Spectral properties in the low-energy limit of one-dimensional Schrödinger operators H = –d2/dx2 + V. The Case 〈1, V1〉 ≠ 0

Melgaard, Michael (2002) Spectral properties in the low-energy limit of one-dimensional Schrödinger operators H = –d2/dx2 + V. The Case 〈1, V1〉 ≠ 0. Mathematische Nachrichten, 238 (1). pp. 113-143. ISSN 0025-584X

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Abstract

In this paper we consider the Schrödinger operator H = –d2/dx2+ V in L2(ℝ), where V satisfies an abstract short-range condition and the (solvability) condition 〈1, V1〉 ≠ 0. Spectral properties of H in the low-energy limit are analyzed. Asymptotic expansions for R(ζ) = (H – ζ)–1 and the S-matrix S(λ) are deduced for ζ → 0 and λ ↓ 0, respectively. Depending on the zero-energy properties of H, the expansions of R(ζ) take different forms. Generically, the expansions of R(ζ) do not contain negative powers; the appearance of negative powers in ζ1/2 is due to the possible presence of zero-energy resonances (half-bound states) or the eigenvalue zero of H (bound state), or both. It is found that there are at most two zero resonances modulo L2-functions.

Item Type: Article
Keywords: Asymptotic expansions; resolvent; bound and half-bound states; S-matrix
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics > QA0299 Analysis. Including analytical methods connected with physical problems
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Depositing User: Michael Melgaard
Date Deposited: 19 Sep 2013 13:52
Last Modified: 19 Sep 2013 13:52
URI: http://sro.sussex.ac.uk/id/eprint/46393
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