Dimension bounds in monotonicity methods for the Helmholtz equation

TL;DR

This paper improves bounds on the finite dimensional space in monotonicity methods for the Helmholtz equation, showing it is trivial when two coefficients have the same number of positive Neumann eigenvalues, enhancing shape detection techniques.

Contribution

It refines the dimension bounds in monotonicity inequalities for the Helmholtz equation, providing conditions under which the finite dimensional space is trivial.

Findings

Finite dimensional space bounds are improved.

Trivial space when coefficients share the same positive Neumann eigenvalues.

Enhances shape detection and inverse boundary problem methods.

Abstract

The article [HPS] established a monotonicity inequality for the Helmholtz equation and presented applications to shape detection and local uniqueness in inverse boundary problems. The monotonicity inequality states that if two scattering coefficients satisfy $q_1 \leq q_2$, then the corresponding Neumann-to-Dirichlet operators satisfy $\Lambda(q_1) \leq \Lambda(q_2)$ up to a finite dimensional subspace. Here we improve the bounds for the dimension of this space. In particular, if $q_1$ and $q_2$ have the same number of positive Neumann eigenvalues, then the finite dimensional space is trivial.

Equations

$${\sigma }_1\le {\sigma }_2⟹\Lambda ({\sigma }_1)\ge \Lambda ({\sigma }_2).$$

$$\left\{\begin{array}[]{rcl}(\Delta+k^{2}q)u\!\!\!&=&\!\!\!0\text{ in $\Omega$},\\[5.0pt] \partial_{\nu}u\!\!\!&=&\!\!\!g\text{ on $\partial\Omega$}.\end{array}\right.$$

$$\Lambda (q):L^2(\partial \Omega )\to L^2(\partial \Omega ),g\mapsto u{|}_{\partial \Omega }.$$

$$q_2-q_1\ge 0⟹\Lambda (q_2)-\Lambda (q_1)\text{ has only finitely many negative eigenvalues}.$$

$$d(q_1,q_2)\le d(q_2).$$

$$d(q_1,q_2)\le d(q_2)-d(q_1).$$

$$q_1\le q_2⟹\Lambda (q_1)\le \Lambda (q_2).$$

$$q(t)=q_1+t(q_2-q_1),t\in [0,1].$$

$${\lambda }_j^{\prime }(t)=k^2{\int }_{\Omega }(q_2-q_1){\phi }_j(t{)}^{2}dx$$

$$N(q)$$

$$D(q)$$

$$(\Delta +k^{2}q)u=F\text{ in Ω},{\partial }_{\nu }u{|}_{\partial \Omega }=g,$$

$${\int }_{\Omega }F\phi dx={\int }_{\partial \Omega }g\phi dS,\phi \in N(q).$$

$$T_q:N(q{)}^{\perp }\times D(q{)}^{\perp }\to H^1(\Omega ),(F,g)\mapsto u_{F,g}$$

$$u=\sum _{j\in J}{a}_j{\phi }_j+\sum _{j\notin J}\frac{1}{{\lambda }_j}\left[{\int }_{\Omega }F{\phi }_{j}dx-{\int }_{\partial \Omega }g{\phi }_{j}dS\right]{\phi }_j,$$

$$(\mathrm{Id}-K-k^2{K}_q)u=-{\iota }^{\ast }F+{\gamma }^{\ast }g$$

$$(-{\iota }^{\ast }F+{\gamma }^{\ast }g,\phi {)}_{H^1(\Omega )}=0$$

$$\Vert u_{F,g}{\Vert }_{H^1(\Omega )}\le C\Vert -{\iota }^{\ast }F+{\gamma }^{\ast }g{\Vert }_{H^1(\Omega )}\le C(\Vert F{\Vert }_{L^2(\Omega )}+\Vert g{\Vert }_{L^2(\partial \Omega )}).$$

$$u=\sum _{j=1}^{\infty }{c}_j{\phi }_j,c_j={\int }_{\Omega }u{\phi }_{j}dx,$$

$${\int }_{\Omega }F{\phi }_{j}dx$$

$$={\int }_{\partial \Omega }g{\phi }_{j}dS+{\lambda }_j{\int }_{\Omega }u{\phi }_j.$$

$$q(t):=q_1+t(q_2-q_1),t\in [0,1].$$

$$H(t):=\Delta +k^{2}q(t),t\in [0,1].$$

$$H(t){\phi }_j(t)={\lambda }_j(t){\phi }_j(t)\text{ in Ω},{\partial }_{\nu }{\phi }_j(t){|}_{\partial \Omega }=0,$$

$$H(z)=\Delta +k^2(q_1+z(q_2-q_1)).$$

$$\mathcal{D}=\{u\in H^1(\Omega );\Delta u\in L^2(\Omega ),{\tilde{\gamma }}_{N}u=0\text{ in H−1/2(∂Ω)}\}$$

$$z\mapsto H(z)u=\Delta u+k^2{q}_{1}u+k^{2}z(q_2-q_1)u$$

$$H(z{)}^{\ast }={\Delta }^{\ast }+B(z{)}^{\ast }=\Delta +B(\bar{z})=H(\bar{z}).$$

$$R_z(\zeta ):=(H(z)-\zeta {)}^{-1},$$

$$R_z({\zeta }_0):=(H(z)-{\zeta }_0{)}^{-1},$$

$$R_z(\zeta )=R_z({\zeta }_0)-({\zeta }_0-\zeta )R_z(\zeta )R_z({\zeta }_0).$$

Full text

††footnotetext:   This is a preprint version of a journal article published in

SIAM J. Math. Anal. 51(4), 2995–3019, 2019 (https://doi.org/10.1137/19M1240708).

1. Introduction

This article is concerned with monotonicity properties arising in inverse problems and applications. As a basic example, if $\sigma_{1}$ and $\sigma_{2}$ are positive functions (representing electrical conductivities) in a bounded domain $\Omega\subset\mathbb{R}^{n}$ and if $\Lambda(\sigma_{1})$ and $\Lambda(\sigma_{2})$ are the corresponding Neumann-to-Dirichlet operators (representing electrical boundary measurements), then one has the monotonicity property

[TABLE]

The last statement means that $\Lambda(\sigma_{1})-\Lambda(\sigma_{2})$ is a positive semidefinite operator on the mean-free functions in $L^{2}(\partial\Omega)$ (the so-called Loewner order). This property, together with a certain nontrivial converse based on localized potentials [Geb08], leads to efficient monotonicity based methods for determining shapes of obstacles or inclusions from electrical or optical boundary measurements, cf. [TR02] for the origin of this idea, [HU13] for the proof of the converse monotonicity property, and the list of references for recent works on monotonicity-based methods at the end of this introduction.

The recent work [HPS] extends monotonicity based methods to imaging problems with positive frequency, in particular acoustic imaging modelled by the Helmholtz equation. It turns out that the basic monotonicity property may fail in this case, but monotonicity still holds up to a finite dimensional space and [HPS] shows that shape detection methods and local uniqueness results can be developed also in this situation. [GH18] extends this idea to farfield inverse scattering and shows numerical reconstructions.

Let us describe the results of [HPS] in more detail. Let $\Omega\subset\mathbb{R}^{n}$, $n\geq 2$ be a bounded Lipschitz domain, and let $q\in L^{\infty}(\Omega)$ be a real valued function with $q\not\equiv 0$. Let $k>0$, and consider the Neumann problem

[TABLE]

We assume that $k>0$ is not a resonance frequency, which means that the Neumann problem has a unique solution $u\in H^{1}(\Omega)$ for any $g\in L^{2}(\partial\Omega)$. Define the Neumann-to-Dirichlet (ND) operator

[TABLE]

Let $\lambda_{1}\geq\lambda_{2}\geq\cdots\to-\infty$ be the Neumann eigenvalues of $\Delta+k^{2}q$ in $\Omega$, and let $d(q)$ be the number of positive Neumann eigenvalues (counted with multiplicity).

In [HPS, Theorem 3.5] it was proved that

[TABLE]

Here we consider $\Lambda(q_{2})-\Lambda(q_{1})$ as a compact self-adjoint operator on $L^{2}(\partial\Omega)$. Let $d(q_{1},q_{2})$ be the number of negative eigenvalues of $\Lambda_{q_{2}}-\Lambda_{q_{1}}$ (counting with multiplicity). In [HPS, Theorem 3.5] it was also proved that $d(q_{1},q_{2})$ satisfies the bound

[TABLE]

The next result gives a more precise estimate for $d(q_{1},q_{2})$.

