Eigenvalues of Robin Laplacians in infinite sectors

TL;DR

This paper analyzes the eigenvalues of Robin Laplacians in infinite sectors, revealing how the spectrum depends on the sector's angle and providing asymptotic behavior as the angle approaches zero.

Contribution

It provides a detailed spectral analysis of Robin Laplacians in sectors, including eigenvalue behavior, dependence on the sector angle, and asymptotic expansions, extending to star graph interactions.

Findings

Discrete spectrum is non-empty iff angle < π/2

Number of eigenvalues grows as angle approaches zero

Eigenvalues have explicit asymptotic formulas as angle tends to zero

Abstract

For $\alpha\in(0,\pi)$, let $U_\alpha$ denote the infinite planar sector of opening $2\alpha$, \[ U_\alpha=\big\{ (x_1,x_2)\in\mathbb R^2: \big|\arg(x_1+ix_2) \big|<\alpha \big\}, \] and $T^\gamma_\alpha$ be the Laplacian in $L^2(U_\alpha)$, $T^\gamma_\alpha u= -\Delta u$, with the Robin boundary condition $\partial_\nu u=\gamma u$, where $\partial_\nu$ stands for the outer normal derivative and $\gamma>0$. The essential spectrum of $T^\gamma_\alpha$ does not depend on the angle $\alpha$ and equals $[-\gamma^2,+\infty)$, and the discrete spectrum is non-empty iff $\alpha<\frac\pi 2$. In this case we show that the discrete spectrum is always finite and that each individual eigenvalue is a continous strictly increasing function of the angle $\alpha$. In particular, there is just one discrete eigenvalue for $\alpha \ge \frac{\pi}{6}$. As $\alpha$ approaches $0$, the number of discrete eigenvalues becomes arbitrary large and is minorated by $\kappa/\alpha$ with a suitable $\kappa>0$, and the $n$th eigenvalue $E_n(T^\gamma_\alpha)$ of $T^\gamma_\alpha$ behaves as \[ E_n(T^\gamma_\alpha)=-\dfrac{\gamma^2}{(2n-1)^2 \alpha^2}+O(1) \] and admits a full asymptotic expansion in powers of $\alpha^2$. The eigenfunctions are exponentially localized near the origin. The results are also applied to $\delta$-interactions on star graphs.