Domains without dense Steklov nodal sets

TL;DR

This paper constructs a dense family of two-dimensional domains with analytic boundaries where Steklov eigenfunctions' nodal sets are not dense at the scale of their eigenvalues, challenging previous expectations about their geometric behavior.

Contribution

It demonstrates the existence of domains with analytic boundaries where Steklov eigenfunctions have non-dense nodal sets at the eigenvalue scale, addressing an open problem in spectral geometry.

Findings

Nodal sets are not dense at scale σ_j^{-1} in certain domains.

Existence of points where eigenfunctions do not vanish within fixed radius.

Contradicts prior assumptions about nodal set density at eigenvalue scale.

Abstract

This article concerns the asymptotic geometric character of the nodal set of the eigenfunctions of the Steklov eigenvalue problem $$ -\Delta \phi_{\sigma_j}=0,\quad\text{ on }\Omega,\qquad\qquad \partial_\nu \phi_{\sigma_j}=\sigma_j \phi_{\sigma_j}\quad \text{ on }\partial\Omega $$ in two-dimensional domains $\Omega$. In particular, this paper presents a dense family $\mathcal{A}$ of simply-connected two-dimensional domains with analytic boundaries such that, for each $\Omega\in \mathcal{A}$, the nodal set of the eigenfunction $\phi_{\sigma_j}$ "is $not$ dense at scale $\sigma_j^{-1}$". This result addresses a question put forth under "Open Problem 10" in Girouard and Polterovich, J. Spectr. Theory, 321-359 (2017). In fact, the results in the present paper establish that, for domains $\Omega\in \mathcal{A}$, the nodal sets of the eigenfunctions $\phi_{\sigma_j}$ associated with the eigenvalue $\sigma_j$ have starkly different character than anticipated: they are not dense at any shrinking scale. More precisely, for each $\Omega\in \mathcal{A}$ there is a value $r_1>0$ such that for each $j$ there is $x_j\in \Omega$ such that $\phi_{\sigma_j}$ does not vanish on the ball of radius $r_1$ around $x_j$.

Tables

Table 1. Table 1. Verification of the Cauchy-theorem-based identity ( 5.8 ) for the domain Ω Ω \Omega bounded by the elliptical curve ( 6.1 ) with a = 2 𝑎 2 a=2 and b = 1 𝑏 1 b=1 .

$n$	$\left|B_n^0(x_1,x_2)\right|$	$\left|e^{-n0.8\lambda }{B}_n(x_1,x_2,0.8\lambda )\right|$	Absolute $B_n^0$ error	Relative $B_n^0$ error
1	5.62e-03	5.62e-03	3.82e-16	6.79e-14
10	2.29e-06	2.29e-06	4.39e-17	1.91e-11
50	6.40e-16	6.57e-16	3.85e-17	5.86e-02
100	3.05e-17	1.30e-28	3.05e-17	2.35e+11
150	1.33e-16	5.95e-41	1.33e-16	2.23e+24
200	2.65e-16	6.58e-53	2.65e-16	4.02e+36

Table 2. Table 2. Same as Figure ( 1 ) but for the kite-shaped domain Ω Ω \Omega bounded by the curve ( 6.2 ).

$n$	$\left|B_n^0(x_1,x_2)\right|$	$\left|e^{-n0.8\lambda }{B}_n(x_1,x_2,0.8\lambda )\right|$	Absolute $B_n^0$ error	Relative $B_n^0$ error
1	5.83e-03	5.83e-03	4.25e-16	7.29e-14
10	5.97e-06	5.97e-06	7.18e-18	1.20e-12
50	2.33e-14	2.34e-14	3.32e-17	1.42e-03
100	1.14e-16	3.05e-25	1.14e-16	3.75e+08
150	1.27e-16	6.78e-36	1.27e-16	1.88e+19
200	2.42e-16	3.05e-45	2.42e-16	7.93e+28

Equations

$$-\Delta {\varphi }_{{\sigma }_j}=0,\text{ on }\Omega ,{\partial }_{\nu }{\varphi }_{{\sigma }_j}={\sigma }_j{\varphi }_{{\sigma }_j}\text{ on }\partial \Omega$$

$$\{\begin{array}{ll}-{\Delta }_g{\varphi }_{\sigma }=0& \text{in }M\\ {\partial }_{\nu }{\varphi }_{\sigma }=\sigma {\varphi }_{\sigma }& \text{on }\partial M.\end{array}$$

$$\partial {\Omega }_1=\{x+\nu g(x)\mid x\in \partial {\Omega }_0\},\Vert g{\Vert }_{C^k(\partial {\Omega }_0)}<\epsilon$$

$$|{\varphi }_{\sigma }|>0\text{ on }B(x_{\sigma },r_1)\subset {\Omega }_1.$$

$$u_{{\sigma }_j}:={\varphi }_{{\sigma }_j}\circ f$$

$${\partial }_r{u}_{{\sigma }_j}(z)=|{\partial }_{z}f(z)|{\partial }_{\nu }{\varphi }_{{\sigma }_j}(f(z))=|{\partial }_{z}f(z)|{\sigma }_j{\varphi }_{{\sigma }_j}(f(z)),$$

$$\{\begin{array}{ll}-\Delta u_{{\sigma }_j}=0& \text{in }D\\ {\partial }_r{u}_{{\sigma }_j}={\sigma }_j|{\partial }_{z}f|u_{{\sigma }_j}& \text{on }\partial D.\end{array}$$

$$\hat{v}(k)=\frac{1}{2\pi }{\int }_0^{2\pi }v(\cos \theta ,\sin \theta )e^{-ik\theta }d\theta$$

$$|\hat{u}_{\sigma}(k)|\leq C_{0}^{|k-m|}\Big{(}\sum_{\ell=m-m_{0}}^{m+m_{0}}|\hat{u}_{\sigma}(\ell)|^{2}\Big{)}^{\frac{1}{2}},\qquad|k|\leq K\sigma.$$

$$e^{-C\sigma}\|\hat{u}\|_{\ell^{2}}\leq\Big{(}\sum_{k=m-m_{0}}^{m_{0}}|\hat{u}(k)|^{2}\Big{)}^{\frac{1}{2}}.$$

$${\tilde{u}}_{\sigma ,\delta }:=u_{\sigma }{|}_{B(0,\delta )}=\sum _{k=-\infty }^{\infty }{\hat{u}}_{{\sigma }_j}(k)r^{|k|}{e}^{ik\theta },{\tilde{u}}_{{\sigma }_j,\delta ,m}:=\sum _{|k|<m}{\hat{u}}_{{\sigma }_j}(k)r^{|k|}{e}^{ik\theta }.$$

