Parabolic transmission eigenvalue-free regions in the degenerate isotropic case

TL;DR

This paper investigates the distribution of transmission eigenvalues in isotropic media with boundary-restricted refraction indices, establishing the existence of parabolic regions free of eigenvalues under certain conditions.

Contribution

It introduces the concept of parabolic eigenvalue-free regions in the degenerate isotropic case, expanding understanding of eigenvalue distribution in this context.

Findings

Existence of parabolic transmission eigenvalue-free regions.

Conditions under which these regions occur.

Insights into eigenvalue distribution in degenerate isotropic media.

Abstract

We study the location of the transmission eigenvalues in the isotropic case when the restrictions of the refraction indices on the boundary coincide. Under some natural conditions we show that there exist parabolic transmission eigenvalue-free regions.

Equations

$$\tau =\frac{{\omega }_d}{(2\pi {)}^d}{\int }_{\Omega }\left(n_1(x{)}^{d/2}+n_2(x{)}^{d/2}\right)dx,$$

$$\left({\mathrm{Op}}_h(a)f\right)(x^{\prime })=(2\pi h{)}^{-d+1}{\int }_{T^{\ast }\Gamma }{e}^{-\frac{i}{h}⟨x^{\prime }-y^{\prime },{\xi }^{\prime }⟩}a(x^{\prime },{\xi }^{\prime })f(y^{\prime })d{y}^{\prime }d{\xi }^{\prime }.$$

$$\left|{\partial }_{x^{\prime }}^{\alpha }{\partial }_{{\xi }^{\prime }}^{\beta }a(x^{\prime },{\xi }^{\prime })\right|\le C_{\alpha ,\beta }{m}^{\ell -{\delta }_1|\alpha |-{\delta }_2|\beta |}$$

$$\left|{\partial }_{x^{\prime }}^{\alpha }{\partial }_{{\xi }^{\prime }}^{\beta }a(x^{\prime },{\xi }^{\prime })\right|\le C_{\alpha ,\beta }{h}^{-\delta (|\alpha |+|\beta |)}⟨{\xi }^{\prime }{⟩}^{k-|\beta |}$$

$$\Vert f{\Vert }_{H_h^s(\Gamma )}:=\Vert {\mathrm{Op}}_h(⟨{\xi }^{\prime }{⟩}^s)f{\Vert }_{L^2(\Gamma )}.$$

$${\mathrm{Op}}_h(a)={\mathcal{O}}_s(1):H_h^s(\Gamma )\to H_h^{s+k}(\Gamma ).$$

$${\mathrm{Op}}_h(a^{+}){\mathrm{Op}}_h(a^{-})-{\mathrm{Op}}_h(a^{+}{a}^{-})={\mathcal{O}}_s(h):H_h^s(\Gamma )\to H_h^s(\Gamma ).$$

$$\mathcal{N}(h,z):H^{m+1}(\Gamma )\to H^m(\Gamma )$$

$$\mathcal{N}(h,z)f:=-ih{\partial }_{\nu }u{|}_{\Gamma }$$

$$\Vert v{\Vert }_{H_h^m(\Omega )}=\sum _{|\alpha |\le m}{h}^{|\alpha |}{\Vert {\partial }_x^{\alpha }v\Vert }_{L^2(\Omega )}.$$

$$n(x)=\sum _{k=0}^{N-1}{x}_1^k{n}_k(x^{\prime })+x_1^N{\mathcal{M}}_N(x)$$

$$\rho (x^{\prime },{\xi }^{\prime },z)=\sqrt{-r_0(x^{\prime },{\xi }^{\prime })+z{n}_0(x^{\prime })},\mathrm{Im}\rho >0.$$

$$\Sigma :=\{(x^{\prime },{\xi }^{\prime })\in T^{\ast }\Gamma :r_{♯}(x^{\prime },{\xi }^{\prime })=1\},r_{♯}=n_0^{-1}{r}_0.$$

Full text

1. Introduction and statement of results

Let $\Omega\subset{\bf R}^{d}$, $d\geq 2$, be a bounded, connected domain with a $C^{\infty}$ smooth boundary $\Gamma=\partial\Omega$. A complex number $\lambda\neq 0$, ${\rm Re}\,\lambda\geq 0$, will be said to be a transmission eigenvalue if the following problem has a non-trivial solution:

[TABLE]

where $\nu$ denotes the Euclidean unit inner normal to $\Gamma$, $n_{j}\in C^{\infty}(\overline{\Omega})$, $j=1,2$ are strictly positive real-valued functions called refraction indices. In the non-degenerate isotropic case when

[TABLE]

it has been recently proved in [16] that there are no transmission eigenvalues in the region

[TABLE]

for some constant $C>0$. Moreover, it follows from the analysis in [6] (see Section 4) that the eigenvalue-free region (1.3) is optimal and cannot be improved in general. In the present paper we will consider the degenerate isotropic case when

[TABLE]

We suppose that there is an integer $j\geq 1$ such that

[TABLE]

[TABLE]

It was proved in [1] (see Theorem 4.2) that in this case the eigenvalue-free region (1.3) is no longer valid. On the other hand, it follows from [5] that under the conditions (1.5) and (1.6) there are no transmission eigenvalues in $|\arg\lambda|\geq\varepsilon$, $|\lambda|\geq C_{\varepsilon}\gg 1$, $\forall\,0<\varepsilon\ll 1$. Our goal in the present paper is to improve this result showing that in this case we have a much larger parabolic eigenvalue-free region. Our main result is the following

Theorem 1.1**.**

Under the conditions (1.5) and (1.6) there exists a constant $C>0$ such that there are no transmission eigenvalues in the region

[TABLE]

where $\kappa_{j}=2(3j+2)^{-1}$.

