Spectral Analysis for Matrix Hamiltonian Operators
Jeremy L. Marzuola, Gideon Simpson
TL;DR
This paper investigates the spectral properties of matrix Hamiltonians related to the nonlinear Schrödinger equation, providing a new algorithm for verifying spectral stability and confirming the absence of embedded eigenvalues in 3D cubic cases.
Contribution
It introduces a novel algorithm for spectral verification and applies it to prove the absence of embedded eigenvalues in 3D cubic Schrödinger Hamiltonians.
Findings
No embedded eigenvalues in 3D cubic Hamiltonians
New algorithm for spectral property verification
Source code available for further research
Abstract
In this work, we study the spectral properties of matrix Hamiltonians generated by linearizing the nonlinear Schr\"odinger equation about soliton solutions. By a numerically assisted proof, we show that there are no embedded eigenvalues for the three dimensional cubic equation. Though we focus on a proof of the 3d cubic problem, this work presents a new algorithm for verifying certain spectral properties needed to study soliton stability. Source code for verification of our comptuations, and for further experimentation, are available at http://www.math.toronto.edu/simpson/files/spec_prop_code.tgz.
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