$$\frac{\Vert {\tilde{u}}_{\sigma ,\delta }-{\tilde{u}}_{\sigma ,\delta ,m}{\Vert }_{C^N(B(0,\delta ))}}{\Vert {\tilde{u}}_{\sigma ,\delta }{\Vert }_{L^2(B(0,\delta ))}}\le C_N({\delta }^{m-N-m_0-1}+e^{-c\sigma })$$

$$u_{\sigma_{j}}=\sum_{|k|<m}\hat{u}_{\sigma_{j}}(k)r^{|k|}e^{ik\sigma}+O\Big{(}(r^{m-m_{0}-1}+e^{-c\sigma_{j}})\sqrt{\sum_{|k|<m}|\hat{u}_{\sigma_{j}}(k)|^{2}\frac{r^{2k+1}}{2k+1}}\Big{)}.$$

$$|{\hat{u}}_{{\sigma }_j}(0)|\ge c\sqrt{\sum _{0<|k|<m}|{\hat{u}}_{{\sigma }_j}(k){|}^2},$$

$$u_{{\sigma }_j}={\hat{u}}_{{\sigma }_j}(0)+O((r+e^{-c{\sigma }_j})|{\hat{u}}_{{\sigma }_j}(0)|)$$

$$f(z)={\int }_0^z{p}^2(w)dw,p(z)=\prod _{i=1}^M(z-{\alpha }_i{)}^{N_i}.$$

$${\partial }_{z}f=\prod _{i=1}^M(z-{\alpha }_i{)}^{2N_i},|{\partial }_{z}f|=\prod _{i=1}^M(|z-{\alpha }_i{|}^2{)}^N.$$

$$|{\partial }_{z}f|(e^{i\theta })=\prod _{i=1}^M(1-e^{i\theta }\overline{{\alpha }_i}-e^{-i\theta }{\alpha }_i+|{\alpha }_i{|}^2{)}^{N_i}$$

$$\Vert g-{\alpha }_0\prod _{i=1}^M(z-{\alpha }_i{)}^{2N_i}{\Vert }_{C^k(\overline{D})}<{\epsilon }_0.$$

$$h(z)={\int }_0^z\frac{g^{\prime }(w)}{g(w)}dw+\log (g(0))$$

$$w(z)=e^{\frac{1}{2}h(z)}$$

$$\Vert w(z)-p_{\epsilon }(z){\Vert }_{C^k(\overline{D})}<\epsilon \min (\Vert w(z){\Vert }_{C^k(\overline{D})},1)$$

$$p_{\epsilon }={\beta }_0\prod _{i=1}^M(z-{\beta }_i{)}^{N_i}$$

$$\Vert g(z)-p_{\epsilon }^2(z){\Vert }_{C^k(\overline{D})}$$

$$\le C_k\Vert (w-p_{\epsilon }){\Vert }_{C^k(\overline{D})}\Vert (w+p_{\epsilon }){\Vert }_{C^k(\overline{D})}$$

$$\le C_k\epsilon (2+\epsilon )\Vert w{\Vert }_{C^k(\overline{D})}$$

$$\partial {\Omega }_{{\epsilon }_0}=\{x+\nu g_{{\epsilon }_0}(x)\mid x\in \partial \Omega \},\Vert g_{{\epsilon }_0}{\Vert }_{C^k(\partial \Omega )}<{\epsilon }_0.$$

$$p_{\epsilon }={\alpha }_0\prod _{i=1}^M(z-{\alpha }_i{)}^{N_i}$$

$$\Vert {\partial }_{z}f(z)-p_{\epsilon }^2(z){\Vert }_{C^{\max (k,1)}(\overline{D})}<\epsilon .$$

$$f_{\epsilon }:={\int }_0^z{p}_{\epsilon }^2(w)dw+f(0)$$

$$\Vert f_{\epsilon }-f{\Vert }_{C^{\max (k+1,2)}(\overline{D})}<\epsilon ,{\partial }_z{f}_{\epsilon }=p_{\epsilon }^2,$$

Full text

[TABLE]

in two-dimensional domains $\Omega$. In particular, this paper presents a dense family $\mathcal{A}$ of simply-connected two-dimensional domains with analytic boundaries such that, for each $\Omega\in\mathcal{A}$, the nodal set of the eigenfunction $\phi_{\sigma_{j}}$ “is not dense at scale $\sigma_{j}^{-1}$”. This result addresses a question put forth under “Open Problem 10” in Girouard and Polterovich, J. Spectr. Theory, 321-359 (2017). In fact, the results in the present paper establish that, for domains $\Omega\in\mathcal{A}$, the nodal sets of the eigenfunctions $\phi_{\sigma_{j}}$ associated with the eigenvalue $\sigma_{j}$ have starkly different character than anticipated: they are not dense at any shrinking scale. More precisely, for each $\Omega\in\mathcal{A}$ there is a value $r_{1}>0$ such that for each $j$ there is $x_{j}\in\Omega$ such that $\phi_{\sigma_{j}}$ does not vanish on the ball of radius $r_{1}$ around $x_{j}$.

1. Introduction

Let $(M,g)$ be a compact Riemannian manifold with piecewise smooth boundary $\partial M$. The Steklov problem is given by

[TABLE]

There is a discrete sequence $0=\sigma_{0}<\sigma_{1}\leq\sigma_{2}\leq\dots$ of values of $\sigma$, with $\sigma_{j}\to\infty$ as $j\to\infty$, for which non-trivial solutions satisfying (1.1) exist [HL01]. These are the Steklov eigenvalues and the corresponding functions $\phi_{\sigma_{j}}$ are the Steklov eigenfunctions. This paper studies the asymptotic character of the nodal set of the eigenfunctions of the Steklov eigenvalue problem in the case $M$ equals a bounded open set $\Omega\in\mathbb{R}^{2}$. In particular the results in this paper show that the nodal set of the eigenfunction $\phi_{\sigma_{j}}$ is not dense at scale $\sigma_{j}^{-1}$ for some such sets $\Omega$—or, more precisely, that there is a dense family $\mathcal{A}$ of simply-connected two-dimensional domains with analytic boundaries such that, for each $\Omega\in\mathcal{A}$, the eigenfunction $\phi_{\sigma_{j}}$ in the domain $\Omega$ remains nonzero on a $j$-dependent ball of $j$-independent radius. This result addresses a question put forth under “Open Problem 10” in [GP17].