To prove this theorem we make use of the semi-classical parametrix for the interior Dirichlet-to-Neumann (DN) map built in [14]. It is proved in [14] that for $|{\rm Im}\,\lambda|\geq\left({\rm Re}\,\lambda+1\right)^{1/2+\epsilon}$, $0<\epsilon\ll 1$, the DN map is an $h-\Psi$DO of class OP${\mathcal{S}}^{1}_{1/2-\epsilon}(\Gamma)$, where $0<h\ll 1$ is a semi-classical parameter such that $h\sim|\lambda|^{-1}$. A direct consequence of this fact is the existence of a transmission eigenvalue-free region of the form

[TABLE]

under the condition (1.2). The most difficult part of the parametrix construction in [14] is near the glancing region (see Section 3 for the definition). Indeed, outside an arbitrary neighbourhood of the glancing region the parametrix construction in [14] works for $|{\rm Im}\,\lambda|\geq\left({\rm Re}\,\lambda+1\right)^{\epsilon}$ and the corresponding parametrix belongs to the class OP${\mathcal{S}}^{1}_{0}(\Gamma)$. In other words, to improve the eigenvalue-free region (1.8) one has to improve the parametrix in the glancing region. Such an improved parametrix has been built in [15] for strictly concave domains and as a consequence (1.8) was improved to

[TABLE]

in this case. In fact, it turns out that to get larger eigenvalue-free regions under the condition (1.2) no parametrix construction in the glancing region is needed. It suffices to show that the norm of the DN map microlocalized in a small neighbourhood of the glancing region gets small if $|{\rm Im}\,\lambda|$ and ${\rm Re}\,\lambda$ are large. Indeed, this strategy has been implemented in [16] to get the optimal transmission eigenvalue-free region (1.3) for an arbitrary domain. In fact, the main point in the approach in [16] is the construction of a parametrix in the hyperbolic region valid for $1\ll C_{\epsilon}\leq|{\rm Im}\,\lambda|\leq\left({\rm Re}\,\lambda\right)^{1-\epsilon}$, ${\rm Re}\,\lambda\geq C^{\prime}_{\epsilon}\gg 1$, $0<\epsilon\ll 1$. The strategy of [16], however, does not work any more when we have the condition (1.4). In this case the parametrix in the glancing region turns out to be essential to get eigenvalue-free regions like (1.7). In Section 3 we revisit the parametrix construction of [14] and we study carefully the way in which it depends on the restriction on the boundary of the normal derivatives of the refraction index (see Theorem 3.1). In Section 4 we improve Theorem 3.1. In Section 5 we show how Theorem 4.1 implies Theorem 1.1. We also show that to improve (1.7) it suffices to improve the parametrix in the glancing region, only (see Proposition 5.2).

As in [9] one can study in this case the counting function $N(r,C)=\#\{\lambda-{\rm trans.\,eig.}:\,C\leq|\lambda|\leq r\}$, where $r\gg C>0$. We have the following

Corollary 1.2**.**

Under the conditions of Theorem 1.1, there exists a constant $C>0$ such that the counting function of the transmission eigenvalues satisfies the asymptotics

[TABLE]

where

[TABLE]

$\omega_{d}$* being the volume of the unit ball in ${\bf R}^{d}$.*

Note that the eigenvalue-free region (1.3) implies (1.10) with $\kappa_{j}$ replaced by $1$. Note also that asymptotics for the counting function $N(r,C)$ with remainder $o(r^{d})$ have been previously obtained in [3], [7], [12] still under the condition (1.2).

2. Basic properties of the $h-\Psi$DOs

In this section we will recall some basic properties of the $h-\Psi$DOs on a compact manifold without boundary. Let $\Gamma$, ${\rm dim}\,\Gamma=d-1$, be as in the previous section and recall that given a symbol $a\in C^{\infty}(T^{*}\Gamma)$, the $h-\Psi$DO, ${\rm Op}_{h}(a)$, is defined as follows

[TABLE]

We have the following criteria of $L^{2}$- boundedness.

Proposition 2.1**.**

Let the function $a$ satisfy the bounds

[TABLE]

for all multi-indices $\alpha$. Then the operator ${\rm Op}_{h}(a)$ is bounded on $L^{2}(\Gamma)$ and satisfies

[TABLE]

with a constant $C>0$ independent of $h$ and $C_{\alpha}$.

Let the function $a$ satisfy the bounds

[TABLE]

for all multi-indices $\alpha$ and $\beta$. Then the operator ${\rm Op}_{h}(a)$ is bounded on $L^{2}(\Gamma)$ and satisfies

[TABLE]

with a constant $C>0$ independent of $h$ and $C_{\alpha,\beta}$, where $s_{d}>0$ is an integer depending only on the dimension.

Given $\ell\in{\bf R}$, $\delta_{1},\delta_{2}\geq 0$ and a function $m>0$ on $T^{*}\Gamma$, we denote by $S^{\ell}_{\delta_{1},\delta_{2}}(m)$ the set of all functions $a\in C^{\infty}(T^{*}\Gamma)$ satisfying

[TABLE]

for all multi-indices $\alpha$ and $\beta$ with constants $C_{\alpha,\beta}>0$ independent of $m$. Given $k\in{\bf R}$, $0\leq\delta<1/2$, we also denote by ${\mathcal{S}}_{\delta}^{k}$ the space of all symbols $a\in C^{\infty}(T^{*}\Gamma)$ satisfying

[TABLE]

for all multi-indices $\alpha$ and $\beta$ with constants $C_{\alpha,\beta}>0$ independent of $h$. It is well-known that the $h-\Psi$DOs of class OP${\mathcal{S}}_{\delta}^{k}$ have nice calculus (e.g. see Section 7 of [2]). The next proposition is very usefull for inverting such operators depending on additional parameters (see also Proposition 2.2 of [14]).

