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.

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:

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

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

• (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.