Domains Without Dense Steklov Nodal Sets

This article concerns the asymptotic geometric character of the nodal set of the eigenfunctions of the Steklov eigenvalue problem -Δϕσj=0,onΩ,∂νϕσj=σjϕσjon∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\Delta \phi _{\sigma _j}=0,\quad \hbox { on }\,\,\Omega ,\quad \partial _\nu \phi _{\sigma _j}=\sigma _j \phi _{\sigma _j}\quad \hbox { on }\,\,\partial \Omega \end{aligned}$$\end{document}in two-dimensional domains Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document}. In particular, this paper presents a dense family A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} of simply-connected two-dimensional domains with analytic boundaries such that, for each Ω∈A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \in \mathcal {A}$$\end{document}, the nodal set of the eigenfunction ϕσj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{\sigma _j}$$\end{document} “is not dense at scale σj-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _j^{-1}$$\end{document}”. This result addresses a question put forth under “Open Problem 10” in Girouard and Polterovich (J Spectr Theory 7(2):321–359, 2017). In fact, the results in the present paper establish that, for domains Ω∈A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \in \mathcal {A}$$\end{document}, the nodal sets of the eigenfunctions ϕσj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{\sigma _j}$$\end{document} associated with the eigenvalue σj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _j$$\end{document} have starkly different character than anticipated: they are not dense at any shrinking scale. More precisely, for each Ω∈A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \in \mathcal {A}$$\end{document} there is a value r1>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_1>0$$\end{document} such that for each j there is xj∈Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_j\in \Omega $$\end{document} such that ϕσj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{\sigma _j}$$\end{document} does not vanish on the ball of radius r1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_1$$\end{document} around xj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_j$$\end{document}.