Proposition 2.2**.**

Let $h^{\ell_{\pm}}a^{\pm}\in{\mathcal{S}}_{\delta}^{\pm k}$, $0\leq\delta<1/2$, where $\ell_{\pm}\geq 0$ are some numbers. Assume in addition that the functions $a^{\pm}$ satisfy

[TABLE]

$\forall\,(x^{\prime},\xi^{\prime})\in T^{*}\Gamma$, for all multi-indices $\alpha_{1}$, $\beta_{1}$, $\alpha_{2}$, $\beta_{2}$ such that $|\alpha_{j}|+|\beta_{j}|\geq 1$, $j=1,2$, with constants $C_{\alpha_{1},\beta_{1},\alpha_{2},\beta_{2}}>0$ independent of $h$ and $\mu$. Then we have

[TABLE]

with a constant $C>0$ independent of $h$ and $\mu$.

Given any real $s$, we define the semi-classical Sobolev norm by

[TABLE]

Using the calculus of the $h-\Psi$DOs one can derive from (2.4) the following

Proposition 2.3**.**

Let $a\in{\mathcal{S}}_{\delta}^{-k}$, $0\leq\delta<1/2$. Then, for every $s$, we have

[TABLE]

Proposition 2.2 implies the following

Proposition 2.4**.**

Let $a^{\pm}\in{\mathcal{S}}_{0}^{\pm k}$. Then, for every $s$, we have

[TABLE]

3. The parametrix construction revisited

In this section we will build a parametrix for the semi-classical Dirichlet-to-Neumann map following [14]. Note that in [14] there is a gap due to a missing term in the transport equations (4.11), which however does not affect the proof of the main results. Here we will correct this gap making some slight modifications.

Given $f\in H^{m+1}(\Gamma)$, let $u$ solve the equation

[TABLE]

where $n\in C^{\infty}(\overline{\Omega})$ is a strictly positive function, $0<h\ll 1$ is a semi-classical parameter and $z\in Z_{1}\cup Z_{2}\cup Z_{3}$, where $Z_{1}=\{z\in{\bf C}:{\rm Re}\,z=1,0<|{\rm Im}\,z|\leq 1\}$, $Z_{2}=\{z\in{\bf C}:{\rm Re}\,z=-1,|{\rm Im}\,z|\leq 1\}$, $Z_{3}=\{z\in{\bf C}:|{\rm Re}\,z|\leq 1,|{\rm Im}\,z|=1\}$. Given $\varepsilon>0$ we also set $Z_{1}(\varepsilon)=\{z\in Z_{1}:h^{\varepsilon}\leq|{\rm Im}\,z|\leq 1\}$. We define the semi-classical Dirichlet-to-Neumann map

[TABLE]

by

[TABLE]

where $\nu$ denotes the Euclidean unit inner normal to $\Gamma$. Given an integer $m\geq 0$, denote by $H_{h}^{m}(\Omega)$ the Sobolev space equipped with the semi-classical norm

[TABLE]

We define similarly the Sobolev space $H_{h}^{m}(\Gamma)$. Note that this norm is equivalent to that one defined in Section 2. Throughout this section we will use the normal coordinates $(x_{1},x^{\prime})$ with respect to the Euclidean metric near the boundary $\Gamma$, where $0<x_{1}\ll 1$ denotes the Euclidean distance to the boundary and $x^{\prime}$ are coordinates on $\Gamma$. We denote by $\Delta_{\Gamma}$ the negative Laplace-Beltrami operator on $\Gamma$ equipped with the Riemannian metric induced by the Euclidean one in $\Omega$. Let $r_{0}(x^{\prime},\xi^{\prime})\geq 0$ be the principal symbol of $-\Delta_{\Gamma}$ written in the coordinates $(x^{\prime},\xi^{\prime})\in T^{*}\Gamma$. Since the function $n$ is smooth up to the boundary we can expand it as

[TABLE]

for every integer $N\geq 1$, where $n_{k}=(k!)^{-1}\partial_{\nu}^{k}n|_{\Gamma}$, $n_{0}>0$, and ${\mathcal{M}}_{N}(x)$ is a real-valued smooth function. Set

[TABLE]

The glancing region for the problem (3.1) is defined by

[TABLE]

Let $\phi\in C_{0}^{\infty}({\bf R})$, $0\leq\phi\leq 1$, $\phi(\sigma)=1$ for $|\sigma|\leq 1$, $\phi(\sigma)=0$ for $|\sigma|\geq 2$, and set $\eta(x^{\prime},\xi^{\prime})=\phi(r_{0}(x^{\prime},\xi^{\prime})/\delta_{0})$. Clearly, taking $\delta_{0}>0$ small enough we can arrange that $|\rho|\geq C\langle r_{0}\rangle^{1/2}\geq C\langle\xi^{\prime}\rangle$ on supp$(1-\eta)$. We also define the function $\chi(x^{\prime},\xi^{\prime})=\phi((1-r_{\sharp}(x^{\prime},\xi^{\prime}))/\delta_{1})$, where $0<\delta_{1}\ll 1$ is independent of $h$ and $z$. Clearly, $\chi=1$ in a neighbourhood of $\Sigma$, $\chi=0$ outside another neighbourhood of $\Sigma$.

We will say that a function $a\in C^{\infty}(T^{*}\Gamma)$ belongs to $S^{\ell_{1}}_{\delta_{1},\delta_{2}}(m_{1})+S^{\ell_{2}}_{\delta_{3},\delta_{4}}(m_{2})$ if $\eta a\in S^{\ell_{1}}_{\delta_{1},\delta_{2}}(m_{1})$ and $(1-\eta)a\in S^{\ell_{2}}_{\delta_{3},\delta_{4}}(m_{2})$. Given any integer $k$, it follows from Lemma 3.2 of [14] that

[TABLE]

In particular, (3.2) implies that

[TABLE]

Since $\rho=i\sqrt{r_{0}}\left(1+{\mathcal{O}}(r_{0}^{-1})\right)$ as $r_{0}\to\infty$, it is easy to check that