Theorem 1.1**.**

*Let $q_{1},q_{2}\in L^{\infty}(\Omega)\setminus\{0\}$ be such that $k$ is not a resonance frequency for $q_{1}$ and $q_{2}$. Assume that $q_{1}\leq q_{2}$ a.e. in $\Omega$. Then *

[TABLE]

This has an immediate consequence: even if $q_{1}$ and $q_{2}$ are positive, the standard monotonicity inequality for the ND operators remains true if $q_{1}$ and $q_{2}$ have the same number of positive Neumann eigenvalues.

Theorem 1.2**.**

Let $q_{1},q_{2}\in L^{\infty}(\Omega)\setminus\{0\}$ be such that $k$ is not a resonance frequency for $q_{1}$ and $q_{2}$. Assume that $d(q_{1})=d(q_{2})$. Then

[TABLE]

Let us describe the main idea of the proof of Theorem 1.1. If $q_{1}$ and $q_{2}$ are in $L^{\infty}(\Omega)\setminus\{0\}$ and satisfy $q_{1}\leq q_{2}$, we define the interpolated potentials

[TABLE]

Denote by $\lambda_{1}(t)\geq\lambda_{2}(t)\geq\cdots\to-\infty$ the Neumann eigenvalues of $\Delta+k^{2}q(t)$ in $\Omega$. Assume for simplicity that each $\lambda_{j}(t)$ is a simple eigenvalue (the proof in Section 2 removes this restriction). Then each map $\lambda_{j}:[0,1]\to\mathbb{R}$ is smooth and strictly increasing. This follows from the variational formula

[TABLE]

where $\{\varphi_{j}(t)\}$ is an $L^{2}$-orthonormal basis consisting of Neumann eigenfunctions corresponding to $\{\lambda_{j}(t)\}$, and from the unique continuation principle. Now, when $t=0$ one starts with $d(q_{1})$ positive eigenvalues, and when $t=1$ one has $d(q_{2})$ positive eigenvalues. Since the maps $t\mapsto\lambda_{j}(t)$ are strictly increasing, exactly $d(q_{2})-d(q_{1})$ eigenvalues cross the real axis as $t$ increases to $1$, and the eigenspace at each crossing gives rise to a one-dimensional subspace of $L^{2}(\partial\Omega)$. Now if $g\in L^{2}(\partial\Omega)$ is orthogonal to all these one-dimensional subspaces, it follows that $((\Lambda(q_{2})-\Lambda(q_{1}))g,g)\geq 0$, proving that the finite-dimensional obstruction has dimension $\leq d(q_{2})-d(q_{1})$.

The next result complements Theorem 1.1 by showing that in certain cases where $q_{1}$ and $q_{2}$ differ by a constant, there are lower bounds on the number of negative eigenvalues. Its proof is based on computing an expression for the quadratic form $((\Lambda(q_{2}-b)-\Lambda(q_{1}+a))g,g)$ in terms of the Neumann eigenfunctions of $\Delta+k^{2}q_{1}$, and showing that the quadratic form is negative for $g$ in a space spanned by finitely many traces of Neumann eigenfunctions.

Theorem 1.3**.**

Let $q_{1},q_{2}\in L^{\infty}(\Omega)\setminus\{0\}$ be such that $k$ is non-resonant for $q_{1}$ and $q_{2}$. Assume that $q_{2}-q_{1}$ is a positive constant and $d(q_{2})>d(q_{1})$. Let $\mu_{1}$ be the largest negative Neumann eigenvalue of $\Delta+k^{2}q_{1}$, and let $\mu_{2}$ be the smallest positive eigenvalue of $\Delta+k^{2}q_{2}$. Let $N_{j}$ be the multiplicity of $\mu_{j}$.

1. (a)

$\Lambda(q_{2}-b)-\Lambda(q_{1}+a)$* has at least $N_{1}$ negative eigenvalues whenever $a<k^{-2}|\mu_{1}|$ is sufficiently close to $k^{-2}|\mu_{1}|$, and $b\in[0,k^{-2}\mu_{2})$.* 2. (b)

$\Lambda(q_{2}-b)-\Lambda(q_{1}+a)$* has at least $N_{2}$ negative eigenvalues whenever $b<k^{-2}\mu_{2}$ is sufficiently close to $k^{-2}\mu_{2}$, and $a\in[0,k^{-2}|\mu_{1}|)$.*

From the previous theorem, we obtain the following special cases where equality is attained in Theorem 1.1.

Theorem 1.4**.**

**

1. (a)

Let $q\in L^{\infty}(\Omega)$ be such that [math] is a Neumann eigenvalue of $\Delta+k^{2}q$ with multiplicity $N$. For $\varepsilon>0$ small enough, $d(q+\varepsilon)-d(q-\varepsilon)=N$ and $\Lambda(q+\varepsilon)-\Lambda(q-\varepsilon)$ has exactly $N$ negative eigenvalues. 2. (b)

Let $\Omega$ be the square $(0,\pi)^{2}\subset\mathbb{R}^{2}$, let $k>0$, and let $N\geq 2$ be even. There is a $c>0$ such that for $\varepsilon>0$ small, $\Lambda(c+\varepsilon)-\Lambda(c-\varepsilon)$ has exactly $N=d(c+\varepsilon)-d(c-\varepsilon)$ negative eigenvalues.

Let us give some more references to earlier and related work, and comment on the relevance of our results. Monotonicity estimates and localized potentials techniques have been used in different ways for the study of inverse problems [Har09, HS10, Har12, AH13, HU13, BHHM17, HU17, BHKS18, GH18, HPS, HL18, HLL18] and several recent works build practical reconstruction methods on monotonicity properties [TR02, HLU15, HU15, HM16, MVVT16, TSV*+*16, Gar17, GS17, SUG*+*17, VMC*+*17, HM18, ZHS18, GS19]. Recently, monotonicity arguments were also discovered to yield Lipschitz stability results, cf. [HM19, SKJ*+*18, Har19]. All of these works consider stationary imaging cases where monotonicity of the ND operators holds in the sense of the Loewner order as explained above. So far, only [HPS, GH18, HL19] cover the case of positive frequency imaging where the monotonicity only holds up to a finite dimensional space. For extending monotonicity-based theoretical uniqueness and stability results, as well as monotonicity-based numerical reconstruction methods, it seems to be of utmost importance to have a good bound on the number of eigenvalues that have to be disregarded. [HPS] showed that this number is smaller than $d(q_{2})$, which might become arbitrarily large for high frequencies $k\to\infty$. Using this bound would result in disregarding a large part of the ND operators for high frequencies, and might make numerical reconstruction methods unfeasible. This article, however, shows that the number is smaller that $d(q_{2})-d(q_{1})$ which might still be small (or even zero) for high frequencies. Note also, that this article indicates that the bound is sharp for $q_{1}$ close to $q_{2}$, but that the bound might get too large when $q_{2}-q_{1}$ increases, cf. section 4.

The rest of this paper is organized as follows. Section 2 gives the proof of Theorem 1.1, and Section 3 proves Theorems 1.3 and 1.4. Section 4 gives an simple alternative proof of Theorem 1.1 for the case where $q_{1}$ and $q_{2}$ are constant, and numerically studies the sharpness of the bound for large $q_{2}-q_{1}$.

Acknowledgements

M.S. was supported by the Academy of Finland (Centre of Excellence in Inverse Modelling and Imaging, grant numbers 312121 and 309963) and by the European Research Council under Horizon 2020 (ERC CoG 770924).

2. Upper bound for the number of negative eigenvalues

For the proof of Theorem 1.1, it will be useful to consider solutions of the Helmholtz equation also when $k$ is a resonant frequency. In this case the Neumann data needs to satisfy finitely many linear constraints.

Lemma 2.1**.**

Let $q\in L^{\infty}(\Omega)$ and $k>0$, and define the sets

[TABLE]

Also let $N(q)^{\perp}$ be the orthogonal complement of $N(q)$ in $L^{2}(\Omega)$, and let $D(q)^{\perp}$ be the orthogonal complement of $D(q)$ in $L^{2}(\partial\Omega)$.