The behavior of both the Steklov eigenvalues (see e.g. [GP17, GPPS14, LPPS17]) and eigenfunctions (see e.g. [PST, GT19, BL15, Zhu16, Zel15, SWZ16, Sha71, HL01]) have been a topic of recent interest. When $M$ has smooth boundary, the Steklov eigenfunctions $\phi_{\sigma_{j}}|_{\partial M}$ behave much like high energy Laplace eigenfunctions with eigenvalue $\sigma_{j}^{2}$. In particular, they oscillate at frequency $\sigma_{j}$. References [PST, BL15, Zhu16, Zel15, SWZ16, WZ15, GRF17, Zhu15] study the nodal sets of $\phi_{\sigma_{j}}|_{M}$, giving both upper and lower bounds on its Hausdorff measure similar to those for Laplace eigenfunctions. In fact, most results regarding Steklov eigenfunctions in the interior of $M$ extract behavior similar to that of high energy Laplace eigenfunctions.

The purpose of this article is to show that, away from the boundary of $M$, Steklov eigenfunctions behave very differently than high energy Laplace eigenfunctions. Not only do they decay rapidly (see [GT19, HL01]) but, at least for a dense class of analytic domains, they oscillate slowly over certain portions of the domain. Girouard–Polterovich [GP17, Open Problem 10(i)] raise the question of whether nodal sets of Steklov eigenfunctions are dense at scale $\sigma_{j}^{-1}$ in $M$. One consequence of the results in the present paper is a negative answer to this question. We show that arbitrarily close to any simply-connected domain with analytic boundary $\Omega_{0}\subset\mathbb{R}^{2}$, there is a domain $\Omega_{1}$ for which the nodal sets are not $\sigma_{j}^{-1}$ dense and, indeed, that there is a region within $\Omega_{1}$ where the nodal set density does not increase as $\sigma_{j}\to\infty$. Moreover, the Steklov eigenfunctions oscillate no faster than a fixed frequency in this region. These results are summarized in the following theorem.

Theorem** 1****.**

Let $\Omega_{0}\subset\mathbb{R}^{2}$ be a bounded simply-connected domain with analytic boundary, and let $k>0$ and $\varepsilon>0$ be given. Then there exist a set $\Omega_{1}\subset\mathbb{R}^{2}$ with analytic boundary given by

[TABLE]

(where ${\nu}$ denotes the outward unit normal to $\partial\Omega_{0}$ and where $g$ is an analytic function defined on $\partial\Omega_{0}$), a point $x_{0}\in\Omega_{1}$ and numbers $0<r_{1}<r_{0}$, ($B(x_{0},r_{0})\subset\Omega_{1}$) such that: for each Steklov eigenvalue $\sigma$ for the domain $\Omega_{1}$ there exists a point $x_{\sigma}\in B(x_{0},r_{0})$ such that $B(x_{\sigma},r_{1})\subset B(x_{0},r_{0})$ and each Steklov eigenfunction $\phi_{\sigma}$ of eigenvalue $\sigma$ for the domain $\Omega_{1}$ satisfies

[TABLE]

Additionally, “$\phi_{\sigma}$ has bounded frequency on $B(x_{0},r_{0})$” (a precise statement follows in Theorem 2).

Theorem 1 is a consequence of the more precise results presented in Theorems 2 and 3 and Corollary 2.2. In particular, these results establish that, for each domain $\Omega$ in a dense class $\mathcal{A}$ of two-dimensional domains, an estimate holds for the truncation error in certain “mapped Fourier expansions” of the eigenfunctions $\phi_{\sigma}$ (i.e., Fourier expansions of $\phi_{\sigma}$ under a change of variables). This estimate is uniformly valid over a subdomain of $\Omega$ for all eigenfunctions $\phi_{\sigma}$ with $\sigma$ large enough. To state these results we first introduce certain conventions and notations, and we review known facts and results from complex analysis.

In what follows, and throughout the reminder of this article, $\mathbb{R}^{2}$ is identified with the complex plane $\mathbb{C}$, $\Omega\subset\mathbb{C}$ denotes a bounded, simply-connected open set with analytic boundary, and $D:=\{z\in\mathbb{C}\mid|z|<1\}$ denotes the open unit disc in the complex plane. Under these assumptions it follows from the Riemann mapping theorem [BK87] that there is a smooth map $f:\overline{D}\to\mathbb{C}$ such that $f|_{D}:D\to\Omega$ is a biholomorphism and $|\partial_{z}f|>0$ on $\overline{D}$—that is to say, $f|_{D}:D\to\Omega$ is a biholomorphic conformal mapping of $\Omega$ up to and including $\partial\Omega$. We call such a function $f$ a mapping function for $\Omega$. Note that, denoting by $\partial_{r}$ and $\partial_{\nu}$ the radial derivative on the boundary of $D$ and the normal derivative on the boundary of $\Omega$, respectively, we have $\partial_{r}=|\partial_{z}f|\partial_{\nu}$ and $|\partial_{z}f|>0$. Thus, for $z\in\partial D$ the function

[TABLE]

satisfies,

[TABLE]

and, hence, the generalized Steklov eigenvalue problem

[TABLE]

Finally we introduce notation for the relevant Fourier analysis. For $v\in C(\overline{D})$ we let

[TABLE]

denote the “boundary Fourier coefficients”, namely, the Fourier coefficients of the restriction $v|_{{}_{\partial D}}$ of $v$ to $\partial D$. Where notationally useful, we write $\mathcal{F}[v]=\hat{v}$.

Definition** 1.1****.**

We say that the Steklov problem on $\Omega$ satisfies the tunneling condition if there is $m_{0}>0$ and a mapping function $f$ for $\Omega$, such that for all $K>0$ there is $C_{0}>0$ satisfying for any $m$

[TABLE]

Lemma 4.1 shows that any tunneling Steklov problem there exist $\sigma_{0}>0$ so that for each $m\in\mathbf{Z}$ there is a constant $C>0$ such that for $\sigma>\sigma_{0}$,

[TABLE]

This estimate and its connections with similar results in quantum mechanics motivate the “tunneling” terminology introduced in Definition 1.1. To explain this, recall that $u$ is an eigenfunction of the Dirichlet to Neumann map which is a pseudodifferential operator on $\partial\Omega$ with symbol $|\xi|_{g}$ where $g$ is the metric on $\partial\Omega$ [Tay11, Sec. 7.11, Vol 2]. Therefore, the classical problem corresponding to the Steklov problem is the Hamiltonian flow for the Hamiltonian $|\xi|_{g}$ on $T^{*}\partial\Omega$ at energy $|\xi|_{g}=\sigma$—which describes the motion of a free particle on $\partial\Omega$. The allowable energies for this classical problem are given by $\{|\xi|_{g}=\sigma\}$ which, in the Fourier series representation correspond to $\sigma=|\xi|_{g}\sim|k|$. Thus, the classically forbidden region is \big{|}\sigma^{-1}|k|-1\big{|}>c>0. Equation  (1.6) tells us that, in cases for which the Steklov problem on $\Omega$ is tunneling, Steklov eigenfunctions carry positive energy even in the classically forbidden region $\sigma^{-1}|k|\ll 1$, with an energy value that is no smaller than exponentially decaying in $\sigma$. (Using the estimates of [GT19] one can also see that Steklov eigenfunctions carry at most exponentially small energy in the forbidden region.)