[TABLE]

for every integer $k$. Since $|\rho|\geq C\sqrt{|{\rm Im}\,z|}$ for $z\in Z_{1}$, $(x^{\prime},\xi^{\prime})\in{\rm supp}\,\chi$ and $|\rho|\geq C>0$ for $z\in Z_{2}\cup Z_{3}$ or $z\in Z_{1}$, $(x^{\prime},\xi^{\prime})\in{\rm supp}\,(1-\chi)$ (see Lemma 3.1 of [14]), it also follows from (3.2) that

[TABLE]

[TABLE]

[TABLE]

for every integer $N\geq 0$ and $0<\epsilon\ll 1$, where $k_{-}=0$ if $k\geq 0$, $k_{-}=|k|$ if $k<0$. Our goal in this section is to prove the following

Theorem 3.1**.**

Let $z\in Z_{1}(1/2-\epsilon)$, $0<\epsilon\ll 1$. Then, for every integer $s\geq 0$ there is a function $b_{s}\in{\mathcal{S}}_{1/2-\epsilon}^{0}$ independent of all $n_{k}$ with $k\geq s$ such that

[TABLE]

where $c_{s}=0$ if $s=0$, and $c_{s}=-is!(-2i)^{-s-1}$ for $s\geq 1$. If $z\in Z_{2}\cup Z_{3}$, then (3.8) holds with $|{\rm Im}\,z|$ replaced by $1$. Moreover, for $z\in Z_{1}(1-\epsilon)$ we have

[TABLE]

where the function $\widetilde{b}_{s}\in{\mathcal{S}}_{0}^{0}$ is independent of all $n_{k}$ with $k\geq s$.

Proof. We will recall the parametrix construction in [14]. We will proceed locally and then we will use partition of the unity to get the global parametrix. Fix a point $x^{0}\in\Gamma$ and let ${\mathcal{U}}_{0}\subset\Gamma$ be a small open neighbourhood of $x^{0}$. Let $(x_{1},x^{\prime})$, $x_{1}>0$, $x^{\prime}\in{\mathcal{U}}_{0}$, be the normal coordinates. In these coordinates the Laplacian can be written as follows

[TABLE]

where $r(x,\xi^{\prime})=\langle R(x)\xi^{\prime},\xi^{\prime}\rangle\geq 0$, $R=(R_{ij})$ being a symmetric $(d-1)\times(d-1)$ matrix-valued function with smooth real-valued entries, $q(x,\xi)=\langle q(x),\xi\rangle=q^{\sharp}(x)\xi_{1}+\langle q^{\flat}(x),\xi^{\prime}\rangle$, $q^{\sharp}$ and $q^{\flat}$ being smooth functions. We can expand them as follows

[TABLE]

[TABLE]

[TABLE]

for every integer $N\geq 1$. Clearly, $r_{0}(x^{\prime},\xi^{\prime})=r(0,x^{\prime},\xi^{\prime})=\langle R_{0}(x^{\prime})\xi^{\prime},\xi^{\prime}\rangle$.

Take a function $\psi^{0}\in C_{0}^{\infty}({\mathcal{U}}_{0})$. In what follows $\psi$ will denote either the function $\psi^{0}$ or the function $\psi^{0}(1-\chi)$. Following [14], we will construct a parametrix $\widetilde{u}_{\psi}$ of the solution of (3.1) with $\widetilde{u}_{\psi}|_{x_{1}=0}={\rm Op}_{h}(\psi)f$ in the form

[TABLE]

where $\Phi_{\delta}=\phi(x_{1}/\delta)\phi(x_{1}/\rho_{1}\delta)$, with $\rho_{1}=|\rho|^{3}$ if $z\in Z_{1}(1/2-\epsilon)$, $\psi=\psi^{0}$, and $\rho_{1}=1$ if $z\in Z_{2}\cup Z_{3}$ or $z\in Z_{1}(1-\epsilon)$, $\psi=\psi^{0}(1-\chi)$. Here $0<\delta\ll 1$ is a parameter independent of $h$ and $z$ to be fixed later on. The phase $\varphi$ is complex-valued such that $\varphi|_{x_{1}=0}=-\langle x^{\prime},\xi^{\prime}\rangle$ and satisfies the eikonal equation mod ${\mathcal{O}}(x_{1}^{N})$:

[TABLE]

where $N\gg 1$ is an arbitrary integer and the function $\Psi_{N}$ is smooth up to the boundary $x_{1}=0$. It is shown in [14], Section 4, that the equation (3.10) has a smooth solution of the form

[TABLE]

safisfying

[TABLE]

More generally, the functions $\varphi_{k}$ satisfy the relations

[TABLE]

for every integer $0\leq K\leq N-1$. Then equation (3.10) is satisfied with

[TABLE]

[TABLE]

where $\varphi_{\nu}=0$ for $\nu\geq N+1$ so that the above sums are finite. Using (3.12) one can prove by induction the following lemma (see Lemma 4.1 of [14]).

Lemma 3.2**.**

We have

[TABLE]

[TABLE]

uniformly in $z$ and $0\leq x_{1}\leq 2\delta\min\{1,|\rho|^{3}\}$. Moreover, if $\delta>0$ is small enough, independent of $\rho$, we have

[TABLE]

One can also easily prove by induction the following

Lemma 3.3**.**

For every integer $k\geq 1$ the functions $\varphi_{k}$ and $\varphi_{k+1}-\frac{zn_{k}}{2(k+1)\rho}$ are independent of all $n_{\ell}$ with $\ell\geq k$.

It follows from (3.13) that $(1-\eta)\varphi_{k}\in{\mathcal{S}}_{0}^{1}$ for all $k$. Define now the functions $\widetilde{\varphi}_{k}$ independent of all $n_{\ell}$, $\ell\geq 0$, satisfying the relations

[TABLE]

$1\leq k\leq N-1$, and $\widetilde{\varphi}_{0}=-\langle x^{\prime},\xi^{\prime}\rangle$, $\widetilde{\varphi}_{1}=i\sqrt{r_{0}}$. Using (3.4) together with (3.12) and (3.16), one can easily prove by induction the following

Lemma 3.4**.**

For every integer $k\geq 1$, we have $(1-\eta)(\varphi_{k}-\widetilde{\varphi}_{k})\in{\mathcal{S}}_{0}^{-1}$.

