-1}$, $z\notin\operatorname{spec}H_{0}$, which is given by

[TABLE]

where the branch of the square root ix fixed by $\Im\sqrt{\lambda}\geq 0$. In terms of $G_{0}$ one has

[TABLE]

and

[TABLE]

2.3 Example 2: Laplacian on a manifold

Let $M$ be a two-dimensional closed manifold and $\Delta=\Delta_{M}$ be the closure of the positive Laplace-Beltrami operator defined initially on $C^{\infty}(M)$. It is known that $\mathcal{D}(\Delta)\subset C^{0}(M)$. Fix some (mutually distinct) points $q_{1},\dots,q_{\mu}\in M$ and denote by $S$ the restriction of $\Delta$ to the functions vanishing at all these points (which makes sense, as all functions from the domain are continuous). The defficiency indices of $S$ are $(\mu,\mu)$.

Let $G(x,y;z)$ be the Green function of $\Delta$, i.e. the integral kernel of the resolvent $(\Delta-z)^{-1}$, then, for $z\notin\operatorname{spec}\Delta$, there holds

[TABLE]

The Green function can be presented as follows:

[TABLE]

where the second term $F$ is continuous in $M\times M$. Using the above decompositions one can show that there exist uniquely defined linear functionals $a_{j}$, $b_{j}$ on $\mathcal{D}(S^{*})$, $j=1,\dots,\mu$, such that

[TABLE]

and as a boundary triple for $S$ one can take $(\mathbb{C}^{\mu},\Gamma_{1},\Gamma_{2})$ with

[TABLE]

Note that the distinguished extension of $S$ corresponding to the boundary condition $\Gamma_{1}f=0$ is just the initial operator $\Delta$. By direct calculation one can show that the $\gamma$-field $\gamma(z)$ and the Weyl function $Q(z)$ for the above boundary triple are given by

[TABLE]

More details to the above constructions can be found in [7]. We also mention the paper [13] treating a trace formula for the operators of the above type.

Below we use the following asymptotic expransion, see [3, Formula (38)]: for all $M\geqslant 1$,

[TABLE]

Here $a_{n}(x,x)$ is the $n$-th local heat kernel coefficient and $\gamma$ is Euler’s constant.

2.4 Hamiltonian on a hybrid manifold

Using the formalism of the previous section, we are going to construct Laplace type operators on hybrid manifolds and study some of their properties.

Consider a hybrid manifold constructed with $m$ compact manifolds $M_{1},\ldots,M_{m}$, with marked points $q_{is}\in M_{i}$, $s=1,\ldots,\mu_{i}$, and $n$ segments $L_{1},\ldots,L_{n}$, as described in the introduction. On each $M_{i}$ consider also the usual Laplace-Beltrami operator $\Delta_{M_{i}}$, which is a self-adjoint operator on $L^{2}(M_{i})$, with domain $\mathcal{D}(\Delta_{M_{i}})$, the second Sobolev space of $M_{i}$. Denote by $D_{i}$ restriction of $\Delta_{M_{i}}$ to the domain

[TABLE]

We parametrize the segments $L_{j}$ by $\{x_{j}\in\mathbb{R}:x_{j}\in[0,l_{j}]\}$, where $j=1,\dots,n$ and denote by $D^{s}_{j}$ the closure in $L^{2}(L_{j})$ of the operator $-\frac{d^{2}}{dx_{j}^{2}}$ defined on $C_{0}^{\infty}(L_{j})$.

The operator $S$ defined by

[TABLE]

is a symmetric operator in $L^{2}(M_{1})\oplus\dots\oplus L^{2}(M_{m})\oplus L^{2}(L_{j})\oplus\dots\oplus L^{2}(L_{n})$ with deficiency indices $(4n,4n)$.

Definition 6**.**

A Laplace type operator on a hybrid manifold is a self-adjoint extension of the operator $S$.