Theorem** 2****.**

Assume that the Steklov problem on $\Omega$ is tunneling and let $\sigma$ denote a Steklov eigenvalue for the set $\Omega$. Let

[TABLE]

Then, there exist a constant $c>0$ such that, for each integer $N>0$, there are constants $C_{N}$, $\sigma_{0}$, $\delta_{0}$, and $m_{0}>0$ so that for all $0<\delta<\delta_{0}$, $m>m_{0}$, and $\sigma_{j}>\sigma_{0}$ the inequality

[TABLE]

holds.

Letting $\{\phi_{\sigma_{j}}\}_{j=1}^{\infty}$ denote an orthonormal basis of Steklov eigenfunctions and calling $u_{\sigma_{j}}=\phi_{\sigma_{j}}\circ f$, Theorem 2 shows in particular that

[TABLE]

In other words, for $r$ small, $u_{\sigma_{j}}$ is well approximated by a function with finitely many Fourier modes. If there is $c>0$ such that

[TABLE]

then we obtain

[TABLE]

and $u_{\sigma_{j}}$ is nearly constant on small balls centered around 0. In general, however, finitely many Fourier modes are necessary to capture the lowest-order asymptotics, as indicated in equation (1.9).

One of the main components of the proof of Theorem 1, in addition to Theorem 2, is the construction of a large class of domains $\Omega$ for which the Steklov problem is tunneling. To this end, we introduce some additional definitions. A function $v\in C(D)$ will be said to be boundary-band-limited provided $\hat{v}(k)=0$ except for a finite number of values of $k\in\mathbb{Z}$. We say that a mapping function $f$ is boundary band limited conformal (BBLC) if $|\partial_{z}f|$ is boundary band-limited. If in addition, $|\partial_{z}f||_{\partial D}$ is non-constant, we will write that $\Omega$ is BBLCN. Finally, we say the domain $\Omega$ is BBLC (BBLCN) if and only if a BBLC (BBLCN) mapping function, $f:D\to\Omega$ exists. We now present the main theorem of this paper.

Theorem** 3****.**

Assume $\Omega$ is BBLCN. Then the Steklov problem on $\Omega$ is tunneling.

Remark** 1.2.** It is not clear whether the elliptical and kite-shaped domains (equations (6.1) and (6.2)) considered in Figures 1, 4 and 5 satisfy the BBLCN condition or, more generally, whether they have tunneling Steklov problems (we have not as yet been able to establish that the tunneling condition holds for domains that are not BBLCN). However, domain-opening observations such as those displayed in Figure 1 and Section 6, suggest that these domains may nevertheless be tunneling. This and other domain-opening observations provide support for Conjecture 1.3 below. (Steklov eigenfunctions on a domain which satisfies the BBLCN condition, and, therefore, in view of Theorem 3, is known to be tunneling, are displayed in Figure 2.)

In view of Remark 3 we conjecture that every Steklov problem on an analytic domain is tunneling unless the Steklov domain $\Omega$ is a disc:

Conjecture** 1.3****.**

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded, simply-connected domain with real analytic boundary that is not equal to $B(x,r)$ for any $x\in\mathbb{R}^{2}$, $r>0$. Then the Steklov problem on $\Omega$ is tunneling.

Outline of the paper

This paper is organized as follows. Section 2 shows that arbitrary analytic, bounded, simply-connected domains can be approximated arbitrarily closely by BBLCN domains. Then, Sections 3 and 4 provide proofs for Theorems 3 and 2, respectively. The numerical methods used in this paper to produce accurate Steklov eigenvalues, eigenfunctions, and associated nodal sets are presented in Section 5. Section 6, finally, illustrates the methods with numerical results for elliptical and kite-shaped domains.

Remark** 1.4.** Throughout this article we abuse notation slightly by allowing $C$ to denote a positive constant that may change from line to line but does not depend on any of the parameters in the problem. In addition $C_{N}$ is a positive constant that may change from line to line and depends only on the parameter $N$.

2. Approximation by tunneling domains

This section shows that any analytic domain can be approximated arbitrarily closely (in a sense made precise in Corollary 2.2) by a BBLCN domain. To do this, first let $M\geq 0$, $\alpha_{i}\in\mathbb{C}\setminus\overline{D}$ for $i=1,\dots,N$, and let $N_{i}\geq 1$, $i=1,\dots M$, and let us seek approximating BBLCN domains whose mappings $f:\overline{D}\to\mathbb{C}$ take the form

[TABLE]

In words: $f$ is the integral of the square of a polynomial with roots outside $\overline{D}$. It follows that

[TABLE]

In particular,

[TABLE]

which manifestly shows that ${|\partial_{z}f|}$ is boundary-band-limited.

We next show that an arbitrary non-vanishing analytic function on $\overline{D}$ can be approximated by the square of a polynomial.

Lemma** 2.1****.**

Let $g:\overline{D}\to\mathbb{C}$ smooth with $g|_{D}$ analytic and $|g|>0$ on $\overline{D}$. Then, for any $\varepsilon_{0}>0$ and $k>0$, there are $M>0$, $\alpha_{0}$, $\{(\alpha_{i},N_{i})\}_{i=1}^{M}$ with $|\alpha_{i}|>1$, $i=1,\dots,M$ such that

[TABLE]

Proof.

