# On the spectrum of leaky surfaces with a potential bias

## TL;DR

This paper investigates the spectral properties of a quantum operator with a surface delta interaction and a potential bias, focusing on the existence of discrete eigenvalues in relation to the geometry and potential bias.

## Contribution

It provides a detailed analysis of the essential and discrete spectra of the operator, especially at the critical potential bias case, revealing conditions for the existence of eigenvalues.

## Key findings

- Discrete spectrum is empty when the bias is supported in the exterior region.
- Isolated eigenvalues may exist if the bias is supported in the interior region.
- The essential spectrum is characterized for the operator with surface delta interaction.

## Abstract

We discuss operators of the type $H = -\Delta + V(x) - \alpha \delta(x-\Sigma)$ with an attractive interaction, $\alpha>0$, in $L^2(\mathbb{R}^3)$, where $\Sigma$ is an infinite surface, asymptotically planar and smooth outside a compact, dividing the space into two regions, of which one is supposed to be convex, and $V$ is a potential bias being a positive constant $V_0$ in one of the regions and zero in the other. We find the essential spectrum and ask about the existence of the discrete one with a particular attention to the critical case, $V_0=\alpha^2$. We show that $\sigma_\mathrm{disc}(H)$ is then empty if the bias is supported in the `exterior' region, while in the opposite case isolated eigenvalues may exist.