Introduction
Let (M, g) be a compact Riemannian manifold with piecewise smooth boundary ∂ M. The Steklov problem is given by (1.1) There is a discrete sequence 0 = σ 0 < σ 1 ≤ σ 2 ≤ · · · of values of σ , with σ j → ∞ as j → ∞, for which non-trivial solutions satisfying (1.1) exist [9]. These are the Steklov eigenvalues and the corresponding functions φ σ 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 ∈ R 2 . In particular the results in this paper show that the nodal set of the eigenfunction φ σ j is not dense at scale σ −1 j for some such sets -or, more precisely, that there is a dense family A of simply-connected two-dimensional domains with analytic boundaries such that, for each ∈ A, the eigenfunction φ σ j in the domain remains nonzero on a j-dependent ball of j-independent radius. This result addresses a question put forth under "Open Problem 10" in [7].
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 [5,9]) but, at least for a dense class of analytic domains, they oscillate slowly over certain portions of the domain. Girouard-Polterovich [7, Open Problem 10(i)] raise the question of whether nodal sets of Steklov eigenfunctions are dense at scale σ −1 j 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 simplyconnected domain with analytic boundary 0 ⊂ R 2 , there is a domain 1 for which the nodal sets are not σ −1 j dense and, indeed, that there is a region within 1 where the nodal set density does not increase as σ j → ∞. 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 0 ⊂ R 2 be a bounded simply-connected domain with analytic boundary, and let k > 0 and ε > 0 be given. Then there exist a set 1 ⊂ R 2 with analytic boundary given by (where ν denotes the outward unit normal to ∂ 0 and where g is an analytic function defined on ∂ 0 ), a point x 0 ∈ 1 and numbers 0 < r 1 < r 0 , (B(x 0 , r 0 ) ⊂ 1 ) such that: for each Steklov eigenvalue σ for the domain 1 there exists a point x σ ∈ B(x 0 , r 0 ) such that B(x σ , r 1 ) ⊂ B(x 0 , r 0 ) and each Steklov eigenfunction φ σ of eigenvalue σ for the domain 1 satisfies Additionally, "φ σ 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 in a dense class A of two-dimensional domains, an estimate holds for the truncation error in certain "mapped Fourier expansions" of the eigenfunctions φ σ (i.e., Fourier expansions of φ σ under a change of variables). This estimate is uniformly valid over a subdomain of for all eigenfunctions φ σ with σ 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 remainder of this article, R 2 is identified with the complex plane C, ⊂ C denotes a bounded, simply-connected open set with analytic boundary, and D := {z ∈ C | |z| < 1} denotes the open unit disc in the complex plane. Under these assumptions it follows from the Riemann mapping theorem [2] that there is a smooth map f : D → C such that f | D : D → is a biholomorphism and |∂ z f | > 0 on D-that is to say, f | D : D → is a biholomorphic conformal mapping of up to and including ∂ . We call such a function f a mapping function for . Note that, denoting by ∂ r and ∂ ν the radial derivative on the boundary of D and the normal derivative on the boundary of , respectively, we have ∂ r = |∂ z f |∂ ν and |∂ z f | > 0. Thus, for z ∈ ∂ D the function and, hence, the generalized Steklov eigenvalue problem (1.4) Finally we introduce notation for the relevant Fourier analysis. For v ∈ C(D) we letv Lemma 4.1 shows that any tunneling Steklov problem there exist σ 0 > 0 so that for each m ∈ Z there is a constant C > 0 such that for σ > σ 0 , (1.6) This estimate and its connections with similar results in quantum mechanics motivate the "tunneling" terminology introduced in Definition 1.1. To explain this, we first review the notion of tunneling from quantum mechanics. In that setting, we consider quantum wave-functions ψ h which are solutions to ∞ and E ∈ R is the energy of ψ h . Since ψ h is the wave-function of a quantum particle with energy E under the influence of the potential V , the corresponding classical Hamilatonian is p = |ξ | 2 + V − E and the corresponding classical particle lies on the surface { p = 0}. The classical tunneling result from quantum mechanics then states that, despite the fact that ψ h is classically localized to { p = 0}, for any U open and bounded, there is c > 0 such that That is, even for U ⊂ {V (x) > E}, the classically forbidden region, there is some positive (albeit exponentially small in h) probability of finding the particle U . (We refer the reader to e.g. [21,Chapters 4,7] for a description of the phase space perspective for studying Schrödinger operators and this type of tunneling estimate.) Returning to the setting of Steklov eigenfunctions, recall that u σ is an eigenfunction of the Dirichlet to Neumann map which is a pseudodifferential operator on ∂ with symbol |ξ | g where g is the metric on ∂ [16,Sec. 7.11,Vol 2]. Therefore, the classical problem corresponding to the Steklov problem is the Hamiltonian flow for the Hamiltonian |ξ | g on T * ∂ at energy σ −1 |ξ | g = 1-which describes the motion of a free particle on ∂ . The allowable energies for this classical problem are given by {σ −1 |ξ | g = 1} which, in the Fourier series representation correspond to σ = |ξ | g ∼ |k|. Thus, the classically forbidden region is σ −1 |k| − 1 > c > 0. Equation (1.6) tells us that, in cases for which the Steklov problem on is tunneling, Steklov eigenfunctions carry positive energy even in the classically forbidden region σ −1 |k| 1, with an energy value that is no smaller than exponentially decaying in σ . (Using the estimates of [5] 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 is tunneling and let σ denote a Steklov eigenvalue for the set . Let Then, there exist a constant c > 0 such that, for each integer N > 0, there are constants C N , σ 0 , δ 0 , and m 0 > 0 so that for all 0 < δ < δ 0 , m > m 0 , and σ > σ 0 the inequality

holds.
Letting {φ σ j } ∞ j=1 denote an orthonormal basis of Steklov eigenfunctions and calling u σ j = φ σ j • f , Theorem 2 shows in particular that (1.9) In other words, for r small, u σ j is well approximated by a function with finitely many Fourier modes. If there is c > 0 such that then we obtain and u σ 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 Eq. (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 for which the Steklov problem is tunneling. To this end, we introduce some additional definitions. A function v ∈ C(D) will be said to be boundary-band-limited providedv(k) = 0 except for a finite number of values of k ∈ Z. We say that a mapping function f is boundary band limited conformal (BBLC) if |∂ z f | is boundary band-limited. If in addition, |∂ z f || ∂ D is nonconstant, we will write that is BBLCN. Finally, we say the domain is BBLC (BBLCN) if and only if a BBLC (BBLCN) mapping function, f : D → exists. We now present the main theorem of this paper. Theorem 3 Assume is BBLCN. Then the Steklov problem on is tunneling.

