Homogenization of an indefinite spectral problem arising in population genetics

TL;DR

This paper analyzes the asymptotic behavior of an indefinite spectral problem in a thin cylinder with sign-changing density, deriving an effective 1D harmonic oscillator model relevant to population genetics.

Contribution

It introduces a novel asymptotic analysis for indefinite spectral problems with sign-changing weights in thin domains, including a new effective spectral problem and convergence results.

Findings

Derived a 1D harmonic oscillator spectral problem as the limit

Proved spectrum convergence as the cylinder thickness tends to zero

Established existence of a positive principal eigenvalue in indefinite problems

Abstract

We study an indefinite spectral problem for a second-order self-adjoint elliptic operator in an asymptotically thin cylinder. The operator coefficients and the spectral density function are assumed to be locally periodic in the axial direction of the cylinder. The key assumption is that the spectral density function changes sign, which leads to infinitely many both positive and negative eigenvalues. The asymptotic behavior of the spectrum, as the thickness of the rod tends to zero, depends essentially on the sign of the average of the density function. We study the positive part of the spectrum in a specific case when the local average is negative. We derive a one-dimensional effective spectral problem that is a harmonic oscillator on the real line, and prove the convergence of spectrum. A key auxiliary result is the existence of a positive principal eigenvalue of an indefinite spectral problem with the Neumann boundary condition on a periodicity cell. This study is motivated by applications in population genetics where spectral problems with sign-changing weight naturally appear.