Define $h:\overline{D}\to\mathbb{C}$ by

[TABLE]

Then, since $U$ is simply-connected and $|g|>0$ on $\overline{D}$, $h$ is analytic in $D$ with smooth extension to $\overline{D}$. In addition,

[TABLE]

is an analytic function on $D$ such that $w^{2}(z)=g(z)$ and $w$ extends smoothly to $\overline{D}$. Then, for all $\varepsilon>0$, there is a polynomial $p_{\varepsilon}$ such that

[TABLE]

In particular, since $|g|>c>0$ on $\overline{D}$, for $0<\varepsilon$ small enough, $p_{\varepsilon}$ has no zeros in $\overline{D}$. Hence,

[TABLE]

for some $|\beta_{0}|>0$, $|\beta_{i}|>1$, $i=1,\dots,M$. Multiplying by $w+p_{\varepsilon}$, we have

[TABLE]

Choosing $\varepsilon=\frac{\varepsilon_{0}}{C_{k}}\min(\frac{1}{3\|w\|_{C^{k}(\overline{D})}},1)$ proves the result with $\alpha_{0}=\beta_{0}^{2}$ and $\alpha_{i}=\beta_{i}$. ∎

This result can be used to approximate any analytic domain by a BBLCN domain:

Corollary** 2.2****.**

For any analytic, bounded, simply-connected domain $\Omega$, $k>0$, and $\varepsilon_{0}>0$ there is a BBLCN domain $\Omega_{\varepsilon_{0}}$ and $g_{\varepsilon_{0}}\in C^{\infty}(\partial\Omega)$ such that with $\nu$ the outward unit normal to $\Omega$,

[TABLE]

Proof.

Since $\Omega$ is analytic, there is $f:\overline{D}\to\mathbb{C}$ analytic such that $f|_{D}:D\to\Omega$ is a biholomorphism and $|\partial_{z}f|>0$ on $D$. Moreover, by [BK87], $\partial_{z}f$ has a smooth extension to $\overline{D}$. Then, applying Lemma 2.1 with $g=\partial_{z}f(z)$ gives

[TABLE]

a polynomial with no roots in $\overline{D}$ such that

[TABLE]

Note also that adjusting $p$ if necessary we may assume that the restriction of $|p_{\varepsilon}|$ to $\partial D$ is not constant. Then, defining

[TABLE]

we have

[TABLE]

so that \big{|}\partial_{z}f_{\varepsilon}\big{|}|_{{}_{\partial D}} is non-constant and band limited. Moreover, since $f$ is a biholomorphism, for $\varepsilon>0$ small enough, $f_{\varepsilon}$ is also a biholomorphism. We next show that since $\|f_{\varepsilon}-f\|_{C^{\max(k+1,2)}(\overline{D})}<\varepsilon$, for $\varepsilon>0$ small enough the curve

[TABLE]

can be expressed in the form (2.1). To do this let

[TABLE]

and note that $F(1,\theta,\omega,s)=0$ if and only if

[TABLE]

Therefore, we aim to find $s=s(\theta)$ and $\omega=\omega(\theta)$ such that $F(1,\theta,\omega(\theta),s(\theta))=0$. Note that

[TABLE]

In particular,

[TABLE]

Therefore, there is $\delta>0$, $\varepsilon_{0}>0$ such that for $0<\varepsilon<\varepsilon_{0}$, $|\omega_{0}|<\delta$, $t_{0}\in(-1,2)$, and $s_{0}\in[-1,1]$ if $F(t_{0},\theta_{0},\omega_{0},s_{0})=0$, then for $|\omega_{0}|<\delta$ and $|t-t_{0}|<\delta$, $\omega=\omega(t,\theta)$ and $s=s(t,s)$ are the unique solutions of $F(t,\theta,\omega,s)=0$. In particular, since $F(0,\theta,0,0)=0$, the solutions $s=s(t,\theta)$ and $\omega=\omega(t,\theta)$ can be continued as functions of $t$ as long as $|\omega(t,\theta)|$ remains small.

We next note that

[TABLE]

and, therefore,

[TABLE]

Hence for $\varepsilon$ small enough the solutions $\omega(t,\theta)$ and $s(t,\theta)$ continue to $t=1$ and satisfy

[TABLE]

Again, using the implicit function theorem, this implies that $\omega(\theta):=\omega(1,\theta)$ and $s(\theta):=s(1,\theta)$ are $2\pi$-periodic. Differentiating $k$ times now yields

[TABLE]

finishing the proof by setting $g_{\varepsilon_{0}}=\pm s$ and shrinking $\varepsilon>0$ as necessary. (Here the $\pm$ corresponds to whether $f(e^{i\theta})$ is positively ($-$) or negatively ($+$) oriented.) ∎

Remark** 2.3.** Since the map $f_{\varepsilon}$ in equation (2.2) may send 0 to a point $z_{0}$ close to the boundary, it is interesting to see how the Steklov eigenfunctions rearrange their nodal sets in such a way that Theorems 1 and 2 are satisfied on the image of $f_{\varepsilon}$. To demonstrate this let $|a|<1$, consider the biholomorphic function $f(z):=\frac{z-a}{\bar{a}z-1}$, and let $f_{\varepsilon}$ denote the approximant of $f$ given by equation (2.2) with

[TABLE]

(This polynomial was obtained as the $N$-th order Taylor polynomial of $\sqrt{\partial_{z}f}$.) In this case, according to Theorems 1 and 2, the Steklov eigenfunctions should be slowly oscillating in a $\sigma$ independent neighborhood of $z_{0}$. Figure 2 displays corresponding Steklov eignfunction or various orders as well as a typical eigenfunction for the exact disc. Note the dramatic change that arises in the Steklov eigenfunctions from a barely visible boundary perturbation of the disc.

3. BBLCN domains and tunneling Steklov problems

This section presents a proof of Theorem 3. In preparation for that proof, let $\Omega\subset\mathbb{C}$ be a BBLCN domain, and denote by $f$ the corresponding mapping function. Define

[TABLE]

Since $\Omega$ is a BBLCN domain, the function \big{|}\partial_{z}f\big{|}|_{{}_{\partial D}} is band limited and \big{|}\partial_{z}f\big{|}|_{{}_{\partial D}} is not identically constant. It follows that

[TABLE]

satisfies $1\leq m_{0}<\infty$.

Denoting by $\hat{u}(n)$ the boundary Fourier coefficients of an eigenfunction $u$, the corresponding boundary Fourier coefficients of $\partial_{r}u$ are given by $|n|\hat{u}(n)$. Thus, a solution to (1.4) is uniquely determined as an $\ell^{2}$ solution to the equation

[TABLE]

In what follows we may, and do, assume that solutions $\hat{u}$ have $\ell^{2}$-norm equal to one.

Proof of Theorem 3

Since

[TABLE]

it follows that (3.1) can be re-expressed in the form

[TABLE]

From (3.2) we obtain

[TABLE]

and, then, for all $|n|\leq K\sigma$,

[TABLE]

The second inequality follows from the fact that $a_{n}\equiv 0$ for $|n|\geq m_{0}$, while the third one results from the relation $a_{0}>0$ and the positivity, $\sigma>0$, of all nontrivial eigenvalues $\sigma$, which imply that

[TABLE]

Making an identical argument, but solving for $\hat{u}(n-m_{0})$, and using that $|a_{m_{0}}|=|a_{-m_{0}}|\neq 0$, we have for all $|n|\leq K\sigma$,

[TABLE]

We now use equation (3.3) to prove the first half of our tunneling estimate.