Remark 1.2
It is not clear whether the elliptical and kite-shaped domains (Eqs, (6.1) and (6.2)) considered in Figs. 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 Fig. 1 and Sect. 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 Fig. 2).
In view of Remark 1.2 we conjecture that every Steklov problem on an analytic domain is tunneling unless the Steklov domain is a disc: simply-connected domain with real analytic boundary that is not equal to B(x, r ) for any x ∈ R 2 , r > 0. Then the Steklov problem on is tunneling. (bottom left). Note that, according to Corollary 2.2 the set is a BBLCN approximation to the disk. As predicted by Theorem 2, oscillations avoid a region around z 0 for high σ . The bottom-right image displays a typical eigenfunction on the exact disc. Note the dramatic change that arises in the Steklov eigenfunctions from a barely visible boundary perturbation of the disc

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 (Fig. 3). Then, Sects. 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 Sect. 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 .

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, and let us seek approximating BBLCN domains whose mappings f : D → C take the form In words: f is the integral of the square of a polynomial with roots outside D. It follows that In particular, We next show that an arbitrary non-vanishing analytic function on D can be approximated by the square of a polynomial.
Then, since D is simply-connected and |g| > 0 on D, h is analytic in D with smooth extension to D. In addition, is an analytic function on D such that w 2 (z) = g(z) and w extends smoothly to D.
Then, for all ε > 0, there is a polynomial p ε such that In particular, since |g| > c > 0 on D, for 0 < ε small enough, p ε has no zeros in D. Hence, This result can be used to approximate any analytic domain by a BBLCN domain: Corollary 2.2 For any analytic, bounded, simply-connected domain , k > 0, and ε 0 > 0 there is a BBLCN domain ε 0 and g ε 0 ∈ C ∞ (∂ ) such that with ν the outward unit normal to , Moreover, by [2], ∂ z f has a smooth extension to D. Then, applying Lemma 2.1 with g = ∂ z f (z) gives Note also that adjusting p if necessary we may assume that the restriction of | p ε | to ∂ D is not constant. Then, defining so that ∂ z f ε | ∂ D is non-constant and band limited. Moreover, since f is a biholomorphism, for ε > 0 small enough, f ε is also a biholomorphism. We next show that since can be expressed in the form (2.1). To do this let Therefore, we aim to find s = s(θ ) and ω = ω(θ) such that F(1, θ, ω(θ), s(θ )) = 0. Note that In particular, In particular, by the implicit function theorem, there is In particular, since F(0, θ, 0, 0) = 0, the solutions s = s(t, θ) and ω = ω(t, θ) can be continued as functions of t as long as |ω(t, θ)| remains small. We next note that and, therefore, Hence for ε small enough the solutions ω(t, θ) and s(t, θ) continue to t = 1 and satisfy

BBLCN Domains and Tunneling Steklov Problems
This section presents a proof of Theorem 3. In preparation for that proof, let ⊂ C be a BBLCN domain, and denote by f the corresponding mapping function. Define Since is a BBLCN domain, the function ∂ z f | ∂ D is band limited and ∂ z f | ∂ D is not identically constant. It follows that Denoting byû(n) the boundary Fourier coefficients of an eigenfunction u, the corresponding boundary Fourier coefficients of ∂ r u are given by |n|û(n). Thus, a solution to (1.4) is uniquely determined as an 2 solution to the equation In what follows we may, and do, assume that solutionsû have 2 -norm equal to one.

Proof of Theorem 3
Since The second inequality follows from the fact that a n ≡ 0 for |n| ≥ m 0 , while the third one results from the relation a 0 > 0 and the positivity, σ > 0, of all nontrivial eigenvalues σ , which imply that Making an identical argument, but solving forû(n − m 0 ), and using that |a m 0 | = |a −m 0 | = 0, we have for all |n| ≤ K σ , Next, if 2m 0 ≤ , then An almost identical argument gives the −K σ − m ≤ n ≤ 0 case.

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 σ . To finish the proof, we use the fact that the harmonic extension of e inθ decays exactly as r |n| . Examining the solution on the ball of radius δ > 0 for some δ small enough, it will be shown that the low frequencies dominate the behavior of u.