We use the constructions of the previous section to describe all self-adjoint extentions of $S$. Namely, for each operator $D_{j}$ take the boundary triple \big{(}\mathbb{C}^{\mu_{j}},\Gamma_{1,M_{j}},\Gamma_{2,M_{j}}\big{)} defined in subsection 2.3 and for each operator $D^{s}_{j}$ take the boundary triple \big{(}\mathbb{C}^{2},\Gamma_{1,L_{j}},\Gamma_{2,L_{j}}\big{)} as defined in subsection 2.2. Then, obviously, their direct sum $(\mathbb{C}^{2N},\Gamma_{1},\Gamma_{2})$, where $\Gamma_{j}=\Gamma_{j,M}\oplus\Gamma_{j,L}$, and $\Gamma_{j,M}=\Gamma_{j,M_{1}}\oplus\Gamma_{j,M_{m}}$, $\Gamma_{j,L}=\Gamma_{j,L_{1}}\oplus\dots\oplus\Gamma_{j,L_{n}}$, $j=1,2$, is a boundary triple for $S$. The corresponding $\gamma$-field $\gamma(z)$ and the Weyl function are also the direct sums of the corresponding objects of each part.

Below we use the enumeration of the points $\{q_{is}\}=:\{q_{j}\}$ in such a way that the point $q_{j}$ is the one attached to the initial point of $L_{k}$ if $j=2k-1$, $k\in\mathbb{N}$, or the the terminal vertex of $L_{k}$ if $j=2k$, $k\in\mathbb{N}$.

Now we are going to restrict the class of admissible boundary conditions. It is natural to suppose that the boundary coniditions at each point of gluing are local, i.e. involves only the boundary values of the functions at this point. In view of the above enumeration of the points $q_{j}$ this means that we consider only the boundary conditions of the form $A\Gamma_{1}=B\Gamma_{2}$ where each of the $2N\times 2N$ matrices $A$ and $B$ consists of four diagonal blocks

[TABLE]

It is useful to introduce the $2\times 2$ matrices

[TABLE]

clearly, these matrices should satisfy the same conditions $A_{j}B_{j}^{*}=AB_{j}^{*}$ and $\det(A_{j}A_{j}^{*}+B_{j}B_{j}^{*})\neq 0$ for each $j$. It is also clear, by proposition 4, that each of the boundary conditions can be rewritten using unitary matrices of the same structure

[TABLE]

with

[TABLE]

and

[TABLE]

This paramatrization will be referred to as canonical. Note that diagonal matrices $\Phi_{j}$ correspond to the absence of coupling at $q_{j}$: the boundary conditions split into one boundary condition on the manifold and another one on the segment attached; we will call such boundary conditions reducible. Below we consider non-reducible boundary conditions only, hence assuming that all the matrices $\Phi_{j}$ in the above constructions are non-diagonal.

3 The resolvent expansion

Warning. From now on, due to technical reasons and in order to be compatible with other works, we will use $-z^{2}$ as our spectral parameter, and not $z$. However, we will still use e.g. the notation $R_{0}(z)$ to denote $(S_{0}+z^{2})^{-1}$, and not $(S_{0}-z)^{-1}$ as in the previous section. Note for further use that in this notation the Green function on manifolds or segments obeys

[TABLE]

(which is just a consequence of the Hilbert resolvent identity).

3.1 Computation of $\operatorname{Tr}R^{2}$

Let us start with the computation of $\operatorname{Tr}R^{2}$ using Theorem 5. Setting

[TABLE]

we obtain

[TABLE]

We recall that the resolvent $R_{0}$ is the direct sum of the resolvents of the ordinary Laplacians on the manifolds $M_{i}$ (denoted by $R_{0,M_{i}}$) and the resolvents for the Neumann Laplacians on the segments $L_{j}$ (denoted by $R_{0,L_{j}}$) forming the hybrid manifold. For the manifolds one has

[TABLE]

and for the segments one uses the same formula with the explicit expression (2.2); we will use it later.