Lemma** 3.1****.**

Let $m\in\mathbb{Z}$, $K>0$, and

[TABLE]

Then, there exists $C_{0}>0$ so that for all $\sigma>0$ and for $-K\sigma\leq n+m\leq K\sigma$ we have

[TABLE]

Proof.

We will assume $m\geq 0$ since the other case follows similarly. The cases of $n=-m_{0},\dots,m_{0}$ are clear if we take $C_{0}\geq 1$. Suppose (3.4) holds for $-m_{0}\leq n\leq\ell$ with $m_{0}\leq\ell$. Then, by (3.3),

[TABLE]

Now, if $m_{0}\leq\ell<2m_{0}$, then

[TABLE]

In particular, taking

[TABLE]

we have

[TABLE]

Next, if $2m_{0}\leq\ell$, then

[TABLE]

Taking ${C_{0}\geq 2|a_{m_{0}}|^{-1}\max(K,\|a_{m}\|_{\ell^{\infty}})+1}$ completes the proof for $-m_{0}\leq n\leq K\sigma-m$.

An almost identical argument gives the $-K\sigma-m\leq n\leq 0$ case. ∎

4. Analysis of Tunneling Steklov Problems

The proof of Theorem 2 now follows in two steps. First, we show that, for eigenfunctions of any tunneling Steklov problem, the boundary Fourier coefficients of low frequency contain a mass no smaller than exponential in $\sigma$. To finish the proof, we use the fact that the harmonic extension of $e^{in\theta}$ decays exactly as $r^{|n|}$. Examining the solution on the ball of radius $\delta>0$ for some $\delta$ small enough, it will be shown that the low frequencies dominate the behavior of $u$.

Lemma** 4.1****.**

Suppose that $\Omega$ has tunneling Steklov problem. Then there exist $\sigma_{0}>0$ so that for all $m>0$ there is $C>0$ such that for $\sigma>\sigma_{0}$,

[TABLE]

Proof.

First, note that by e.g. [GT19, Corollary 1.3], for $\sigma>3m$ there is $C>0$ so that

[TABLE]

By Lemma 3.1

[TABLE]

In particular,

[TABLE]

Taking $\sigma_{0}$ large enough so that $Ce^{-C\sigma}\leq\frac{1}{2}$, finishes the proof. ∎

Proof of Theorem 2.

In what follows we utilize the definitions (1.7) for a given eigenvalue $\sigma_{j}=\sigma$, and, for that eigenvalue we denote $\hat{u}(k)=\hat{u}_{\sigma_{j}}(k)=\hat{u}_{\sigma}(k)$. Then, applying the relation

[TABLE]

which is valid for all sequences $\{b_{k}\}_{k\in\mathbb{Z}}\subset\mathbb{C}$, to the right-hand equation in (1.7), for $m\geq m_{0}$ we obtain

[TABLE]

To estimate the error in approximating $u_{\sigma,\delta}$ by $\tilde{u}_{\sigma,\delta,m}$, first note that

[TABLE]

Applying Lemma 4.1 with $m=0$, and absorbing the $\sigma^{N}$ into the exponential factor we then obtain

[TABLE]

where

[TABLE]

We can now estimate

[TABLE]

Thus, using the definition of tunneling (Definition 1.1), we obtain

[TABLE]

provided that $\delta<\frac{1}{2}C_{0}^{-1}$. Therefore, using (4.2),

[TABLE]

Thus, choosing $\delta>0$ such that $\delta<e^{-2C}$ and taking $\sigma_{0}>N{+m_{0}+1}$ the claim follows. ∎

We can now present a proof of Theorem 1.

Proof of Theorem 1.

From Corollary 2.2 we know that there exists a tunneling domain $\Omega_{1}\subset\mathbb{C}$ satisfying (1.2) for the given value $\varepsilon>0$. Let $\sigma_{0}$ be as in Theorem 2. Clearly, it suffices to prove the statement of the theorem for $\sigma>\sigma_{0}$, since for $\sigma\leq\sigma_{0}$ the statement follows from the fact that there are finitely many Steklov eigenvalues below $\sigma_{0}$ and that $\psi_{\sigma}$ cannot vanish in any open set. Therefore, we may and do assume $\sigma>\sigma_{0}$ along with the other assumptions in Theorem 2, so that, in particular, inequality (1.8) holds. In what follows we write

[TABLE]

Fixing $m\geq m_{0}+2$, and letting $\tilde{u}_{\sigma,\delta}$ and $\tilde{u}_{\sigma,\delta,m}$ be given by (1.7) (with $u_{\sigma}$ related to $\phi_{\sigma}$ via  (1.3)) we note that

[TABLE]

It follows that there exists $x_{0}\in B(0,\delta)$ such that

[TABLE]

Now, since $\|\tilde{u}_{\sigma,\delta,m}\|_{C^{1}}\leq C_{m,\delta}\|\tilde{u}_{\sigma,\delta,m}\|_{L^{2}}$, it follows from (4.4) that there is $r_{m,\delta}\in\mathbb{R}$, $0<r_{m,\delta}<\delta$ (in particular, independent of $\sigma$) such that

[TABLE]

But, since $m\geq m_{0}+2$, the estimate (1.8) with $N=0$ yields

[TABLE]

and

[TABLE]

(To establish the rightmost inequality in (4.7) the relation $\|\tilde{u}_{\sigma,\delta}-\tilde{u}_{\sigma,\delta,m}\|_{L^{2}_{\delta}}\leq\sqrt{\pi}\delta\|\tilde{u}_{\sigma,\delta}-\tilde{u}_{\sigma,\delta,m}\|_{L^{\infty}_{\delta}}$ was used before the inequality (1.8) was applied.) From (4.7) we obtain

[TABLE]

It follows from (4.5), (4.6) and (4.8) that

[TABLE]

and, therefore

[TABLE]

Taking $\delta_{1}$ sufficiently small and $\delta\leq\delta_{1}$ the inequality

[TABLE]

holds, and it therefore follows that for a certain constant $D>0$ we have

[TABLE]

provided $\delta<\delta_{1}$. In particular,

[TABLE]

Since the derivative of $f$ never vanishes, for $\delta<\delta_{1}$ and for a certain $E>0$ there is a ball $\mathcal{B}$ of radius $Er_{m,\delta}$ such that $\phi_{\sigma}$ does not vanish on $\mathcal{B}$. The proof is now complete.

