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Mathematics Colloquium: Spectral Analysis of Schrödinger Operators with Two-Scale Potentials

Oct 24, 2025, 3:30 - 4:30 PM

 

Speaker:  Emmanuel Fleurantin, GMU


Title:  Spectral Analysis of Schrödinger Operators with Two-Scale Potentials
 

Abstract:  We study Schrödinger operators of the form Δ − Wε on the space of radially symmetric square-integrable functions. While such operators appear in quantum mechanics, our interest is in their spectral properties as differential operators. We consider potentials Wε = V0 + V1, ε that decompose into two components acting at distinct spatial scales. The component V0 acts at a fixed scale, while V1, ε(|x|) = ε2V1(ε|x|) represents a scaled potential, where the scaling parameter ε controls the separation between the two scales. Both potentials decay appropriately at infinity. Our main result addresses how eigenvalues accumulate on the positive real axis when these two scales are present. In the limit ε → 0, the total number of positive eigenvalues of Δ − Wε equals the sum of positive eigenvalues from the two separate problems: Δ − V0 and Δ − V1. Our analysis combines dynamical systems techniques with separation of scales arguments, providing a framework for understanding how spectral properties of multi-scale operators decompose into contributions from individual scales.

 

Time:  Friday, October 24, 3:30pm – 4:30pm

Place:  Exploratory Hall, room 4106 or Zoom

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