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Positive Solutions of Schrödinger Equations in Product form and Martin Compactifications of the Plane, II

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Abstract

We determine the structure of all positive solutions to a Schrödinger equation (−Δ + V1(x1) + V2(x2))u = 0 on \(\mathbb R^{2}\), where real potentials Vj, j = 1,2, satisfy \(V_{j}\in {L^{1}_{1}} =\{V; (1+|t|)V(t)\in L^{1}(\mathbb R)\}\). We also treat the case where \(V_{1}\in {L^{1}_{1}}\) and V2 belongs to a wide class of functions including model potentials V2(t) = |t|a, a > 0. We show that non-minimal Martin boundary points appear generically. On analysis of asymptotics of the Green functions of the Schrödinger equations, the Jost solutions of one dimensional Schrödinger equations with potential functions in \({L_{1}^{1}}\) play a central role.

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Authors and Affiliations

  1. 5-30-1 Shinyoshida-higashi, Kohoku-ku, Yokohama, 223-0058, Japan

    Minoru Murata

  2. Department of Mathematics, Meijo University, Shiogamaguchi, Tenpaku-ku, Nagoya, 468-8502, Japan

    Tetsuo Tsuchida

Authors
  1. