Curved planar quantum wires with Dirichlet and Neumann boundary conditions

TL;DR

This paper studies the spectral properties of a quantum particle confined in a curved two-dimensional strip with mixed boundary conditions, revealing how the total bending angle influences the existence of bound states.

Contribution

It provides a detailed analysis of the discrete spectrum for curved quantum wires with Dirichlet and Neumann boundary conditions, highlighting the role of bending in spectral properties.

Findings

Discrete eigenvalues depend on the total bending angle.

Bound states exist if the total bending angle is positive.

The spectral threshold is influenced by the strip's curvature.

Abstract

We investigate the discrete spectrum of the Hamiltonian describing a quantum particle living in the two-dimensional curved strip. We impose the Dirichlet and Neumann boundary conditions on opposite sides of the strip. The existence of the discrete eigenvalue below the essential spectrum threshold depends on the sign of the total bending angle for the asymptotically straight strips.