1. (a)

$N(q)$* and $D(q)$ are finite-dimensional spaces whose dimension is the multiplicity of [math] as the Neumann eigenvalue of $\Delta+k^{2}q$ in $\Omega$.* 2. (b)

For any $F\in L^{2}(\Omega)$ and $g\in L^{2}(\partial\Omega)$, the equation

[TABLE]

has a solution $u\in H^{1}(\Omega)$ if and only if one has the compatibility conditions

[TABLE]

In particular, a solution exists whenever $F\in N(q)^{\perp}$ and $g\in D(q)^{\perp}$. The solution is unique up to addition of a function in $N(q)$, and one has a bounded map

[TABLE]

where $u_{F,g}$ is the unique solution with $(u,\varphi)_{H^{1}(\Omega)}=0$ for $\varphi\in N(q)$. 3. (c)

Let $\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq\ldots\to-\infty$ be the Neumann eigenvalues of $\Delta+k^{2}q$ in $\Omega$ and let $(\varphi_{j})_{j=1}^{\infty}$ be a corresponding orthonormal basis of $L^{2}(\Omega)$ consisting of Neumann eigenfunctions. If $F\in L^{2}(\Omega)$ and $g\in L^{2}(\partial\Omega)$ satisfy (2.2), then any solution $u\in H^{1}(\Omega)$ of (2.1) may be represented as the $L^{2}(\Omega)$-convergent sum

[TABLE]

where $J=\{j\geq 1\,;\,\lambda_{j}=0\}$ is finite, and $a_{j}\in\mathbb{R}$ are some constants.

Remark**.**

The sum in part (c) may not converge in higher norms in general. In fact, if it did converge in some space where the normal derivative operator is bounded, then one would get that $\partial_{\nu}u|_{\partial\Omega}=0$, which is not true if $g\neq 0$.

Proof.

As in [HPS, Section 2.1] we use the compact inclusion map $\iota:H^{1}(\Omega)\to L^{2}(\Omega)$, to define $K=\iota^{*}\iota$ and $K_{q}=\iota^{*}M_{q}\iota$, where $M_{q}:L^{2}(\Omega)\to L^{2}(\Omega)$ is the multiplication operator by $q$. Both $K$ and $K_{q}$ are compact self-adjoint operators from $H^{1}(\Omega)$ to $H^{1}(\Omega)$. A function $u\in H^{1}(\Omega)$ is a weak solution of (2.1) if and only if

[TABLE]

where $\gamma:H^{1}(\Omega)\to L^{2}(\partial\Omega)$ is the trace operator. By Fredholm theory (see e.g. [RR04, Corollary 8.95]), this problem has a solution $u\in H^{1}(\Omega)$ for given $F\in L^{2}(\Omega)$, $g\in L^{2}(\partial\Omega)$ if and only if

[TABLE]

for all $\varphi\in H^{1}(\Omega)$ in the kernel of $\mathrm{Id}-K-k^{2}K_{q}$. But this finite dimensional kernel is equal to $N(q)$, showing that (2.1) is solvable if and only if (2.2) holds. The representation in [RR04, Corollary 8.95] shows that there is a unique solution $u_{F,g}$ with $(u_{F,g},\varphi)_{H^{1}(\Omega)}=0$ for $\varphi\in N(q)$, and that

[TABLE]

Finally, the map $N(q)\to D(q)$, $\varphi\mapsto\varphi|_{\partial\Omega}$ is bijective by the unique continuation principle. This proves (a) and (b).

To prove (c) let $u$ be a solution of (2.1). Since $(\varphi_{j})$ is an orthonormal basis of $L^{2}(\Omega)$ we have

[TABLE]

with convergence in $L^{2}(\Omega)$. Testing the weak form of (2.1) against $\varphi_{j}$ and integrating by parts gives that

[TABLE]

This yields the representation for $u$ in (c). ∎

For $q_{1},q_{2}\in L^{\infty}(\Omega)\setminus\{0\}$, we define

[TABLE]

We also define the family of operators

[TABLE]

The following result from analytic perturbation theory is needed to describe the behaviour of the eigenvalues of $H(t)$ as $t$ changes.

Lemma 2.2**.**

Let $q_{1},q_{2}\in L^{\infty}(\Omega)$ and $k>0$, assume that $q_{1}\leq q_{2}$ and $q_{1}\not\equiv q_{2}$, and let $\lambda_{1}\geq\lambda_{2}\geq\ldots\to-\infty$ be the Neumann eigenvalues of $\Delta+k^{2}q_{1}$ in $\Omega$. There exist real-analytic functions $\lambda_{j}:[0,1]\to\mathbb{R}$ and $\varphi_{j}:[0,1]\to L^{2}(\Omega)$ with the following properties:

1. (a)

$\lambda_{j}(0)=\lambda_{j}$* for $j\geq 1$, and for any $t\in[0,1]$, the numbers $\lambda_{j}(t)$ represent the repeated*††We call $\lambda_{1}\geq\lambda_{2}\geq\ldots$ the repeated eigenvalues iff the value $\lambda_{j}$ is repeated the times of its multiplicity in the sequence.* Neumann eigenvalues of $H(t)$ in $\Omega$. Zero is a Neumann eigenvalue of $H(t_{0})$ with multiplicity $N$ if and only if precisely $N$ functions $\lambda_{j}(t)$ vanish at $t=t_{0}$.* 2. (b)

Each $\lambda_{j}(t)$ is strictly increasing on $[0,1]$. 3. (c)

Each $\varphi_{j}(t)$ is a Neumann eigenfunction in $H^{1}(\Omega)$ satisfying

[TABLE]

and $(\varphi_{j}(t))_{j=1}^{\infty}$ is an orthonormal basis of $L^{2}(\Omega)$ for any $t\in[0,1]$. Each map $[0,1]\to H^{1}(\Omega),\ t\mapsto\varphi_{j}(t)$ is real-analytic.

Proof.

This result will be proved by analytic perturbation theory, and hence in this proof we will assume the function spaces to be complex valued.

(a) For $z\in\mathbb{C}$, define the operator

[TABLE]

Then $H(t)=\Delta+k^{2}q(t)$ for $t\in\mathbb{R}$. We wish to use [Kat95, Theorem VII.3.9 on p. 392] to show that the Neumann eigenvalues of $H(t)$ can be parametrized analytically with respect to $t$ (see [BS12, Theorem 3.1, p. 442] and [RS78, Theorem XII.13] for related results). In order to do this, we need to realize $H(z)$ with Neumann boundary values as a self-adjoint analytic family of unbounded operators on $L^{2}(\Omega)$. In the present case where $\Omega$ has Lipschitz boundary, the required results may be found in [GM08, Section 2] (in fact the easier abstract results in [GM08, Appendix B] would suffice).

Define the set

[TABLE]

where $\tilde{\gamma}_{N}$ is the weak Neumann trace operator in [GM08, formula (2.40) and (2.41)]. We consider $H(z)$ as an unbounded linear operator on $L^{2}(\Omega)$ with domain $\mathrm{dom}(H(z))=\mathcal{D}$. The family $(H(z))_{z\in\mathbb{C}}$ has the following properties:

1. (i)

Each $H(z)$ is closed and densely defined. This follows since $\Delta$ with domain $\mathcal{D}$ is self-adjoint by [GM08, Theorem 2.6], hence $\Delta$ and consequently also $H(z)$ is closed and densely defined. 2. (ii)

The family $(H(z))_{z\in\mathbb{C}}$ is holomorphic of type (A) (see [Kat95, Section VII.2.1]) since for each $u\in\mathcal{D}$, the map

[TABLE]

is holomorphic. 3. (iii)

The family $(H(z))_{z\in\mathbb{C}}$ is a self-adjoint holomorphic family, i.e. a holomorphic family of operators satisfying $H(z)^{*}=H(\bar{z})$ (see [Kat95, Section VII.3.1]): since $\Delta$ with domain $\mathcal{D}$ is self-adjoint [GM08, Theorem 2.6] and the map $B(z):u\mapsto k^{2}(q_{1}+z(q_{2}-q_{1}))u$ is bounded on $L^{2}(\Omega)$, by [DS67, Lemma XII.1.6] we have

[TABLE] 4. (iv)

$H(z)$ has compact resolvent, when $z\in\mathbb{C}$. This can be seen as follows. Let

[TABLE]

denote the resolvent. Arguing as in the proof of [GM08, Corollary 2.7] using [GM08, Remark 2.19], we have that

[TABLE]

is compact, when $\zeta_{0}\in\mathbb{R}_{+}$ is large enough. By the resolvent identity in [RS72, Theorem VIII.2] we have that

[TABLE]

The resolvent $R_{z}(\zeta)$ is by definition continuous on $L^{2}(\Omega)$, when $\zeta$ is in the resolvent set. The above formula implies hence that $R_{z}(\zeta)$ is compact on $L^{2}(\Omega)$, since $R_{z}(\zeta_{0})$ is compact.