The linear operator $T(z)$ can be also rewritten as an integral operator as follows. Denote the entries of the $2N\times 2N$ matrix $[BQ(z)-A]^{-1}B$ by $c_{ij}(z)$. Using $G(x,y,z)=\overline{G(y,x,\bar{z})}$ we get, for any $f\in L^{2}$:

[TABLE]

The integral here is in fact the sum of the integrals over manifolds and segments. Using this integral representation of the operator $A$, we calculate the remaining terms of $\operatorname{Tr}R^{2}$ as follows. First, for any $f\in L^{2}$:

[TABLE]

Hence, the operator $R_{0}(z)T(z)$ has the integral kernel

[TABLE]

therefore

[TABLE]

or,

[TABLE]

Let us now find $\operatorname{Tr}T^{2}(z)$. We will use the same method and represent the operator $T^{2}(z)$ as an integral operator:

[TABLE]

Now we find:

[TABLE]

Summarizing the results obtained, we state

Theorem 7**.**

Consider a hybrid manifold $H$, consisting of the manifolds $M_{i}$ and the segments $L_{j}$, and the Laplace operator corresponding to the boundary conditions $A\Gamma_{1}=B\Gamma_{2}$ on $H$. Then the following formula for the trace of the square of the resolvent of this operator holds:

[TABLE]

Here $G(x,y,z)$ is the Green function of $S_{0}$ on the hybrid manifold and the entries of the matrix $[BQ(z)-A]^{-1}B$ are denoted by $c_{ij}(z)_{1\leqslant i,j\leqslant 2N}$.

3.2 The asymptotic expansion

The result of Theorem 7 is given in terms of the Green functions for the Laplacians on the smooth parts of the hybrid manifold. But using it in applications is practically impossible for two reasons: first of all, inverting the $2N\times 2N$ matrix $BQ(z)-A$ can be difficult, and secondly, an explicit computation of the Green function is almost never possible. Nevertheless, there are ways to get some simplifications if we restrict attention to large $z$. As we will see below, the special structure of the matrix $(BQ(z)-A)$ allows us to find its inverse asymptotically as $z\rightarrow\infty$. Likewise, one can use the representation of $R_{0}$ as an asymptotic series in powers of $z$ (analog of heat kernel expansion). Thus we may attempt to generalize (1.3) to the case of hybrid manifolds.

For further calculations the following lemma is crucial.

Lemma 8**.**

Let $M$ be a compact two-dimensional manifold and $G(x,y;z)$ be the integral kernel of $(\Delta+z^{2})^{-1}$. For any $x,y\in M$, $x\neq y$, and any $k=0,1,2,\dots$, there exists $C_{k}=C_{k,x,y}>0$ such that for $z\to+\infty$ one has

[TABLE]

Proof.

Let $P(x,y;t)$ be the heat kernel of $\Delta$, i.e. the integral kernel for the semigroup $e^{-t\Delta}$. It is known [9, Corollary 6] that there are positive constants $a$ and $\delta$ such that

[TABLE]

Clearly,

[TABLE]

hence

[TABLE]

where $p_{k}$ is a certain polynomial. One can estimate

[TABLE]

where $q_{k}$ is another polynomial. Now let us note the identity

[TABLE]

where $K_{\nu}$ is the modified Bessel function, see [16, Supplement 4.3], which gives, with some polynomials $q_{j}$,

[TABLE]

Now it is sufficient to recall the asymptotic behavior $K_{\nu}(s)=O(e^{-s})$ for $s\to+\infty$. ∎

Thus, if the points $x$ and $y$ do not coincide, the Green function $G(x,y,z)$, as well as its derivatives $G^{\prime}_{z}(x,y,z)$ and $G^{\prime\prime}_{zz}(x,y,z)$ decay exponentially as $z$ tends to infinity.