Proof of Theorem 2
In what follows we utilize the definitions (1.7) for a given eigenvalue σ j = σ , and, for that eigenvalue we denoteû(k) =û σ j (k) =û σ (k). Then, applying the relation which is valid for all sequences {b k } k∈Z ⊂ C, to the right-hand equation in (1.7), for m ≥ m 0 we obtain To estimate the error in approximating u σ,δ byũ σ,δ,m , first note that Applying Lemma 4.1 with m = 0, and absorbing the σ N into the exponential factor we then obtain |k|≥2σû (k)r |k| e ikθ where A = A 0 is given by We can now estimate provided that δ < 1 2 C −1 0 . Therefore, using (4.2), Thus, choosing δ > 0 such that δ < e −2C and taking σ 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 1 ⊂ C satisfying (1.2) for the given value ε > 0. Let σ 0 be as in Theorem 2. Clearly, it suffices to prove the statement of the theorem for σ > σ 0 , since for σ ≤ σ 0 the statement follows from the fact that there are finitely many Steklov eigenvalues below σ 0 and that φ σ cannot vanish in any open set. Therefore, we may and do assume σ > σ 0 along with the other assumptions in Theorem 2, so that, in particular, inequality (1.8) holds. In what follows we write Fixing m ≥ m 0 + 2, and lettingũ σ,δ andũ σ,δ,m be given by (1.7) (with u σ related to φ σ via (1.3)) we note that It follows that there exists x 0 ∈ B(0, δ) such that Now, since ũ σ,δ,m C 1 ≤ C m,δ ũ σ,δ,m L 2 , it follows from (4.4) that there is r m,δ ∈ R, 0 < r m,δ < δ (in particular, independent of σ ) such that But, since m ≥ m 0 + 2, the estimate (1.8) with N = 0 yields and (4.7) (To establish the rightmost inequality in (4.7) the relation ũ σ,δ −ũ σ,δ,m L 2 δ ≤ √ πδ ũ σ,δ −ũ σ,δ,m L ∞ δ was used before the inequality (1.8) was applied.) From (4.7) we obtain It follows from (4.5), (4.6) and (4.8) that (4.9) and, therefore Taking δ 1 sufficiently small and δ ≤ δ 1 the inequality holds, and it therefore follows that for a certain constant D > 0 we have provided δ < δ 1 . In particular, Since the derivative of f never vanishes, for δ < δ 1 and for a certain E > 0 there is a ball B of radius Er m,δ such that φ σ does not vanish on B. The proof is now complete.

Integral Representation
Let ⊂ R 2 denote a domain with, say, a C 2 boundary, and let denote the Single Layer Potential (SLP) for a given density φ : ∂ → 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 [10,12]. 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 ∂ can be expressed in terms of well known "jump conditions" that involve the single and double layer boundary integral operators respectively.
In view of the jump conditions for the SLP [10], use of the representation for the eigenfunction u, the Steklov boundary condition in Eq. (1.1) gives rise to the generalized eigenvalue problem Unfortunately, however, the single layer operator 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 (where φ denotes the average of φ over ∂ ), which leads to the modified eigenvalue equation [1] The right-hand operator in this equation is invertible [10, Theorem 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 φ of the corresponding generalized eigenvalue problem (5.2) or (5.4), respectively, at all required points x ∈ .

Remark 5.1
Note that for a given harmonic function u in , φ in (5.2) and that in (5.4) are not the same.