Thus the family $(H(z))_{z\in\mathbb{C}}$ satisfies the conditions in [Kat95, Theorem VII.3.9 on p. 392], and there are real-analytic functions $\lambda_{j}(t)$ and real-analytic vector functions $\varphi_{j}(t)$, for $t\in[0,1]$, such that $\lambda_{j}(t)$ represent all the repeated eigenvalues of $H(t)$, $\varphi_{j}(t)$ are the corresponding eigenfunctions, and $(\varphi_{j}(t))$ is an orthonormal basis of $L^{2}(\Omega)$. Since $\varphi_{j}(t)\in\mathcal{D}$, these are exactly the standard Neumann eigenvalues and eigenfunctions of $H(t)$ (see [GM08, formula (2.41)]). We may reorder $\lambda_{j}(t)$ and $\varphi_{j}(t)$ so that $\lambda_{j}(0)=\lambda_{j}$.

(b) We compute $\lambda_{j}^{\prime}(t)$ using a variational formula: by [Kat95, formula (VII.3.18), p. 391] and by the fact that $H^{\prime}(t)u=k^{2}(q_{2}-q_{1})u$, we have

[TABLE]

Using the assumption that $q_{2}\geq q_{1}$, we have $\lambda_{j}^{\prime}(t)\geq 0$. Moreover, since $q_{1}\not\equiv q_{2}$, we have $q_{2}-q_{1}\geq c>0$ in some set $E$ of positive measure in $\Omega$. Thus we see that $\lambda_{j}^{\prime}(t)>0$ (otherwise if $\lambda_{j}^{\prime}(t)=0$, then $\int_{E}\varphi_{j}(t)^{2}\,dx=0$ which would contradict the unique continuation principle). This implies that each $\lambda_{j}(t)$ is a strictly increasing function on $[0,1]$.

(c) All other statements in (c) have been proved, except that $t\mapsto\varphi_{j}(t)$ is real-analytic as a $H^{1}(\Omega)$-valued function. To prove this, note first that for any $v\in L^{2}(\Omega)$ the map

[TABLE]

is real-analytic on $[0,1]$. Now, if $\psi\in H^{1}(\Omega)$, we compute

[TABLE]

Each term on the last line is real-analytic for $t\in[0,1]$. Thus $t\mapsto\varphi_{j}(t)$ is weakly, and hence strongly, analytic as a $H^{1}(\Omega)$-valued function. ∎

We will next combine Lemmas 2.1 and 2.2 to obtain solutions of

[TABLE]

that depend Lipschitz continuously on $t\in[0,1]$ as long as $g$ is orthogonal to a finite-dimensional subspace of $L^{2}(\partial\Omega)$.

Lemma 2.3**.**

Assume the conditions in Lemma 2.2. Let $t_{1}<\ldots<t_{K}$ be the times when [math] is a Neumann eigenvalue of $H(t)$, let $g\in L^{2}(\partial\Omega)$, and let $u_{t}$ be the unique solution of (2.4) for $t\in[0,1]\setminus\{t_{1},\ldots,t_{K}\}$.

1. (a)

The map

[TABLE]

is real-analytic. The derivative $\partial_{t}u_{t}$ is the unique $H^{1}(\Omega)$ solution of $(\Delta+k^{2}q(t))v=-k^{2}q^{\prime}(t)u_{t}$ in $\Omega$ with $\partial_{\nu}v|_{\partial\Omega}=0$. For any compact $F\subset[0,1]\setminus\{t_{1},\ldots,t_{K}\}$, there is $C_{F}>0$ such that

[TABLE] 2. (b)

Let $t_{0}$ be one of $t_{1},\ldots,t_{K}$ and let $I=\{j\geq 1\,;\,\lambda_{j}(t_{0})=0\}$. Then

[TABLE]

uniformly for $t$ close to $t_{0}$. 3. (c)

If J=\{j\geq 1\,;\,\lambda_{j}(t_{l})=0\text{ for some l\in{1,2,\ldots,K}}\}, then

[TABLE]

uniformly over $t\in[0,1]\setminus\{t_{1},\ldots,t_{K}\}$. 4. (d)

With the notation of Lemma 2.1 let

[TABLE]

If additionally $g\in D^{\perp}$, then $t\mapsto u_{t}$ extends uniquely as a Lipschitz continuous map

[TABLE]

Proof.

We first show that [math] is a Neumann eigenvalue of $H(t)$ at only finitely many times $t$. Note that $\Delta+k^{2}q_{2}$ has at most finitely many positive Neumann eigenvalues. Since the functions $\lambda_{j}(t)$ are strictly increasing and since the Neumann eigenvalues of $\Delta+k^{2}q_{2}$ are given by $\lambda_{j}(1)$, we see that only finitely many of the functions $\lambda_{j}(t)$ have a zero in $[0,1]$. Thus there are only finitely many times $0\leq t_{1}<\ldots<t_{K}\leq 1$ in the interval $t\in[0,1]$ so that [math] is a Neumann eigenvalue of $H(t)$.

Proof of part (a).

Fix any $g\in L^{2}(\partial\Omega)$ and let $t\in[0,1]\setminus\{t_{1},\ldots,t_{K}\}$. Then the Neumann problem for $\Delta+k^{2}q(t)$ in $\Omega$ is well-posed, and we define $u_{t}=T_{q(t)}(0,g)$ as the unique solution of (2.4). Fix $t_{0}\in[0,1]\setminus\{t_{1},\ldots,t_{K}\}$, and note that for $t$ close to $t_{0}$ one has

[TABLE]

and $\partial_{\nu}(u_{t}-u_{t_{0}})|_{\partial\Omega}=0$. It follows that

[TABLE]

where $G(q(t_{0})):F\mapsto T_{q(t_{0})}(F,0)$ is bounded $L^{2}(\Omega)\to H^{1}(\Omega)$ by Lemma 2.1. Choosing $t$ close to $t_{0}$, we can solve the last equation by Neumann series so that

[TABLE]

Thus $t\mapsto u_{t}$ is real-analytic in $[0,1]\setminus\{t_{1},\ldots,t_{K}\}$. Moreover, for any $t_{0}\in[0,1]\setminus\{t_{1},\ldots,t_{K}\}$ there is $\varepsilon(t_{0})>0$ so that

[TABLE]

This proves the uniform bound for $\lVert u_{t}\rVert_{H^{1}(\Omega)}$ over any compact subset $F$ of $[0,1]\setminus\{t_{1},\ldots,t_{K}\}$. Finally, differentiating the power series and evaluating at $t_{0}$ yields

[TABLE]

so that $\partial_{t}u_{t}|_{t=t_{0}}$ is the unique solution of $(\Delta+k^{2}q(t_{0}))v=-k^{2}(q_{2}-q_{1})u_{t_{0}}$ in $\Omega$ with $\partial_{\nu}v|_{\partial\Omega}=0$.

Proof of part (b).

Let $t_{0}$ be one of $t_{1},\ldots,t_{K}$. Let $I=\{j\geq 1\,;\,\lambda_{j}(t_{0})=0\}$ (so that $I$ is finite), and write

[TABLE]

where $P_{t}$ and $Q_{t}$ are the orthogonal projections on $L^{2}(\Omega)$ given by

[TABLE]

We need to prove that $\lVert w_{t}\rVert_{H^{1}(\Omega)}\leq C\lVert g\rVert_{L^{2}(\partial\Omega)}$ for $t$ close to $t_{0}$.

Fix some $\bar{t}\in[0,1]\setminus\{t_{1},\ldots,t_{K}\}$, and write $u_{t}=r+h_{t}$ where $r\in H^{1}(\Omega)$ is the unique solution of

[TABLE]

Then $h_{t}$ solves

[TABLE]

where

[TABLE]

Now $w_{t}=Q_{t}u_{t}=Q_{t}r+Q_{t}h_{t}$, so that

[TABLE]

Testing the equation for $h_{t}$ against $\varphi_{j}(t)$ and integrating by parts gives that

[TABLE]

and consequently

[TABLE]

Now, the main point is that $\lvert\lambda_{j}(t_{0})\rvert\geq c>0$ for $j\notin I$. Moreover, the formula (2.3) implies that

[TABLE]

These facts imply that there is $\varepsilon>0$ so that

[TABLE]

It follows that

[TABLE]

Thus $\lVert w_{t}\rVert_{L^{2}(\Omega)}\leq C\lVert g\rVert_{L^{2}(\partial\Omega)}$ uniformly over $t\in[t_{0}-\varepsilon,t_{0}+\varepsilon]$, since $\lVert r\rVert_{H^{1}(\Omega)}\leq C\lVert g\rVert_{L^{2}(\partial\Omega)}$ and $F_{t}=-k^{2}(t-\bar{t})(q_{2}-q_{1})r$.

