Families of spherical caps: spectra and ray limit

TL;DR

This paper analyzes the spectral properties of spherical caps, deriving trace formulas for their eigenvalues, exploring the degenerate spectrum near hemispheres, and calculating first perturbative corrections due to curvature and boundaries.

Contribution

It introduces a new trace formula for spherical caps, examines the spectrum near the hemispherical limit, and computes the first perturbative corrections for curved domains with boundaries.

Findings

Spectral density exhibits degeneracy near the hemisphere.

Derived trace formula breaks down as caps approach the hemisphere.

Calculated leading perturbative correction for curved domains with boundaries.

Abstract

We consider a family of surfaces of revolution ranging between a disc and a hemisphere, that is spherical caps. For this family, we study the spectral density in the ray limit and arrive at a trace formula with geodesic polygons describing the spectral fluctuations. When the caps approach the hemisphere the spectrum becomes equally spaced and highly degenerate whereas the derived trace formula breaks down. We discuss its divergence and also derive a different trace formula for this hemispherical case. We next turn to perturbative corrections in the wave number where the work in the literature is done for either flat domains or curved without boundaries. In the present case, we calculate the leading correction explicitly and incorporate it into the semiclassical expression for the fluctuating part of the spectral density. To the best of our knowledge, this is the first calculation of such perturbative corrections in the case of curvature and boundary.