The amplitude $a$ is of the form

[TABLE]

where the functions $a_{j}$ satisfy the transport equations mod ${\mathcal{O}}(x_{1}^{N})$:

[TABLE]

$a_{0}|_{x_{1}=0}=\psi$, $a_{j}|_{x_{1}=0}=0$ for $j\geq 1$, where $a_{-1}=0$ and the functions $A_{N}^{(j)}$ are smooth up to the boundary $x_{1}=0$. We will be looking for the solutions to (3.17) in the form

[TABLE]

We can write

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

where $\varphi_{\nu}=0$ for $\nu\geq N+1$. Similarly

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $a_{\nu,j-1}=0$ for $\nu\geq N+1$. We also have

[TABLE]

with

[TABLE]

[TABLE]

with

[TABLE]

[TABLE]

with

[TABLE]

where $\varphi_{\nu}=0$, $a_{\nu,j}=0$ for $\nu\geq N+1$ so that the above sums are finite. Inserting the above identities into equation (3.17) and comparing the coefficients of all powers $x_{1}^{k}$, $0\leq k\leq N-1$, we get that the functions $a_{k,j}$ must satisfy the relations

[TABLE]

[TABLE]

and $a_{0,0}=\psi$, $a_{0,j}=0$, $j\geq 1$, $a_{k,-1}=0$, $k\geq 0$. Then equation (3.17) is satisfied with

[TABLE]

Let us calculate $a_{1,0}$. By (3.18) with $j=0$, $k=0$, we get

[TABLE]

On the other hand, by (3.12) with $K=1$ we get

[TABLE]

Using the identity

[TABLE]

we can write $\varphi_{2}$ in the form

[TABLE]

[TABLE]

By (3.19) and (3.20),

[TABLE]

[TABLE]

[TABLE]

By (3.2) and (3.21) we conclude

[TABLE]

The next lemma follows from Lemma 3.2 and (3.22) together with equations (3.18) and can be proved in the same way as Lemma 4.2 of [14]. We will sketch the proof.

Lemma 3.5**.**

We have

[TABLE]

[TABLE]

uniformly in $z$ and $0\leq x_{1}\leq 2\delta\min\{1,|\rho|^{3}\}$.

Proof. Recall that $\nabla_{x^{\prime}}\varphi_{0}=-\xi^{\prime}$. By (3.13) we have

[TABLE]

[TABLE]

We will prove (3.23) by induction. In view of (3.22) we have (3.23) with $k=1$, $j=0$. Suppose now that (3.23) is true for all $j\leq J-1$ and all $k\geq 1$, and for $j=J$ and $k\leq K$. We have to show that it is true for $j=J$ and $k=K+1$. To this end, we will use equation (3.18) with $j=J$ and $k=K$. Indeed, the LHS is equal to $2i(K+1)\rho a_{K+1,J}$ modulo $S^{-3K-4J-2}_{2,2}(|\rho|)+S^{-J+1}_{0,1}(|\rho|)$, while the RHS belongs to $S^{-3K-4J-2}_{2,2}(|\rho|)+S^{-J+1}_{0,1}(|\rho|)$. In other words, $\rho a_{K+1,J}$ belongs to $S^{-3K-4J-2}_{2,2}(|\rho|)+S^{-J+1}_{0,1}(|\rho|)$. This implies that $a_{K+1,J}$ belongs to $S^{-3K-4J-3}_{2,2}(|\rho|)+S^{-J}_{0,1}(|\rho|)$, as desired. Furthermore, (3.24) follows from (3.13) and (3.23) since the functions $A_{N}^{(j)}$ are expressed in terms of $\varphi_{k}$ and $a_{k,j}$. One needs the simple observation that

[TABLE]

implies

[TABLE]

$\Box$

Using Lemma 3.3 we will prove the following

Lemma 3.6**.**

For all $k\geq 1,\,j\geq 0$, the function

[TABLE]

is independent of all $n_{\ell}$ with $\ell\geq k+j$.

Proof. It follows from Lemma 3.3 that the function

[TABLE]

is independent of all $n_{\ell}$ with $\ell\geq k+1$. We will first prove the assertion for $j=0$ and all $k\geq 1$ by induction in $k$. In view of (3.21) it is true for $k=1$. Suppose it is true for all integers $k\leq K$ with some integer $K\geq 1$. We will prove it for $k=K+1$. To this end, we will use equation (3.18) with $j=0$ and $k=K$. Since the RHS is zero, we get that the function

[TABLE]

is independent of all $n_{\ell}$ with $\ell\geq K+1$. Hence, so is the function

[TABLE]

as desired. We will now prove the assertion for all $k\geq 1$, $j\geq 0$ by induction in $j$. Suppose it is true for $j\leq J$ and all $k\geq 1$ with some integer $J\geq 1$. We will prove it for $j=J+1$ and all $k\geq 1$. To this end, we will use equation (3.18) with $j=J+1$ and $k$ replaced by $k-1$, $k\geq 1$. We have that, modulo functions independent of all $n_{\ell}$ with $\ell\geq k+J+1$, the LHS is equal to $2ik\rho a_{k,J+1}$, while the RHS is equal to $-k(k+1)a_{k+1,J}$. Hence the function

[TABLE]

is independent of all $n_{\ell}$ with $\ell\geq k+J+1$, which clearly implies the desired assertion.   $\Box$

It follows from (3.23) that $(1-\eta)a_{k,j}\in{\mathcal{S}}_{0}^{-j}$ for all $k\geq 1$, $j\geq 0$. Define now the functions $\widetilde{a}_{k,j}$ independent of all $n_{\ell}$, $\ell\geq 0$, satisfying the relations

[TABLE]

[TABLE]

and $\widetilde{a}_{0,0}=\psi$, $\widetilde{a}_{0,j}=0$, $j\geq 1$, $\widetilde{a}_{k,-1}=0$, $k\geq 0$, where $\widetilde{\varphi}_{k}^{\Delta}$ is defined by replacing in the definition of $\varphi_{k}^{\Delta}$ all functions $\varphi_{j}$ by $\widetilde{\varphi}_{j}$. Using Lemma 3.4 we will prove the following

Lemma 3.7**.**

For all $k\geq 1,\,j\geq 0$, we have $(1-\eta)(a_{k,j}-\widetilde{a}_{k,j})\in{\mathcal{S}}_{0}^{-j-1}$.