Finally we estimate the $H^{1}$ norm. By Lemma 2.1, we have that

[TABLE]

Now since $w_{t}$ solves the equation

[TABLE]

where by (2.6)

[TABLE]

we obtain that

[TABLE]

Consequently

[TABLE]

Using Cauchy’s inequality with $\varepsilon$ in the boundary integral and the trace result $\lVert w_{t}\rVert_{L^{2}(\partial\Omega)}\leq C\lVert w_{t}\rVert_{H^{1}(\Omega)}$, we obtain that

[TABLE]

We have seen above that $\lVert w_{t}\rVert_{L^{2}(\Omega)}\leq C\lVert g\rVert_{L^{2}(\partial\Omega)}$ uniformly over $\lvert t-t_{0}\rvert\leq\varepsilon$, and the same is true for $\lVert G_{t}\rVert_{L^{2}(\Omega)}$. Thus $\lVert w_{t}\rVert_{H^{1}(\Omega)}\leq C\lVert g\rVert_{L^{2}(\partial\Omega)}$ uniformly over $\lvert t-t_{0}\rvert\leq\varepsilon$.

Proof of part (c).

This is completely analogous to the proof of part (b), upon using the fact that $\lvert\lambda_{j}(t)\rvert\geq c>0$ uniformly over $j\notin J$ and $t\in[0,1]$.

Proof of part (d).

Let now $g\in D^{\perp}$, let $t_{0}=t_{l}$ where $1\leq l\leq K$, and let $t\neq t_{0}$ be close to $t_{0}$. As in part (b), we write $u_{t}=v_{t}+w_{t}$ where $v_{t}=P_{t}u_{t}$ and $w_{t}=Q_{t}u_{t}$.

We first prove that under the assumption $g\in D^{\perp}$, the map $t\mapsto v_{t}$ is a real-analytic from $[0,1]$ to $H^{1}(\Omega)$. By (2.6) we have that

[TABLE]

Now $\lambda_{j}(t_{0})=0$, so $v_{t}$ could potentially blow up as $t\to t_{0}$. However, this is prevented by the fact that $g\in D^{\perp}$, which ensures that $v_{t}$ may be written as

[TABLE]

Since $\lambda_{j}$ and $\varphi_{j}$ are real-analytic, one has

[TABLE]

where $\mu_{j}$ and $\psi_{j}$ are real-analytic near $t_{0}$ with $\psi_{j}$ taking values in $H^{1}(\Omega)$, and $\mu_{j}(t_{0})=\lambda_{j}^{\prime}(t_{0})>0$. Thus

[TABLE]

where $\mu_{j}(t)\geq c>0$ near $t_{0}$. The map $[0,1]\to L^{2}(\partial\Omega)$, $t\mapsto\psi_{j}(t)|_{\partial\Omega}$ is real-analytic since the trace operator is bounded from $H^{1}(\Omega)$ to $L^{2}(\partial\Omega)$. Thus $t\mapsto v_{t}$ is real-analytic near $t_{0}$, and one has $\lVert v_{t}\rVert_{H^{1}(\Omega)}\leq C\lVert g\rVert_{L^{2}(\partial\Omega)}$ near $t_{0}$. Combined with part (b), this implies that

[TABLE]

Next we define $u_{t_{0}}$ so that the map $s\mapsto u_{s}$ is Lipschitz continuous at $t_{0}$. Recalling the operator $D(q(t_{0}))^{\perp}\to H^{1}(\Omega),\ g\mapsto u_{g}$ where $u_{g}=T_{q(t_{0})}(0,g)$ from Lemma 2.1, we define

[TABLE]

Then $u_{t_{0}}$ solves $(\Delta+k^{2}q(t_{0}))u_{t_{0}}=0$ in $\Omega$ with $\partial_{\nu}u_{t_{0}}|_{\partial\Omega}=g$, and one has $P_{t_{0}}u_{t_{0}}=v_{t_{0}}$.

It remains to prove that $s\mapsto u_{s}$ is Lipschitz continuous at $t_{0}$. Note that

[TABLE]

and $\partial_{\nu}(u_{t}-u_{t_{0}})|_{\partial\Omega}=0$. It follows from Lemma 2.1 that the function $-k^{2}(t-t_{0})(q_{2}-q_{1})u_{t}$ is in $N(q(t_{0}))^{\perp}$, and that

[TABLE]

for some $\varphi\in N(q(t_{0}))$. Define the operator

[TABLE]

Then $G(q(t_{0}))$ is bounded (since $Q_{t_{0}}=\mathrm{Id}-P_{t_{0}}$ is bounded on $H^{1}(\Omega)$), and

[TABLE]

Using the uniform bound (2.9), we get that

[TABLE]

To analyze the last term, we note that by the assumption that $g\in D^{\perp}$ and by (2.7) and (2.8)

[TABLE]

where $\mu_{j}$ and $\psi_{j}$ are real-analytic near $t_{0}$ with $\psi_{j}$ taking values in $H^{1}(\Omega)$. Thus in particular

[TABLE]

Using (2.9) again, this concludes the proof that $s\mapsto u_{s}$ is Lipschitz continuous near $t_{0}$. Since this is true near $t_{1},\ldots,t_{K}$, and since $s\mapsto u_{s}$ is real-analytic away from $\{t_{1},\ldots,t_{K}\}$, we have proved (d). ∎

We are now ready to prove Theorem 1.1.

Proof of Theorem 1.1.

We will do the proof in three steps.

Step 1: Definition of a finite-dimensional space $D$.

We can assume that $q_{2}\geq q_{1}$ and $q_{2}\not\equiv q_{1}$, since the case $q_{2}\equiv q_{1}$ is immediate. Write $d_{1}=d(q_{1})$, $d_{2}=d(q_{2})$ and $N=d_{2}-d_{1}$, and let $q(t)$ and $\lambda_{j}(t)$ be as in Lemma 2.2. Now the positive Neumann eigenvalues of $q(0)$ are $\lambda_{1}(0),\ldots,\lambda_{d_{1}}(0)$. Since the functions $\lambda_{j}(t)$ are strictly increasing, the positive Neumann eigenvalues related to $q(1)$ are \lambda_{1}(1),\ldots,\lambda_{d_{1}}(1),$$\lambda_{j_{1}}(1),\ldots, $\lambda_{j_{N}}(1)$ for some indices $j_{1},\ldots,j_{N}$ (here it is possible that $N=0$). We reorder the indices for $j\geq d_{1}+1$ so that the positive Neumann eigenvalues related to $q(1)$ are in descending order \lambda_{1}(1),\ldots,\lambda_{d_{1}}(1),$$\lambda_{d_{1}+1}(1),\ldots,\lambda_{d_{2}}(1). It follows that $\lambda_{j}(t)$ for $j\leq d_{1}$ are positive on $[0,1]$, $\lambda_{j}(t)$ for $d_{1}+1\leq j\leq d_{2}$ have a unique zero and cross from negative to positive on $[0,1]$, and $\lambda_{j}(t)$ for $j\geq d_{2}+1$ are always negative on $[0,1]$.

Let $0<t_{1}<\ldots<t_{K}<1$ be the times when [math] is a Neumann eigenvalue of $H(t)$, and let

[TABLE]

as in Lemma 2.3. By Lemmas 2.1 and 2.2, $\dim(D(q(t_{l})))$ is the multiplicity of [math] as a Neumann eigenvalue of $H(t_{l})$, which is precisely the number of functions $\lambda_{j}(t)$ that vanish at $t_{l}$. Since exactly $N=d_{2}-d_{1}$ functions $\lambda_{j}$ have a zero in $[0,1]$, it follows that $\dim(D)\leq N$. (The dimension of $D$ would be equal to $N$ if all the spaces $D(q(t_{1})),\ldots,D(q(t_{K}))$ would be linearly independent, but this may not be true in general.)

