2015/05/23 by Hayato Chiba, Chiba, Hayato
Mathematics · Physics and Astronomy · #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory in Mathematical Physics #math.FA
paper · pdf · doi:10.48550/arxiv.1505.06267
arxiv created 2015/05/23 · openalex publication_date 2015/05/23 · arxiv updated 2015/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A spectral theory of linear operators on a rigged Hilbert space is applied to Schrödinger operators with exponentially decaying potentials and dilation analytic potentials. The theory of rigged Hilbert spaces provides a unified approach to resonances (generalized eigenvalues) for both classes of potentials without using any spectral deformation techniques. Generalized eigenvalues for one dimensional Schrödinger operators (ordinary differential operators) are investigated in detail. A certain holomorphic function \mathbbD(λ) is constructed so that \mathbbD(λ) = 0 if and only if λ is a generalized eigenvalue. It is proved that \mathbbD(λ) is equivalent to the analytic continuation of the Evans function. In particular, a new formulation of the Evans function and its analytic continuation is given.