# Singular limits of sign-changing weighted eigenproblems

## TL;DR

This paper investigates how the spectrum of differential eigenvalue problems with sign-changing weights behaves as the negative weight region is scaled to negative infinity, leading to a limiting problem with Dirichlet conditions on the interface.

## Contribution

It establishes the convergence of spectra for sign-changing weighted eigenproblems under extreme rescaling, extending to various PDEs with different boundary conditions.

## Key findings

- Spectrum converges to a restricted problem with Dirichlet conditions.
- Results apply to second and fourth order PDEs with various boundary conditions.
- Provides a unified framework for eigenproblems with sign-changing weights.

## Abstract

Consider the eigenvalue problem generated by a fixed differential operator with a sign-changing weight on the eigenvalue term. We prove that as the negative part of the weight is rescaled towards negative infinity on some subregion, the spectrum converges to that of the original problem restricted to the complementary region. On the interface between the regions the limiting problem acquires Dirichlet-type boundary conditions. Our main theorem concerns eigenvalue problems for sign-changing bilinear forms on Hilbert spaces. We apply our results to a wide range of PDEs: second and fourth order equations with both Dirichlet and Neumann-type boundary conditions, and a problem where the eigenvalue appears in both the equation and the boundary condition.