Step 2: We will next show that

[TABLE]

Fix $g\in D^{\perp}$, and let $[0,1]\to H^{1}(\Omega),\ t\mapsto u_{t}$ be the map in Lemma 2.3. Since $q(0)=q_{1}$ and $q(1)=q_{2}$, it follows that

[TABLE]

We write, for $t\in[0,1]$,

[TABLE]

Then $F$ is Lipschitz continuous in $[0,1]$ since $t\mapsto u_{t}$ is:

[TABLE]

We compute the derivative of $F$ using the fact from Lemma 2.3 that $u_{t}$ is real-analytic in $[0,1]\setminus\{t_{1},\ldots,t_{K}\}$, and $\partial_{t}u_{t}$ is the unique solution of

[TABLE]

Thus

[TABLE]

Since $F(t)$ is Lipschitz continuous and hence absolutely continuous, we may use the fundamental theorem of calculus to compute

[TABLE]

Since $q_{2}-q_{1}\geq 0$ a.e., we get that $((\Lambda(q_{2})-\Lambda(q_{1}))g,g)\geq 0$ for $g\in D^{\perp}$ as required.

Step 3: One has $d(q_{1},q_{2})\leq d(q_{2})-d(q_{1})$.

By the previous step one has $((\Lambda(q_{2})-\Lambda(q_{1}))g,g)\geq 0$ for all $g\in D^{\perp}$. By [HPS, Corollary 3.3] this implies that $\Lambda(q_{1})\leq_{\dim(D)}\Lambda(q_{2})$, i.e. that $\Lambda(q_{2})-\Lambda(q_{1})$ has $\leq\dim(D)\leq d(q_{2})-d(q_{1})$ negative eigenvalues. ∎

3. Lower bounds for the number of negative eigenvalues

In this section we will prove Theorems 1.3 and 1.4. We will work under the assumption that $q_{2}-q_{1}$ is a positive constant, which ensures that the Neumann eigenvalues and eigenfunctions of $\Delta+k^{2}q(t)$ behave in a very simple way as $t$ varies (in particular, analytic perturbation theory is not required).

Proof of Theorem 1.3.

The proof proceeds in several steps.

Step 1: Notation for eigenvalues and eigenfunctions.

Let $d_{1}=d(q_{1})$, and let

[TABLE]

be the Neumann eigenvalues of $\Delta+k^{2}q_{1}$ in $\Omega$. (Here it is possible that $d_{1}=0$, and all eigenvalues are negative.) Let $(\varphi_{j})_{j=1}^{\infty}$ be a corresponding orthonormal basis of $L^{2}(\Omega)$ consisting of Neumann eigenfunctions, i.e.

[TABLE]

Define the potentials $q(t)=(1-t)q_{1}+tq_{2}$. Since by assumption $c=q_{2}-q_{1}$ is a positive constant, we have

[TABLE]

Now, one has

[TABLE]

Thus the Neumann eigenvalues of $\Delta+k^{2}q(t)$ are given by

[TABLE]

and the corresponding $L^{2}$-orthonormal Neumann eigenfunctions $\varphi_{j}(t)=\varphi_{j}$ are independent of $t$. We note that the functions $\lambda_{j}(t)$, $t\in[0,1]$, are strictly increasing. They are positive if $j\leq d_{1}$, cross from negative to positive and satisfy $\lambda_{j}(t_{j})=0$ at times

[TABLE]

if $d_{1}+1\leq j\leq d_{2}$, and stay negative if $j\geq d_{2}+1$. Here $d_{2}=d(q_{2})$.

Step 2: Formula for $((\Lambda(q_{2})-\Lambda(q_{1}))g,g)$.

Fix $g\in L^{2}(\partial\Omega)$, and let $t\in[0,t_{d_{1}+1})\cup(t_{d_{2}},1]$. Let $u_{t}$ be the solution of

[TABLE]

Note that the Neumann problem is well-posed for $t$ in this range, and as in Lemma 2.1 one has the $L^{2}(\Omega)$-convergent representation

[TABLE]

with

[TABLE]

As in Lemma 2.3 (but with slightly different notation), we write $u_{t}=v_{t}+w_{t}$ where

[TABLE]

Thus we have

[TABLE]

Note that the coefficient $\frac{k^{2}c}{\lambda_{j}(1)\lambda_{j}(0)}$ is negative exactly when $d_{1}+1\leq j\leq d_{2}$, so that the sum in (3.4) is $\leq 0$ while the last integral may be positive.

Step 3: Formula for $((\Lambda(q_{2}-b)-\Lambda(q_{1}+a))g,g)$.

We will now replace $q_{1}$ by $q_{1}+a$ and $q_{2}$ by $q_{2}-b$ and show that for suitable choices of $a$ and $b$, the negative contributions in (3.4) dominate the positive ones. This will imply that the corresponding quadratic form is negative on some finite-dimensional space, yielding a lower bound for the number of negative eigenvalues. We do the rescalings

[TABLE]

where $\alpha,\beta\in[0,1]$ and

[TABLE]

The equation (3.4) now becomes

[TABLE]

In the notation of Theorem 1.3, one has $\mu_{1}=\lambda_{d_{1}+1}$ and $\mu_{2}=\lambda_{d_{2}}(1)=\lambda_{d_{2}}+k^{2}c$. Then $t_{d_{1}+1}=c^{-1}k^{-2}|\mu_{1}|$ (since $\lambda_{d_{1}+1}(t_{d_{1}+1})=0$) and $t_{d_{2}}=1-c^{-1}k^{-2}\mu_{2}$ (since $\lambda_{d_{2}}(t_{d_{2}})=0$). It follows that

[TABLE]

The next step is to show that the last integral in (3.5) is uniformly bounded over $\alpha\in[0,t_{d_{1}+1})$ and $\beta\in(t_{d_{2}},1]$. This will follow since $w_{t}$ is related only to those eigenfrequencies that are uniformly bounded away from zero.

Step 4: $\lVert w_{t}\rVert_{H^{1}(\Omega)}\leq C\lVert g\rVert_{L^{2}(\partial\Omega)}$ uniformly over $t\in[0,t_{d_{1}+1})\cup(t_{d_{2}},1]$.

This follows directly from Lemma 2.3(c).

Step 5: Proof of part (a).

We will show that there is a subspace $V$ of $L^{2}(\partial\Omega)$ with $\dim(V)=N_{1}$ such that (3.5) is negative when $g\in V\setminus\{0\}$, $\alpha<t_{d_{1}+1}$ is close to $t_{d_{1}+1}$, and $\beta\in(t_{d_{2}},1]$. Combined with (3.6) and [HPS, Lemma 3.2(b)] applied to $A=-(\Lambda(q_{2}-b)-\Lambda(q_{1}+a))$ with $r=0$, this will prove part (a).

By the trace theorem and Step 4, we have

[TABLE]

uniformly over $\alpha\in[0,t_{d_{1}+1})$ and $\beta\in(t_{d_{2}},1]$. Thus

[TABLE]

If $j\in[d_{1}+1,d_{2}]$, then $\alpha<t_{j}$ and $\beta>t_{j}$, and $\lambda_{j}(\beta)=\lambda_{j}(t_{j})+k^{2}c(\beta-t_{j})=k^{2}c(\beta-t_{j})>0$. Thus one has

[TABLE]

Since $\lambda_{j}(\alpha)=\lambda_{j}(t_{j})-k^{2}c(t_{j}-\alpha)=-k^{2}c(t_{j}-\alpha)<0$, we obtain that

[TABLE]

uniformly over $\alpha\in[0,t_{d_{1}+1})$ and $\beta\in(t_{d_{2}},1]$.

Recall now the assumption that $\lambda_{d_{1}+1}$ has multiplicity $N_{1}$, and define

[TABLE]

Here $\varphi_{d_{1}+1},\ldots,\varphi_{d_{1}+N_{1}}$ are Neumann eigenfunctions corresponding to $\lambda_{d_{1}+1}$. We claim that $\dim(V)=N_{1}$. For if $a_{1}\varphi_{d_{1}+1}|_{\partial\Omega}+\ldots+a_{N_{1}}\varphi_{d_{1}+N_{1}}|_{\partial\Omega}=0$, then the function $\varphi=a_{1}\varphi_{d_{1}+1}+\ldots+a_{N_{1}}\varphi_{d_{1}+N_{1}}$ satisfies

[TABLE]

By the unique continuation principle this implies that $\varphi\equiv 0$, and since $\{\varphi_{j}\}$ are orthonormal in $L^{2}(\Omega)$ we obtain $a_{1}=\ldots=a_{N_{1}}=0$. This proves that $\dim(V)=N_{1}$.