Proof. By Lemma 3.4 together with (3.18) and (3.25) we obtain that the relations

[TABLE]

[TABLE]

[TABLE]

are satisfied modulo ${\mathcal{S}}_{0}^{-j-1}$. We will proceed by induction. Suppose now that the assertion is true for all $j\leq J-1$ and all $k\geq 1$, and for $j=J$ and $k\leq K$. This implies that the LHS of (3.26) with $k=K$ and $j=J$ is equal to $2i(K+1)(1-\eta)\widetilde{\varphi}_{1}(a_{K+1,J}-\widetilde{a}_{K+1,J})$ modulo ${\mathcal{S}}_{0}^{-J}$, while the RHS belongs to ${\mathcal{S}}_{0}^{-J}$. Hence, $(1-\eta)(a_{K+1,J}-\widetilde{a}_{K+1,J})$ belongs to ${\mathcal{S}}_{0}^{-J-1}$, as desired.   $\Box$

In view of (3.11) we have

[TABLE]

where

[TABLE]

Lemma 3.8**.**

For every integer $m\geq 0$ there are $N_{m}>1$ and $\ell_{m}>0$ such that for all $N\geq N_{m}$ we have the estimate

[TABLE]

if $\psi=\psi^{0}$, $z\in Z_{1}(1/2-\epsilon)$, or $\psi=\psi^{0}(1-\chi)$, $z\in Z_{1}(1-\epsilon)$. If $\psi=\psi^{0}$, $z\in Z_{2}\cup Z_{3}$, then (3.27) holds with $\epsilon$ replaced by $1$.

Proof. Denote by $G_{D}$ the Dirichlet self-adjoint realization of the operator $-n^{-1}\Delta$ on the Hilbert space $L^{2}(\Omega;n(x)dx)$. It is easy to see that

[TABLE]

where $\theta(z)=|{\rm Im}\,z|$ if $z\in Z_{1}$, $\theta(z)=1$ if $z\in Z_{2}\cup Z_{3}$. Clearly, under the conditions of Lemma 3.8, we have $h<\theta(z)\leq 1$. The above bound together with the coercivity of $G_{D}$ imply

[TABLE]

for every integer $s\geq 0$. We also have the identity

[TABLE]

where $\gamma$ denotes the restriction on $\Gamma$, and

[TABLE]

[TABLE]

where $K=K_{1}+K_{2}$ with

[TABLE]

[TABLE]

By the trace theorem we get from (3.28) and (3.29),

[TABLE]

[TABLE]

To bound the norm of $V$ we need to bound the kernel of the operator

[TABLE]

By Lemma 3.1 of [14] we have

[TABLE]

[TABLE]

where $C>0$ is some constant. Hence, by (3.15), for $0\leq x_{1}\leq 2\delta\min\{1,|\rho|^{3}\}$ we have

[TABLE]

[TABLE]

On the other hand, by Lemmas 3.2 and 3.5, for $|\rho|^{4}\geq h$ and $0\leq x_{1}\leq 2\delta\min\{1,|\rho|^{3}\}$ we have

[TABLE]

[TABLE]

for every multi-index $\alpha$ with some $\ell_{\alpha}>0$ independent of $N$. By (3.31), (3.32) and (3.33), using that $|\rho|^{2}\geq C|{\rm Im}\,z|$, $C>0$, on supp$\,\eta$, we conclude

[TABLE]

for $z\in Z_{1},\,(x^{\prime},\xi^{\prime})\in{\rm supp}\,\eta$, and

[TABLE]

otherwise, with possibly a new $\ell_{\alpha}>0$ independent of $N$. Similar estimates hold for the function $K_{1}$, too. Indeed, observe that on supp$\,\left[\Delta,\Phi_{\delta}\right]$ we have $\delta\min\{1,|\rho|^{3}\}\leq x_{1}\leq 2\delta\min\{1,|\rho|^{3}\}$, and hence

[TABLE]

[TABLE]

with some constant $C>0$. Using (3.36) one can easily get that the estimates (3.34) and (3.35) are satisfied with $K_{2}$ replaced by $K_{1}$. Therefore, the function $K$ satisfies the bounds

[TABLE]

Moreover, since $|\rho|\geq Const>0$ on supp$(1-\chi)$, in the case when $\psi=\psi^{0}(1-\chi)$ we obtain that (3.37) holds with $Z_{1}(1/2-\epsilon)$ replaced by $Z_{1}(1-\epsilon)$ and $2\epsilon$ replaced by $\epsilon$. Note now that the kernel, $L_{\alpha}$, of the operator ${\mathcal{K}}_{\alpha}$ is given by

[TABLE]

If $N$ is taken large enough, (3.37) implies the bounds

[TABLE]

with a new $\ell_{\alpha}>0$ independent of $N$. When $\psi=\psi^{0}(1-\chi)$, (3.38) holds with $Z_{1}(1/2-\epsilon)$ replaced by $Z_{1}(1-\epsilon)$ and $2\epsilon$ replaced by $\epsilon$. Clearly, (3.27) follows from (3.30) and (3.38).   $\Box$

