TY - JOUR

T1 - A spectral theory of linear operators on rigged hilbert spaces under analyticity conditions II

T2 - Applications to SchrÖdinger operators

AU - Chiba, Hayato

N1 - Funding Information:
This work was supported by a Grant-in-Aid for Young Scientists (B), no. 22740069 from MEXT Japan.
Publisher Copyright:
© 2018 Faculty of Mathematics, Kyushu University.

PY - 2018

Y1 - 2018

N2 - 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 D(λ) is constructed so that D(λ) = 0 if and only if λ is a generalized eigenvalue. It is proved that D(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.

AB - 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 D(λ) is constructed so that D(λ) = 0 if and only if λ is a generalized eigenvalue. It is proved that D(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.

KW - Generalized spectrum

KW - Resonance pole

KW - Rigged hilbert space

KW - Schrödinger operator

KW - Spectral theory

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U2 - 10.2206/kyushujm.72.375

DO - 10.2206/kyushujm.72.375

M3 - Article

AN - SCOPUS:85057598149

SN - 1340-6116

VL - 72

SP - 375

EP - 405

JO - Kyushu Journal of Mathematics

JF - Kyushu Journal of Mathematics

IS - 2

ER -