∎

5. Numerical Formulation

5.1. Integral representation

Let $\Omega\subset\mathbb{R}^{2}$ denote a domain with, say, a $C^{2}$ boundary, and let

[TABLE]

denote the Single Layer Potential (SLP) for a given density $\phi:\partial\Omega\to\mathbb{R}$ in a certain Banach space $H$ of functions. Both Sobolev and continuous spaces $H$ of functions lead to well developed Fredholm theories in this context [Kre14, MM00]. It is useful to recall that, as shown e.g. in the aforementioned references, the limiting values of the potential $S$ and its normal derivative on $\partial\Omega$ can be expressed in terms of well known “jump conditions” that involve the single and double layer boundary integral operators

[TABLE]

respectively.

In view of the jump conditions for the SLP [Kre14], use of the representation

[TABLE]

for the eigenfunction $u$, the Steklov boundary condition in equation (1.1) gives rise to the generalized eigenvalue problem

[TABLE]

Unfortunately, however, the single layer operator $\mathcal{S}$ on the right side of this equation is not always invertible. In order to avoid singular right-hand sides and the associated potential sensitivity to round-off errors, in what follows we utilize the Kress potential

[TABLE]

(where $\overline{\phi}$ denotes the average of $\phi$ over $\partial\Omega$), which leads to the modified eigenvalue equation [Akh16]

[TABLE]

The right-hand operator in this equation is invertible [Kre14, Thm. 7.41], as desired. For either formulation, the evaluation of a given eigenfunction $u$ requires evaluation of the SLP, in accordance with either (5.1) or (5.3), for the solution $\phi$ of the corresponding generalized eigenvalue problem (5.2) or (5.4), respectively, at all required points $x\in\Omega$.

Remark** 5.1.** Note that for a given harmonic function $u$ in $\Omega$, $\phi$ in (5.2) and that in (5.4) are not the same.

5.2. Fourier expansion and exponential decay

In terms of a given $2\pi$-periodic parametrization $C(t)$ of $\partial\Omega$, the Steklov eigenfunction $u$ corresponding to a given solution $(\phi,{\sigma})$ of the regularized eigenvalue problem (5.4), which is given by the single layer expression (5.3), can be expressed, for a given point $x=(x_{1},x_{2})\in\Omega$,

[TABLE]

where $C(t)=(C_{1}(t),C_{2}(t))$ and where $\overline{\phi}$ denotes the average of $\phi$ over the curve $\partial\Omega$. Unfortunately, a direct use of this expression does not capture important elements in the eigenfunction within $\Omega$, such as the nodal sets, since, for analytic domains, the eigenfunctions decay exponentially fast within $\Omega$ as the frequency increases [PST, GT19]. In regions where the actual values of the eigenfunction may be significantly below machine precision the expression (5.5) must be inaccurate: this expression can only achieve the exponentially small values via the cancellations that occur as the the solution $\phi$ becomes more and more oscillatory. But such cancellations cannot take place numerically below the level of machine precision. In order to capture the decay explicitly within the numerical algorithm we proceed in a manner related to the construction used in [PST].

To accurately obtain the exponentially decaying values of the Steklov eigenfunction we proceed as follows. We first consider the Fourier expansion

[TABLE]

of the product $\left[\phi(C(t))-\overline{\phi}\,\right]\left|\dot{C}(t)\right|$; note that, as is easily checked, the $n=0$ term in the Fourier expansion (5.6) is indeed equal to zero. Inserting this expansion in (5.5) we obtain

[TABLE]

[TABLE]

Then, assuming an analytic boundary, as is relevant in the context of this paper, and further assuming, for simplicity, that $C(t)$ is in fact an entire function of $t$ (as are, for example, all parametrizations $C(t)$ given by vector Fourier series containing finitely many terms), we introduce, for $x=(x_{1},x_{2})\in\Omega$, the quantities

[TABLE]

and

[TABLE]

Using Cauchy’s Theorem for $x=(x_{1},x_{2})\in\Omega$ and any $s\in\mathbb{R}$ satisfying $|s|\leq{\lambda}(x)$, we obtain

[TABLE]

and, thus, letting $s=\alpha{\lambda}(x)$ for any $\alpha\in\mathbb{R}$ satisfying $|\alpha|\leq 1$, the eigenfunction $u$ is given by

[TABLE]

Lemma** 5.2****.**

There is $C>0$ such that for all $n>0$,

[TABLE]

Moreover, there is $c>0$ and a sequence $\{n_{k}\}_{k=1}^{\infty}$ with $|n_{k}|\to\infty$ such that

[TABLE]

A proof of Lemma 5.2 is given in Appendix B. It follows from Lemma 5.2 that equation (5.8) optimally captures the exponential decay of the $B_{n}$ terms as $\sigma\to\infty$. Note that this setup does not capture the exponential decay of the coefficients $A_{n}$ below machine precision away from $|n|\sim\sigma$, and, therefore, the accuracy of the resulting interior eigenfunction reconstructions does not exceed that accuracy level. But the function $\lambda(x_{1},x_{2})$ does capture the exponential decay and the geometrical character of the eigenfunction as long as the (spatially constant) coefficients $A_{n}$ for low $n$ remain above machine precision.

For general curves $C(t)$ no closed form expressions exist for the function ${\lambda}(x)$, and a numerical algorithm must be used for the evaluation of this quantity, as part of a numerical implementation of the eigenfunction expression (5.9). In our implementation the function ${\lambda}$ was evaluated via an application of Newton’s method to the nonlinear equation

[TABLE]

Explicit expressions can be obtained for circles and ellipses, however:

1. (1)

For a circle of radius 1:

[TABLE] 2. (2)

For an ellipse of semiaxes $a>b$:

[TABLE]

The derivation of the expression (5.11) is outlined in Appendix A.

5.3. Exponential decay and verification of Cauchy’s

theorem

Tables 1 and 2 demonstrate the validity of equation (5.8) (since in both cases the results in the second and third columns closely agree with each other for $n\leq 50$), as well as the exponential decay of the exact coefficients $B_{n}^{0}$—as born by the results in the third column of these tables. The disagreement observed for $n>50$ is caused by the lack of precision of the results in the second column beyond machine accuracy, a problem that is eliminated in the third column via an application of the relation (5.8).

6. Numerical Results

Figures 4 and 5 present density plots and fixed-sign sets for Steklov eigenfunctions over domains bounded by the elliptical and kite-shaped curves parametrized by the vector functions

[TABLE]

with $a=2$ and $b=1$, and

[TABLE]

respectively. These figures demonstrate, in particular, domain-opening and non-density of nodal sets as discussed in Remark 3.