In the case when $\psi=\psi^{0}$, by (3.23) we have

[TABLE]

on supp$\,\eta$, and

[TABLE]

on supp$\,(1-\eta)$. Since $|\rho|\geq C\sqrt{|{\rm Im}\,z|}$ for $z\in Z_{1}$, $(x^{\prime},\xi^{\prime})\in{\rm supp}\,\chi$ and $|\rho|\geq C>0$ for $z\in Z_{2}\cup Z_{3}$ or $z\in Z_{1}$, $(x^{\prime},\xi^{\prime})\in{\rm supp}\,(1-\chi)$, we get

[TABLE]

for $z\in Z_{1}$, $(x^{\prime},\xi^{\prime})\in{\rm supp}\,\chi$, and

[TABLE]

otherwise. Hence $(1-\chi)a_{k,j}\in{\mathcal{S}}_{0}^{-j}$, $h^{k+j}a_{k,j}\in{\mathcal{S}}_{1/2-\epsilon}^{-j}$ uniformly in $z\in Z_{1}(1/2-\epsilon)$, $a_{k,j}\in{\mathcal{S}}_{0}^{-j}$ for $z\in Z_{2}\cup Z_{3}$. Therefore, we have

[TABLE]

for every integer $m\geq 0$, where $\eta_{1}\in C_{0}^{\infty}(T^{*}\Gamma)$ is such that $\eta_{1}=1$ on supp$\,\eta$. In view of (2.4) this implies

[TABLE]

[TABLE]

for every integer $m\geq 0$. In view of Proposition 2.3 we also have

[TABLE]

By (3.39) and (3.40) we conclude

[TABLE]

By Lemma 3.7 we also have

[TABLE]

In the case when $\psi=\psi^{0}(1-\chi)$, the functions $a_{k,j}$ vanish on supp$\,\chi$, and hence $a_{k,j}\in{\mathcal{S}}_{0}^{-j}$ for $z\in Z_{1}$. Therefore, in this case the estimate (3.41) holds with $|{\rm Im}\,z|$ replaced by $1$ and $Z_{1}(1/2-\epsilon)$ replaced by $Z_{1}$.

We are ready now to prove Theorem 3.1. If $s=0$ we put $b_{0}^{\psi}=-ih(1-\eta)\widetilde{a}_{1,0}$, and if $s\geq 1$ we put

[TABLE]

In view of Lemma 3.6, the function $b_{s}^{\psi}$ is independent of all $n_{\ell}$ with $\ell\geq s$. If we take $N$ big enough, we can decompose the function $\tau_{\psi}$ as

[TABLE]

where

[TABLE]

By (3.39), (3.41) and (3.42) we have

[TABLE]

Moreover, if $\psi=\psi^{0}(1-\chi)$, the estimate (3.43) holds with $|{\rm Im}\,z|$ replaced by $1$ and $Z_{1}(1/2-\epsilon)$ replaced by $Z_{1}$. We would like to apply Lemma 3.8 with $m=s+1$. To this end we take $N$ big enough to arrange that

[TABLE]

By (3.27) and (3.43) we get

[TABLE]

[TABLE]

if $\psi=\psi^{0}$. Moreover, if $\psi=\psi^{0}(1-\chi)$, the estimate (3.44) holds with $|{\rm Im}\,z|$ replaced by $1$ and $Z_{1}(1/2-\epsilon)$ replaced by $Z_{1}(1-\epsilon)$.

We will now use a partition of the unity on $\Gamma$. We can find functions $\{\psi_{j}^{0}\}_{j=1}^{J}$ such that $\sum_{j=1}^{J}\psi_{j}^{0}=1$ and (3.44) is valid with $\psi$ replaced by each $\psi_{j}$, where $\psi_{j}$ is defined by replacing in the definiton of $\psi$ the function $\psi^{0}$ by $\psi_{j}^{0}$. Summing up all the estimates we get (3.8) and (3.9), respectively.   $\Box$

4. Improved estimates

To prove Theorem 1.1 we actually need the following improved version of Theorem 3.1.

Theorem 4.1**.**

Let $z\in Z_{1}(1/2-\epsilon)$, $0<\epsilon\ll 1$. Then, for every integer $s\geq 1$ there are an operator ${\mathcal{B}}_{s}$ independent of all $n_{k}$ with $k\geq s$ and an operator

[TABLE]

independent of all $n_{k}$ with $k\geq 1$ such that

[TABLE]

where $I$ denotes the identity. If $z\in Z_{2}\cup Z_{3}$, then (4.2) holds with $|{\rm Im}\,z|$ replaced by $1$.

Proof. Recall that by (3.5), (3.6), (3.7), we have that for every integer $k$, $h^{\frac{k_{-}}{4}}\rho^{k}\in{\mathcal{S}}_{1/2-\epsilon}^{k}$ uniformly in $z\in Z_{1}(1/2-\epsilon)$ and $\rho^{k}\in{\mathcal{S}}_{0}^{k}$ if $z\in Z_{2}\cup Z_{3}$. We would like to apply Proposition 2.2 with

[TABLE]

Using (3.2) one can easily check that (2.5) is satisfied with $\mu=h|{\rm Im}\,z|^{-2}={\mathcal{O}}(h^{2\epsilon})$. By (2.6) we get

[TABLE]

if $h$ is taken small enough. It follows from (4.3) that the operator ${\rm Op}_{h}(a^{-}_{s})$ is invertible with an inverse

[TABLE]

Since $h^{s}a^{+}_{s}\in{\mathcal{S}}_{1/2-\epsilon}^{s+1}$ uniformly in $z$, by Proposition 2.3 we have

[TABLE]

which implies (4.1). By (3.27) and (4.1),

[TABLE]

if $N$ is taken large enough, where ${\mathcal{T}}=\sum_{j=1}^{J}{\mathcal{T}}_{\psi_{j}}$. On the other hand, we can write

[TABLE]

where

[TABLE]

[TABLE]