Let now $g\in V\setminus\{0\}$. Since $\lambda_{d_{1}+1}$ has multiplicity $N_{1}$ and since $t_{j}$ is the unique zero of $t\mapsto\lambda_{j}(t)$, by (3.1) one has $t_{d_{1}+1}=\ldots=t_{d_{1}+N_{1}}$, and thus

[TABLE]

uniformly over $\alpha\in[0,t_{d_{1}+1})$ and $\beta\in(t_{d_{2}},1]$. The middle term on the right is $\leq 0$, and writing

[TABLE]

where $\delta>0$ (the infimum is over the unit sphere in a finite dimensional normed space and the quantity inside the infimum is positive for $g\in V\setminus\{0\}$), we obtain that

[TABLE]

where $C$ is uniform over $\alpha\in[0,t_{d_{1}+1})$ and $\beta\in(t_{d_{2}},1]$. Thus choosing $\alpha<t_{d_{1}+1}$ sufficiently close to $t_{d_{1}+1}$, one has $((\Lambda(q_{2}-b)-\Lambda(q_{1}+a))g,g)<0$ for $g\in V\setminus\{0\}$. This concludes the proof of part (a).

Step 6: Proof of part (b).

This is completely analogous to Step 5: one defines the subspace

[TABLE]

and shows that $((\Lambda(q_{2}-b)-\Lambda(q_{1}+a))g,g)<0$ for $g\in V\setminus\{0\}$ when $\beta\in(t_{d_{2}},1]$ is sufficiently close to $t_{d_{2}}$. ∎

Proof of Theorem 1.4.

(a) If $\lambda_{j}$ are the Neumann eigenvalues of $\Delta+k^{2}q$ in $\Omega$, then $\lambda_{j}\pm k^{2}\varepsilon$ are the eigenvalues of $\Delta+k^{2}(q\pm\varepsilon)$ in $\Omega$. Thus if $\varepsilon_{0}>0$ is small enough and $\varepsilon\leq\varepsilon_{0}$, one has $d(q+\varepsilon)-d(q-\varepsilon)=N$, and $\Lambda(q+\varepsilon)-\Lambda(q-\varepsilon)$ has at most $N$ negative eigenvalues by Theorem 1.1. Moreover, by Theorem 1.3 with $q_{1}=q-\varepsilon_{0}$, $q_{2}=q+\varepsilon_{0}$, $\mu_{1}=-k^{2}\varepsilon_{0}$ and $\mu_{2}=k^{2}\varepsilon_{0}$, we obtain that $\Lambda(q+\varepsilon)-\Lambda(q-\varepsilon)$ has at least $N$ negative eigenvalues for $\varepsilon$ small.

(b) Recall that we now assume that $\Omega:=(0,1)^{2}$. It is enough to show that for any even $N\geq 2$, there is an eigenvalue $\mu$ of $\Delta$ in $\Omega$ with multiplicity $N$. If this holds, then choosing $c=-k^{-2}\mu$ gives that [math] is an eigenvalue of $\Delta+k^{2}c$ of multiplicity $N$, and the result follows from part (a).

An orthonormal basis of $L^{2}(\Omega)$ consisting of Neumann eigenfunctions of $\Delta$ in $\Omega$ is given by $(\varphi_{l_{1},l_{2}})_{l_{1},l_{2}=0}^{\infty}$, where

[TABLE]

for some normalizing constants $c_{l_{1},l_{2}}$. The eigenvalue corresponding to $\varphi_{l_{1},l_{2}}$ is $-(l_{1}^{2}+l_{2}^{2})$. See e.g. [GN13].

We set $\lambda=5^{r}$ where $r\geq 1$ is an odd integer, and write $N=r+1$. Since $5$ is a prime of the form $4m+1$, there are $4N$ pairs $(s_{1},s_{2})\in\mathbb{Z}^{2}$ such that $\lambda=s_{1}^{2}+s_{2}^{2}$ [HW08, Theorem 278]. Now $r$ is odd, so $\lambda$ is not a square and both $s_{1}$ and $s_{2}$ must be nonzero, and thus there are exactly $N$ pairs $(l_{1},l_{2})\in(\mathbb{N}\cup\{0\})^{2}$ so that $\lambda=l_{1}^{2}+l_{2}^{2}$. This shows that the multiplicity of $-\lambda$ as a Neumann eigenvalue of $\Delta$ is exactly $N$. ∎

4. The Helmholtz equation with constant parameter

In this section, we will treat the Neumann problem for the Helmholtz equation

[TABLE]

with a constant coefficient $q(x)=\mathrm{const}$. In this case, the Helmholtz solution operator can be expressed using the Neumann eigenfunctions of the Laplace equation, which allows us to give a simple independent proof of Theorem 1.1, and show that the dimension bound in Theorem 1.1 is sharp for the Helmholtz solution operators.

For the special case of a constant coefficient in a two-dimensional unit square we also derive an infinite matrix representation of the Neumann-Dirichlet-operator and study numerically the question whether the bound in Theorem 1.1 is sharp for the Neumann-Dirichlet-operators.

4.1. The dimension bound for the constant parameter case

Definition 4.1**.**

For $0\not\equiv q\in L^{\infty}(\Omega)$, and a non-resonant wavenumber $k>0$ we define the Helmholtz solution operator

[TABLE]

where $v\in H^{1}(\Omega)$ solves

[TABLE]

Note that, the Neumann-Dirichlet-operator

[TABLE]

where $u\in H^{1}(\Omega)$ solves (4.1), obviously fulfills

[TABLE]

where $\gamma$ denotes the compact trace operator

[TABLE]

Theorem 4.2**.**

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded Lipschitz domain. Let $a,b\in\mathbb{R}$ with $a<b$, and let $k>0$ be non-resonant for $q(x)=a$ and $q(x)=b$. Then

1. (a)

$S(b)-S(a)$* has exactly $d(b)-d(a)$ negative eigenvalues.* 2. (b)

$\Lambda(b)-\Lambda(a)$* has at most $d(b)-d(a)$ negative eigenvalues.*

Note that (b) follows from Theorem 1.1, but our proof of Theorem 4.2 is independent of this result and rather elementary, so we believe that this is of independent interest.

As in the proof of Lemma 2.1 (see also [HPS, Section 2.1]), we let $\mathrm{Id}:\ H^{1}(\Omega)\to H^{1}(\Omega)$ denote the identity operator, $\iota:\ H^{1}(\Omega)\to L^{2}(\Omega)$ denote the compact embedding, and $M_{q}:\ L^{2}(\Omega)\to L^{2}(\Omega)$ denote the multiplication operator by $q$. Then $K:=\iota^{*}\iota$ and $K_{q}:=\iota^{*}M_{q}\iota$ are compact self-adjoint linear operators from $H^{1}(\Omega)$ to $H^{1}(\Omega)$, and

[TABLE]

where the inverse exists if and only if $k>0$ is non-resonant for the potential $q$, cf., e.g., [HPS, Lemma 2.2].

For constant coefficients $q(x)=a\in\mathbb{R}$ this simplifies to

[TABLE]

Since $K:\ H^{1}(\Omega)\to H^{1}(\Omega)$ is a compact self-adjoint, positive definite operator, there exists an orthonormal basis $(v_{l})_{l\in\mathbb{N}}$ of $H^{1}(\Omega)$ of eigenfunctions corresponding to eigenvalues $\lambda_{l}>0$,

[TABLE]

Note that in this section, $\lambda_{l}$ are the eigenvalues of the compact operator $K$ which converge to zero (unlike in the earlier sections, where $\lambda_{j}$ were Neumann eigenvalues converging to $-\infty$).

Lemma 4.3**.**

1. (a)

A function $v\in H^{1}(\Omega)$ is an eigenfunction of $K$ with eigenvalue $\lambda$ if and only if $v$ is a Neumann eigenfunction of the Laplace equation with Neumann eigenvalue $\frac{1}{\lambda}-1$, i.e.,

[TABLE] 2. (b)

$k>0$* is non-resonant for the potential $q(x)=a\in\mathbb{R}$ if and only if*

[TABLE]

i.e., if and only if $ak^{2}$ is not a Neumann eigenvalue. Moreover,

[TABLE] 3. (c)

If $k>0$ is non-resonant for the potential $q(x)=a\in\mathbb{R}$ then

[TABLE] 4. (d)

Let $a,b\in\mathbb{R}$ with $a<b$, and $k>0$ be non-resonant for $q(x)=a$ and $q(x)=b$. Then

[TABLE]

where $0\neq c_{l}\in\mathbb{R}$, and the number of negative $c_{l}$ is exactly $d(b)-d(a)$.

Proof.

1. $Kv=\lambda v$ is equivalent to

[TABLE]

which is the variational formulation equivalent to

[TABLE]

This proves (a).

