Spectrum of the Dirichlet Laplacian in sheared waveguides

Abstract

Let \(\Omega \subset {\mathbb {R}}^3\) be a sheared waveguide, i.e., \(\Omega \) is built by translating a cross section in a constant direction along an unbounded spatial curve. Consider \(-\Delta _{\Omega }^D\) the Dirichlet Laplacian operator in \(\Omega \). Under the condition that the tangent vector of the reference curve admits a finite limit at infinity, we find the essential spectrum of \(-\Delta _{\Omega }^D\). Then, we state sufficient conditions that give rise to a non-empty discrete spectrum for \(-\Delta _{\Omega }^D\); in particular, we show that the number of discrete eigenvalues can be arbitrarily large since the waveguide is thin enough.

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Correspondence to Alessandra A. Verri.

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A Appendix

A Appendix

Stability of the essential spectrum

This appendix is dedicated to the proof of Proposition 2. We use the arguments of [4]. In that work, the authors employed a different characterization of the essential spectrum which can be adapted to our problem. In fact,

Lemma 1

A real number \(\lambda \) belongs to the essential spectrum of \(H_{f', g'}\) if, and only if, there exists a sequence \(\{\psi _n\}_{n=1}^\infty \subset {\mathrm {dom}~}b_{f',g'}\) such that the following conditions hold:

  1. (i)

    \(\Vert \psi _n\Vert =1\), for all \( n \ge 1\);

  2. (ii)

    \((H_{f', g'}-\lambda {\mathbf {1}})\psi _n \rightarrow 0\), as \(n \rightarrow \infty \), in the norm of the dual space \(({\mathrm {dom}~}b_{f', g'})^*\);

  3. (iii)

    \(\mathrm{supp}\,\psi _n \subset Q \backslash (-n,n) \times S\), for all \(n \ge 1\).

The proof of Lemma 1 is very similar to the proof of Lemma 4.1 of [4], and it will be omitted in this text.

Proof of Proposition 2

Let \(\lambda \in \sigma _{ess}(H_{\beta _1, \beta _2})\). By Lemma 1, there exists a sequence \(\{\psi _n\}_{n=1}^\infty \subset {\mathrm {dom}~}b_{\beta _1, \beta _2}\) such that the conditions \((i)-(iii)\) are satisfied. Denote the norm in \(({\mathrm {dom}~}b_{\beta _1, \beta _2})^*\) by \(\Vert \cdot \Vert _-\); one has \(\Vert \cdot \Vert _- = \Vert (H_{\beta _1, \beta _2}+{\mathbf {1}})^{-1/2} \cdot \Vert \). For each \(n \in {\mathbb {N}}\), write

$$\begin{aligned} \psi _n = (H_{\beta _1, \beta _2}+{\mathbf {1}})^{-1} (H_{\beta _1, \beta _2}-\lambda {\mathbf {1}}) \psi _n +(1+\lambda ) (H_{\beta _1, \beta _2}+ {\mathbf {1}})^{-1} \psi _n. \end{aligned}$$

By (ii), we can see that the sequence \(\{\psi _n\}_{n=1}^\infty \) is bounded in \({\mathrm {dom}~}b_{\beta _1, \beta _2}\).

Now, for simplicity, write

$$\begin{aligned} \tau _1(x):= f'(x) - \beta _1, \quad \tau _2(x):= g'(x) - \beta _2. \end{aligned}$$