In the statement of Theorem 7 the terms $G(q_{i},q_{j},z)$ appear, where the distance between the points $q_{j}$ and $q_{j}$ is fixed by the configuration of our hybrid space. Hence all terms of such type for non-coinciding points $q_{i}\neq q_{j}$ are exponentially small as $z\rightarrow\infty$, so we need to take into account only the diagonal terms $G(q_{i},q_{i},z)$. Using this observation and performing calculations similar to the proof of Theorem 7, but neglecting terms of exponential small order of $z$ we prove that for some positive constant $c$, as $z\rightarrow\infty$

[TABLE]

This result can be rewritten in a compact matrix-form, as we are going to see now. Let us denote by $G^{\prime}_{z}$ the diagonal matrix with entries $G^{\prime}_{z}(q_{i},q_{i},z)$. Denote also by $G^{\prime\prime}_{zz}$ the diagonal matrix with entries $G^{\prime\prime}_{zz}(q_{i},q_{i},z)$. This is a slight abuse of notation, since $G^{\prime}_{z}$ and $G^{\prime\prime}_{zz}$ are also well defined operators; it will be clear from the context what is meant by this notation. Then it is easy to see that (3.2) can be formulated as follows.

Theorem 9**.**

We have the following asymptotic relation as $z\rightarrow\infty$:

[TABLE]

Now we need only to compute $(BQ(z)-A)^{-1}B$, and fortunately, this is possible in this approximation.

As we have already mentioned, the matrix $Q(z)$ has a direct product representation, $Q(z)=Q_{1}\oplus\dots Q_{m}\oplus G$ where $Q_{i}$ is the $Q$-matrix for the $i$-th manifold $M_{i}$ in the hybrid manifold (see subsection 2.3), for example,

[TABLE]

and $G=(G_{kl})$ is an $N\times N$-matrix consisting of the Green functions (denoted by $G_{s}$) for the Neumann Laplacian on the segments:

[TABLE]

Nevertheless, the matrix $BQ(z)-A$ is still too complicated to find its inverse explicitly. But letting $z\rightarrow\infty$ and applying Lemma 8 the situation improves radically. More precisely, all non-diagonal terms of $Q(z)$ are either zero or of type $G(q_{i},q_{j},z),\,i\neq j$, and decay exponentially as $z$ tends to infinity. Therefore, we can write $Q(z)=Q_{d}(z)+\tilde{Q}(z)$, where $\tilde{Q}(z)$ is exponentially small as $z$ goes to infinity, and the leading order approximation $Q_{d}(z)$ to $Q(z)$ is given by

[TABLE]

Then (B\big{(}Q(z)-A)^{-1}B to first order, i.e. $(BQ_{d}(z)-A)^{-1}$, consists of four diagonal $N\times N$ blocks. The inverse matrix can be found using the Frobenius formula for a block matrix consisting of matrices $A,B,C,D$:

[TABLE]

Hence one can obtain the block representation

[TABLE]

where $U$, $W$, $V$, $X$ are diagonal $N\times N$ matrices with

[TABLE]

and

[TABLE]

Note that if we change the parameters $A\to LA$, $B\to LB$, with a non-degenrate matrix $L$ of the same structure as $A$ or $B$, then all of the above expressions acquire the factor $\det L_{j}$.

It is important to mention that the matrix $X_{i}$ is invertible for large $z$. To see this, let us note first that all four summands in (3.1) have different asymptotics for large $z$: by (2.3) one has $F_{i}\sim\ln z^{2}$, and by direct calculation,

[TABLE]

hence it is sufficient to show that at least one of them is non-zero. To show this, it is useful to assume that the matrices $A$ and $B$ are taken in the canonical form, see subsection 2.4. In this case, assuming that all four summands in (3.1) vanish one arrives for each $i$ to the incompatible system

[TABLE]

Lemma 10**.**

With the notation above, we have

[TABLE]

for some positive constant $c$.

Proof.