The first part of (b) and (c) are obvious. The second part of (b) has been proven in [HPS, Lemma 2.1].

To prove (d) note that for all $F\in H^{1}(\Omega)$

[TABLE]

where the sum is convergent in $H^{1}(\Omega)$. Hence,

[TABLE]

For the coefficients

[TABLE]

we obviously have that $c_{l}\neq 0$ and that $c_{l}<0$ if and only if

[TABLE]

By the second part of (b), the number of negative $c_{l}$ is exactly $d(b)-d(a)$.

∎

Proof of Theorem 4.2..

Using Lemma 4.3 we have that

[TABLE]

is a subspace of dimension $\mathrm{dim}(V)=d(b)-d(a)$,

[TABLE]

Using, e.g., [HPS, Lemma 3.2] this shows that $S(b)-S(a)$ has exactly $d(b)-d(a)$ negative eigenvalues and thus proves Theorem 4.2(a).

Using that $\Lambda(b)-\Lambda(a)=\gamma(S(b)-S(a))\gamma^{*}$, we also have that

[TABLE]

for all $g$ with $\gamma^{*}g\perp_{H^{1}}V$, which is equivalent to

[TABLE]

and thus

[TABLE]

Using $\mathrm{dim}(\gamma(V))\leq\mathrm{dim}(V)=d(b)-d(a)$ and [HPS, Cor. 3.3]), this shows that $\Lambda(b)-\Lambda(a)$ has at most $d(b)-d(a)$ negative eigenvalues and thus proves Theorem 4.2(b). ∎

4.2. Helmholtz equation on the two-dimensional unit square

We now consider the special case of the Helmholtz equation with constant parameter $q(x)=a\in\mathbb{R}$ on the two-dimensional unit square

[TABLE]

and derive an infinite matrix representation for the Neumann-Dirichlet-operator $\Lambda(a)$.

For the unit square the Neumann eigenfunctions are well known:

Lemma 4.4**.**

For $l,m\in\mathbb{N}_{0}$ we define

[TABLE]

with $d_{0}:=1$ and $d_{j}=\sqrt{2}$ for $j\in\mathbb{N}$. The functions $v_{l,m}$ are Neumann eigenfunctions of the Laplacian

[TABLE]

and eigenfunctions of $K:\ H^{1}(\Omega)\to H^{1}(\Omega)$

[TABLE]

$(v_{l,m})_{l,m\in\mathbb{N}}$* is an orthonormal basis of $L^{2}(\Omega)$, and $(\sqrt{\lambda_{l,m}}v_{l,m})_{l,m\in\mathbb{N}}$ is an orthonormal basis of $H^{1}(\Omega)$.*

Proof.

It is easily checked that the functions $(v_{l,m})_{l,m\in\mathbb{N}}$ are Neumann eigenfunctions and that they form an orthonormal basis of $L^{2}(\Omega)$. Lemma 4.3, that the $(v_{l,m})_{l,m\in\mathbb{N}}$ are also eigenfunctions of $K$, and this yields that

[TABLE]

which shows that $(\sqrt{\lambda_{l,m}}v_{l,m})_{l,m\in\mathbb{N}}$ is an orthonormal basis of $H^{1}(\Omega)$. ∎

We can now expand the Neumann-Dirichlet operator $\Lambda(a)$ in an orthonormal basis of cosine functions on the four sides of $\partial\Omega$.

Lemma 4.5**.**

Define $g_{s}:\ \partial\Omega\to\mathbb{R}$ ($s\in\mathbb{N}_{0}$) by setting for all $j\in\mathbb{N}_{0}$

[TABLE]

Then $(g_{s})_{s\in\mathbb{N}_{0}}\subseteq L^{2}(\partial\Omega)$ is an orthonormal basis of $L^{2}(\partial\Omega)$.

The infinite matrix representation of $\Lambda(a)$ with respect to this basis is given by $4\times 4$-blocks of the form

[TABLE]

where (for $i,j\in\mathbb{N}_{0}$)

[TABLE]

Proof.

Clearly, $(g_{s})_{s\in\mathbb{N}_{0}}\subseteq L^{2}(\partial\Omega)$ is an orthonormal basis of $L^{2}(\partial\Omega)$, and

[TABLE]

Using that $(\sqrt{\lambda_{lm}}v_{lm})_{l,m\in\mathbb{N}}$ is an orthonormal basis of $H^{1}(\Omega)$ that diagonalizes the solution operator (cf. lemma 4.3(c)) we have that

[TABLE]

The assertion then follows from a simple calculation using the sum formulas

[TABLE]

(see e.g. [Rem91, formulas (1) on p. 327 and (4) on p. 329]). ∎

4.3. Numerical evaluation of the dimension bound

We still consider the special case of the Helmholtz equation on the unit square $\Omega=(0,1)^{2}$ with constant parameter $q(x)=a\in\mathbb{R}$, resp., $q(x)=b\in\mathbb{R}$, and fix $k:=1$ without loss of generality. It follows from lemma 4.3 and lemma 4.4 that resonances occur when $a$ or $b$ equals $\pi^{2}(l^{2}+m^{2})$ with $l,m\in\mathbb{N}_{0}$, and that

[TABLE]

We know from Theorem 4.2 (and the more general Theorem 1.1) that $\Lambda(b)-\Lambda(a)$ will have at most $d(b)-d(a)$ negative eigenvalues. Moreover, we know from Theorem 1.4 that this bound is achieved, when $a$ and $b$ are sufficiently close together and only slightly smaller, resp., larger than a Neumann eigenvalue $\pi^{2}(l^{2}+m^{2})$, and $d(b)-d(a)$ will then be the multiplicity of this Neumann eigenvalue which can attain any even positive integer.

We will now numerically evaluate how the number of negative eigenvalues of $\Lambda(b)-\Lambda(a)$ behaves. For this end we use the numerical programming language Matlab to calculate a $1000\times 1000$ matrix approximating $\Lambda(b)-\Lambda(a)$ using the matrix representation formula in lemma 4.5 for $i,j=0,\ldots,249$. We estimated the error in this finite dimensional approximation to be below $\delta=10^{-5}$ in the spectral norm by comparing $\Lambda(b)-\Lambda(a)$ to its upper left $500\times 500$ entries (filled up by zeros to a $1000\times 1000$ matrix). Accordingly, we considered eigenvalues below $-\delta$ to be negative and counted their number (with multiplicity).

Figure 1 shows this numerically computed number of negative eigenvalues of $\Lambda(b)-\Lambda(a)$ and the theoretical bound $d(b)-d(a)$ as a function of $b\in\mathbb{R}$ for $a=-10$ (top left), $a=10$ (top right), $a=100$ (bottom left), and $a=200$ (bottom right). Whenever $b$ crosses an eigenvalue $\pi^{2}(l^{2}+m^{2})$ with $l,m\in\mathbb{N}_{0}$ the theoretical bound $d(b)-d(a)$ increases by the multiplicity of this eigenvalue. The plots indicate that the number of negative eigenvalue $\Lambda(b)-\Lambda(a)$ also increases by the multiplicity of this eigenvalue but that there is an additional effect decreasing the number of negative eigenvalues when $b-a$ increases.

To further investigate this additional effect, figure 2 shows the values of the eigenvalues of $\Lambda(b)-\Lambda(a)$ as a function of $b$ for fixed $a=-10$. More precisely, for each integer $b=-10,-9,\ldots,200$ (excluding the resonance $b=0$), the black dots are plotted at the position $(b,\lambda_{j}(a,b))$ where $\lambda_{j}(a,b)$, $j=1,\ldots,1000$, are the numerically calculated eigenvalues of $\Lambda(b)-\Lambda(a)$. The red dashed lines show the positions of the Neumann eigenvalues. Whenever $b$ crosses an eigenvalue, new negative eigenvalues of $\Lambda(b)-\Lambda(a)$ appear. But at the same time the values of the eigenvalues increase with $b-a$, and it seems that negative eigenvalues can become positive again which would explain the drops in the number of negative eigenvalues observed in figure 1.

Let us stress however that this numerical experiment is only an indication of what might happen to stipulate further research. We do not have a rigorous proof that the observed drop in the number of negative eigenvalues really exists. $\Lambda(b)-\Lambda(a)$ is a compact operator with an infinite number of eigenvalues accumulating at zero, and we cannot rigorously rule out the possibility that there exist more negative eigenvalues (up to the theoretically proven bound $d(b)-d(a)$) that we did not find due to their absolute values being below the numerical precision level.