Some calculations show that

$$\begin{aligned} b_{f',g'}(\varphi , \psi _n) - \lambda \langle \varphi , \psi _n \rangle&= b_{\beta _1, \beta _2} (\varphi , \psi _n) - \lambda \langle \varphi , \psi _n \rangle \\&\quad + \int \limits _\Lambda \left( -\tau _1 \frac{\partial \varphi }{\partial y_1} -\tau _2 \frac{\partial \varphi }{\partial y_2}\right) \left( \psi _n' - \beta _1 \frac{\partial \psi _n}{\partial y_1} - \beta _2 \frac{\partial \psi _n}{\partial y_2}\right) {\mathrm {d}}x{\mathrm {d}}y\\&\quad + \int \limits _\Lambda \left( \varphi ' - (\beta _1 + \tau _1) \frac{\partial \varphi }{\partial y_1} - (\beta _2+\tau _2) \frac{\partial \varphi }{\partial y_2}\right) \left( -\tau _1 \frac{\partial \psi _n}{\partial y_1} -\tau _2 \frac{\partial \psi _n}{\partial y_2}\right) {\mathrm {d}}x{\mathrm {d}}y. \end{aligned}$$

Since \(\{\psi _n\}_{n=1}^\infty \) is bounded in \(H_0^1(\Lambda )\), and \(\Vert \partial \varphi / \partial y_1\Vert ^2, \Vert \partial \varphi / \partial y_2\Vert ^2 \le b_{f',g'}(\varphi ) = \Vert \varphi \Vert _+^2\), one has

$$\begin{aligned}&\sup _{0 \ne \varphi \in H_0^1(\Lambda )} \left\{ \int \limits _\Lambda \left( -\tau _1 \frac{\partial \varphi }{\partial y_1} -\tau _2 \frac{\partial \varphi }{\partial y_2}\right) \left( \psi _n' - \beta _1 \frac{\partial \psi _n}{\partial y_1} - \beta _2 \frac{\partial \psi _n}{\partial y_2}\right) {\mathrm {d}}x{\mathrm {d}}y/ \Vert \varphi \Vert _+ \right\} \\&\qquad \le \left( \Vert \tau _1\Vert _{L^\infty ({\mathbb {R}} \backslash (-n,n))} + \Vert \tau _2\Vert _{L^\infty ({\mathbb {R}} \backslash (-n,n))} \right) \left( \Vert \psi _n'\Vert + \beta _1 \left\| \frac{\partial \psi _n}{\partial y_1} \right\| + \beta _2 \left\| \frac{\partial \psi _n}{\partial y_2} \right\| \right) \rightarrow 0, \end{aligned}$$

and

$$\begin{aligned}&\sup _{0 \ne \varphi \in H_0^1(\Lambda )} \left\{ \int \limits _\Lambda \left( \varphi ' - (\beta _1 + \tau _1) \frac{\partial \varphi }{\partial y_1} - (\beta _2+\tau _2) \frac{\partial \varphi }{\partial y_2}\right) \left( -\tau _1 \frac{\partial \psi _n}{\partial y_1} -\tau _2 \frac{\partial \psi _n}{\partial y_2}\right) {\mathrm {d}}x{\mathrm {d}}y/ \Vert \varphi \Vert _+ \right\} \\&\qquad \le \Vert \tau _1\Vert _{L^\infty ({\mathbb {R}} \backslash (-n,n))} \left\| \frac{\partial \psi _n}{\partial y_1} \right\| + \Vert \tau _2\Vert _{L^\infty ({\mathbb {R}} \backslash (-n,n))} \left\| \frac{\partial \psi _n}{\partial y_2} \right\| \rightarrow 0, \end{aligned}$$

as \(n \rightarrow \infty \). Then, \(\lambda \in \sigma _{ess}(H_{f',g'})\). The inclusion \(\sigma _{ess}(H_{f',g'}) \subset \sigma _{ess}(H_{\beta _1, \beta _2})\) can be obtained in a similar way.

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Verri, A.A. Spectrum of the Dirichlet Laplacian in sheared waveguides. Z. Angew. Math. Phys. 72, 23 (2021). https://doi.org/10.1007/s00033-020-01444-z

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Keywords

  • Sheared waveguides
  • Dirichlet Laplacian
  • Essential spectrum
  • Discrete spectrum

Mathematics Subject Classification

  • Primary: 35R45
  • 81Q10
  • Secondary: 35P20
  • 47F05
  • 58J50