Clearly, the operator ${\mathcal{B}}_{s}$ is independent of all $n_{k}$ with $k\geq s$ because so is the function $b_{s}$. Therefore, it follows from (4.4) that to prove (4.2) it suffices to prove the bound

[TABLE]

In view of Lemmas 3.5 and 3.7, we have $\widetilde{b}_{s}=h^{s+1}g_{s}$ with $g_{s}\in S_{2,2}^{-3-4s}(|\rho|)+S_{0,1}^{-s-1}(|\rho|)$ uniformly in $h$ as long as $|\rho|^{4}\geq h$. Thus, (4.5) is equivalent to

[TABLE]

To prove (4.6) observe that $\rho^{s+1}g_{s}\in S_{2,2}^{-2-3s}(|\rho|)+S_{0,1}^{0}(|\rho|)$ uniformly in $h$, which yields the bounds

[TABLE]

By (2.4) and (4.7) we get

[TABLE]

On the other hand, applying Proposition 2.2 with $a^{+}=\rho^{s+1}$ and $a^{-}=g_{s}$ yields the bound

[TABLE]

[TABLE]

Clearly, (4.6) follows from (4.8) and (4.9).   $\Box$

5. Proof of Theorem 1.1

Define the DN maps ${\mathcal{N}}_{j}(\lambda)$, $j=1,2$, by

[TABLE]

where $\nu$ is the Euclidean unit inner normal to $\Gamma$ and $u_{j}$ is the solution to the equation

[TABLE]

and consider the operator

[TABLE]

Clearly, $\lambda$ is a transmission eigenvalue if there exists a non-trivial function $f$ such that $T(\lambda)f=0$. Thus Theorem 1.1 is a consequence of the following

Theorem 5.1**.**

Under the conditions of Theorem 1.1, the operator $T(\lambda)$ sends $L^{2}(\Gamma)$ into $H^{j+1}(\Gamma)$. Moreover, there exists a constant $C>0$ such that $T(\lambda)$ is invertible for $|{\rm Im}\,\lambda|\geq C({\rm Re}\,\lambda+1)^{1-\kappa_{j}}$ with an inverse satisfying in this region the bound

[TABLE]

where the Sobolev space is equipped with the classical norm.

Proof. We make our problem semi-classical by putting $h=|{\rm Re}\,\lambda^{2}|^{-1/2}$, $z=h^{2}\lambda^{2}=\pm 1+i{\rm Im}\,z$, if $|{\rm Re}\,\lambda^{2}|\geq|{\rm Im}\,\lambda^{2}|$, $\pm{\rm Re}\,\lambda^{2}>0$, and $h=|{\rm Im}\,\lambda^{2}|^{-1/2}$, $z=h^{2}\lambda^{2}={\rm Re}\,z+i$, if $|{\rm Re}\,\lambda^{2}|\leq|{\rm Im}\,\lambda^{2}|$. Clearly, $h\sim|\lambda|^{-1}$. We set ${\mathcal{N}}_{j}(h,z)=-ih{\mathcal{N}}_{j}(\lambda)$ and

[TABLE]

We now apply Theorem 4.1 with $s=j\geq 1$. In view of the conditions (1.5) and (1.6), we get

[TABLE]

for $z\in Z_{1}(1/2-\epsilon)$, where

[TABLE]

When $z\in Z_{2}\cup Z_{3}$, the estimate (5.3) holds with $|{\rm Im}\,z|$ replaced by $1$. It follows from (5.3) that the operator $(n^{(1)}_{j}-n^{(2)}_{j})^{-1}{\mathcal{A}}_{j}T(h,z)$ is invertible for $z\in Z_{1}(1/2-\epsilon)$, $|{\rm Im}\,z|\geq(C^{\prime}_{j}h)^{1/(3j/2+1)}$, and for $z\in Z_{2}\cup Z_{3}$, $h$ small enough. Hence so is $T(h,z)$ and we have the bound

[TABLE]

Now (5.2) follows from (5.4) after passing from $(h,z)$ to $\lambda$ and using the fact that the semi-classical norm in $H_{h}^{j+1}(\Gamma)$ is bounded from above by the classical norm in $H^{j+1}(\Gamma)$.   $\Box$

It is worth noticing that it follows from the estimate (3.9) that the operator $T(h,z)$ can be inverted outside the glancing region for much smaller $|{\rm Im}\,z|$. In other words, to improve the eigenvalue-free region (1.7) one has to improve the parametrix in the glancing region, only. More precisely, we have the following

Proposition 5.2**.**

Let $z\in Z_{1}(1-\epsilon)$. Then, under the conditions of Theorem 1.1, there exists an operator

[TABLE]

such that

[TABLE]

When $z\in Z_{2}\cup Z_{3}$, the estimate (5.5) holds with $\chi$ replaced by [math].

Proof. By (3.9) with $s=j$ we have

[TABLE]

for $z\in Z_{1}(1-\epsilon)$. Let $\chi_{1}\in C_{0}^{\infty}(T^{*}\Gamma)$ be such that $\chi=1$ on supp$\,\chi_{1}$, $\chi_{1}=1$ in a neighbourhood of $\Sigma$, and set

[TABLE]

We have $\widetilde{a}_{j}^{+}\in{\mathcal{S}}_{0}^{-j-1}$, $\widetilde{a}_{j}^{-}\in{\mathcal{S}}_{0}^{j+1}$ and $\widetilde{a}_{j}^{+}\widetilde{a}_{j}^{-}=1-\chi$. We now apply Proposition 2.4 with $\widetilde{a}_{j}^{+}$ and $\widetilde{a}_{j}^{-}$ in place of $a^{+}$ and $a^{-}$. We have

[TABLE]

Clearly, (5.5) follows from (5.6) and (5.7) with $\widetilde{\mathcal{A}}_{j}=h^{-j}{\rm Op}_{h}(1-\chi){\rm Op}_{h}(\widetilde{a}_{j}^{-})$.   $\Box$