First, we write

[TABLE]

As can be easily seen from the expressions for $U_{ii}$, $W_{ii}$ and $V_{ii}$, one has $[BQ_{d}(z)-A]^{-1}B=O(z^{2})$. As $\tilde{Q}(z)$ decays exponentially fast for large $z$, the product $[BQ_{d}(z)-A]^{-1}\tilde{Q}(z)$ will then also be exponentially small. By using the Neumann series to compute $[1+[BQ_{d}(z)-A]^{-1}\tilde{Q}(z)]^{-1}$, we see that this term is of the form $1+O(e^{-cz})$, and this immediately implies the conclusion of the Lemma.

∎

Lemma 10 allows us to reformulate theorem 9 as follows:

Theorem 11**.**

For $z\to+\infty$ there holds, up to $O(e^{-cz})$ terms,

[TABLE]

This theorem describes the large spectral parameter asymptotic of the trace of the second power of the resolvent for a Laplace operator through the Green functions of the manifolds and segments forming the hybrid manifold. In the next section, we will write this expansion in terms of heat kernel coefficients for the manifolds forming the hybrid manifold.

3.3 Pseudoasymptotic resolvent expansion in terms of heat kernel coefficients

To compute the above expression for the trace in terms of heat kernel coefficients, we need to consider the term

[TABLE]

In fact, this term consists of two types of integrals: over the manifolds and over the segments. As we already know (see equation (1.2)), for manifolds the following expansion holds as $z\rightarrow\infty$: for all $q\geqslant 0$

[TABLE]

where $R_{0}$ is the resolvent of the Laplace operator on this manifold and the constants $a_{k}$ are the heat kernel coefficients for it. The integral over segments can be easily computed asymptotically using the explicit expressions (2.2):

[TABLE]

Definition 12**.**

Assume that there exist rational functions $c_{k}$, $k=1,2,\dots$, such that for any $q\geq 0$ and $\alpha\in(0,1)$ a function $f=f(z)$ admits the following representation for $z\to+\infty$:

[TABLE]

An expansion of the form (3.6) will be called pseudoasymptotic, and in this case we write

[TABLE]

Lemma 13**.**

If a function $f$ admits a pseudoasymptotic expansion, then this expansion is unique.

Proof.

First, we notice that if $P$ and $Q$ are two polynomials and $\alpha>0$, then, for large $z$,

[TABLE]

Suppose that there are two pseudoasymptotic expansions of $f$, i.e. for any $q\geq 0$ and any $\alpha>0$

[TABLE]

For $q=0$ we obtain

[TABLE]

which implies $c_{0}=d_{0}$. By an immediate induction on $k$, we show similarly that we will always obtain $c_{k}(\ln z^{2})-d_{k}(\ln z^{2})=O(z^{-\alpha})$ and consequently the uniqueness of the expansion. ∎

Remark 14*.*

Let us notice that if a function $f$ possesses a pseudoasymptotic expansion, and if each coefficient $c_{k}$ is a rational function with nonpositive degree, then each $c_{k}$ can be uniquely expanded as a $\ln z^{2}$-asymptotic series: for all $K\geq 0$,

[TABLE]

for some constants $c_{kl}$.

Theorem 15**.**

The trace $\operatorname{Tr}R(z)^{2}$ obtained in Theorem 11 admits a pseudoasymptotic expansion of the form

[TABLE]

where c_{4}(\lambda)=\dfrac{\sum_{j}\chi(M_{j})}{6}+\dfrac{N}{4}+\dfrac{N_{0}}{\lambda}+o\Big{(}\dfrac{1}{\lambda}\Big{)},\quad\lambda\to+\infty and

[TABLE]

Moreover, if $N_{0}=N$, then $c_{k}=O(1)$ as $z\to+\infty$ for all $k$.

Proof.

Consider first more carefully the expression (3.1) for $X_{i}$. Let us show first that at least one of the last two terms on the right-hand side, $\alpha_{j}F_{i}$ with $\alpha_{j}=a_{i,i+N}b_{i+N,i}-a_{i+N,i+N}b_{i,i}$