Fourier Expansion and Exponential Decay
In terms of a given 2π -periodic parametrization C(t) of ∂ , the Steklov eigenfunction u corresponding to a given solution (φ, σ ) of the regularized eigenvalue problem (5.4), which is given by the single layer expression (5.3), can be expressed, for a given point where C(t) = (C 1 (t), C 2 (t)) and where φ denotes the average of φ over the curve ∂ . Unfortunately, a direct use of this expression does not capture important elements in the eigenfunction within , such as the nodal sets, since, for analytic domains, the eigenfunctions decay exponentially fast within as the frequency increases [5,13]. 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 solution φ 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 [13].
To accurately obtain the exponentially decaying values of the Steklov eigenfunction we proceed as follows. We first consider the Fourier expansion A n e int .
(5.6) of the product φ(C(t)) − φ Ċ (t) ; 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 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 ) ∈ , the quantities λ(x) = sup {s ≥ 0 : x = C(t + ir) for all r with |r | ≤ s and for all t ∈ [0, 2π ]} and B n (x 1 , x 2 , s) Using Cauchy's Theorem for x = (x 1 , x 2 ) ∈ and any s ∈ R satisfying |s| ≤ λ(x), we obtain and, thus, letting s = αλ(x) for any α ∈ R satisfying |α| ≤ 1, the eigenfunction u is given by Lemma 5.2 There is C > 0 such that for all n > 0, Moreover, there is c > 0 and a sequence {n k } ∞ k=1 with |n k | → ∞ such that A proof of Lemma 5.2 is given in Appendix B. It follows from Lemma 5.2 that Eq. (5.8) optimally captures the exponential decay of the B n terms as σ → ∞. Note that this setup does not capture the exponential decay of the coefficients A n below machine precision away from |n| ∼ σ , and, therefore, the accuracy of the resulting interior eigenfunction reconstructions does not exceed that accuracy level. But the function λ(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 λ(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 λ was evaluated via an application of Newton's method to the nonlinear equation Explicit expressions can be obtained for circles and ellipses, however: (1) For a circle of radius 1: (2) For an ellipse of semiaxes a > b: The derivation of the expression (5.11) is outlined in Appendix A.

Exponential Decay and Verification of Cauchy's Theorem
Tables 1 and 2 demonstrate the validity of Eq. (5.8) (since in both cases the results in the second and third columns closely agree with each other for n ≤ 50), as well as the exponential decay of the exact coefficients B 0 n -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). Table 1 Verification of the Cauchy-theorem-based identity (5.8) for the domain bounded by the elliptical curve (6.1) with a = 2 and b = 1 with a = 2 and b = 1, and respectively. These figures demonstrate, in particular, domain-opening and nondensity of nodal sets as discussed in Remark 1.2.
As suggested in the caption to Fig. 4, the question might be considered as to whether the Steklov problem for the ellipse can be reduced to solution of decoupled separatedvariables one-dimensional problems in elliptical coordinates. As mentioned in that caption, however, a simple calculation shows that the Steklov problem does not separate in those variables. Note also that the observed domain-opening and non-density of nodal sets are inconsistent with separation of variables-since, if variables separate, a simple eigenfunction can be written as where μ and τ are as in Appendix A. But, since T (τ ) must have 1/σ dense zeros set and |G(μ 0 )| > 0, we obtain that the nodal set is 1/σ dense throughout the ellipse which is indeed inconsistent with the fixed sign domains presented in the figure. An analogous argument can be used for high energy eigenfunctions, and for an arbitrary domain for which reducibility to separation of variables is assumed, even in cases involving multiple eigenvalues, once the work of Shamma [14] describing the asymptotic structure of Steklov eigenfunctions is incorporated. Thus, any bounded analytic and simply-connected domain for which the Steklov eigenproblem can be reduced to problems of separated variables cannot exhibit asymptotic domain-openings. It follows that any such domain cannot be a tunneling domain and must thus equal a disc-at least if Conjecture 1.3 holds. by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Similarly, if h vanishes to order k at z
where is any contour starting at z 0 − ε 2 , ending at z 0 + ε 1 , and lying in In particular, let Letting ε 1 and ε 2 tend to zero completes the proof for the case z 0 = t 0 + iλ. The proof for z 0 = t 0 − iλ follows by substituting z by −z.
Similarly if h vanishes to order k at z 0 = t 0 − iλ, we have S 1 Proof We consider the first case, the second follows similarly. Selecting χ(t) with sufficiently small support we ensure that, within the support of χ , h(t + iλ) vanishes only at t = t 0 . We then have  where in the last equality Lemma B.1 was used.
We may now complete the proof of Lemma 5.2. Let 0 ≤ t 1 < t 2 < . . . < t M < 2π denote the zeroes of h(t + iλ) as a function of t, and let k j (0 ≤ j ≤ M) denote the vanishing order at t = t j . Then, by Lemma B.2, for χ j supported close enough to t j with χ j ≡ 1 near t j , and n > 0, χ j (t) log h(t + iλ)e int dt = − 2π k j e int j |n| + O(n −2 ).