Acknowledgements. Thanks to Agustin Fernandez Lado for writing the code numerical Steklov-eigenfunction solver and for providing the derivation presented in Appendix A. Thanks also to Jared Wunsch for suggesting part of the proof of Lemma 5.2 The authors are grateful to the American Institute of Mathematics where this research began. J.G. is grateful to the National Science Foundation for support under the Mathematical Sciences Postdoctoral Research Fellowship DMS-1502661 and under DMS-1900434. O.B. gratefully acknowledges support by NSF, AFOSR and DARPA through contracts DMS-1411876, FA9550-15-1-0043 and HR00111720035, and the NSSEFF Vannevar Bush Fellowship under contract number N00014-16-1-2808.

Appendix A Function ${\lambda}(x)$ For an ellipse of semiaxes

$a>b$

Let $\gamma=\sqrt{a^{2}-b^{2}}$ and $\mu_{0}=\mathrm{arcosh}(a/\gamma)$. Using elliptical coordinates with foci $(\pm\gamma,0)$ to represent the point $x=(x_{1},x_{2})$, so that $x_{1}=\gamma\cosh(\mu)\cos(\tau)$ and $x_{2}=\gamma\sinh(\mu)\sin(\tau)$, and letting the boundary of the ellipse be given by $C_{1}(t)=\gamma\cosh(\mu_{0})\cos(t)$, $C_{2}(t)=\gamma\sinh(\mu_{0})\sin(t)$, in view of the relations $x_{1}+ix_{2}=\gamma\cosh(\mu+i\tau)$ and $C_{1}(t)+iC_{2}(t)=\gamma\cosh(\mu_{0}+it)$ we obtain

[TABLE]

It follows that the left-hand side of this equation vanishes for some value of $t$ if and only if either $s=(\mu_{0}-\mu)$ or $s=(\mu_{0}+\mu)$. Thus, ${\lambda}(x)$ equals the smallest of these two positive numbers, namely ${\lambda}(x)=(\mu_{0}-\mu)$, which is equivalent to the desired relation (5.11).

Appendix B Proof of Lemma 5.2

First, let

[TABLE]

Then, for $|{\operatorname{Im}\,}z|<\lambda(x_{1},x_{2})$, the expression

[TABLE]

defines the principal branch of $\log h(z)$—which is, then, an analytic function in the strip $|{\operatorname{Im}\,}z|<\lambda$. On $\pm{\operatorname{Im}\,}z=\lambda$, we define

[TABLE]

Lemma** B.1****.**

Let $h(z)$ denote an analytic function defined on an open neighborhood of the set $\{z:|{\operatorname{Im}\,}z|\leq\lambda\}$ which does not vanish for $|{\operatorname{Im}\,}z|<\lambda$, but which vanishes to order $k$ at $z_{0}=t_{0}+i\lambda$. Then,

[TABLE]

Similarly, if $h$ vanishes to order $k$ at $z_{0}=t_{0}-i\lambda$,

[TABLE]

Proof.

Note that for $\varepsilon>0$ small enough $\{h(z)=0\}\cap\{|z-z_{0}|<\varepsilon\}=z_{0}$. Therefore

[TABLE]

where $\Gamma$ is any contour starting at $z_{0}-\varepsilon_{2}$, ending at $z_{0}+\varepsilon_{1}$, and lying in

[TABLE]

In particular, let

[TABLE]

and $\Gamma=\Gamma_{1}\cup\Gamma_{2}$. Then, since

[TABLE]

[TABLE]

Letting $\varepsilon_{1}$ and $\varepsilon_{2}$ tend to zero completes the proof for the case $z_{0}=t_{0}+i\lambda$. The proof for $z_{0}=t_{0}-i\lambda$ follows by substituting $z$ by $-z$. ∎

Lemma** B.2****.**

Let $h(z,x_{1},x_{2})$ denote an analytic function on $|{\operatorname{Im}\,}z|\leq\lambda$ which vanishes to order $k$ at $z_{0}=t_{0}+i\lambda$. Then for $\chi\in C_{c}^{\infty}(S^{1})$ supported in a sufficiently small neighborhood of $t_{0}$, with $\chi\equiv 1$ near $t_{0}$, we have

[TABLE]

Similarly if $h$ vanishes to order $k$ at $z_{0}=t_{0}-i\lambda$, we have

[TABLE]

Proof.

We consider the first case, the second follows similarly.

Selecting $\chi(t)$ with sufficiently small support we ensure that, within the support of $\chi$, $h(t+i\lambda)$ vanishes only at $t=t_{0}$. We then have

[TABLE]

and

[TABLE]

Since $|t-t_{0}|^{-k}|h(t+i\lambda)|$ is smooth and bounded away from zero on the support of $\chi$, the second term in (B.2) is $O(n^{-\infty})$.

Taking real parts in the asymptotic formula [BO99, p. 381] we obtain

[TABLE]

Then, using (B.3) together with the fact that $\log 1=0$ we may approximate the first term on the right-hand side of (B.2) by

[TABLE]

Let us now estimate the second term on the right-hand side of (B.1). We have

[TABLE]

where in the last equality Lemma B.1 was used. ∎

We may now complete the proof of Lemma 5.2. Let $0\leq t_{1}<t_{2}<\dots<t_{M}<2\pi$ denote the zeroes of $h(t+i\lambda)$ as a function of $t$, and let $k_{j}$ ($0\leq j\leq M$) denote the vanishing order at $t=t_{j}$. Then, by Lemma B.2, for $\chi_{j}$ supported close enough to $t_{j}$ with $\chi_{j}\equiv 1$ near $t_{j}$, and $n>0$,

[TABLE]

By shrinking the support of $\chi_{j}$, we may assume that ${\operatorname{supp\,}}\chi_{j}\cap\chi_{\ell}=\emptyset$ for $\ell\neq j$. Then, since $\chi_{j}\equiv 1$ near $t_{j}$, $(1-\sum_{j}\chi_{j}(t)))\log h(t+i\lambda)\in C^{\infty}(S^{1})$ and hence

[TABLE]

Thus in view of equation (5.7) we obtain

[TABLE]

Proceeding by contradiction, assume that

[TABLE]

Then in particular,

[TABLE]

But we note that

[TABLE]

Recalling that

[TABLE]

we obtain

[TABLE]

which contradicts (B.4).

If $h(t+i\lambda)$ does not vanish anywhere, then $h(t-i\lambda)$ vanishes at some $0\leq t_{1}<t_{2}<\dots<t_{M}<2\pi$ and we may repeat the argument this time considering

[TABLE]

and taking the limit as $n\to-\infty$.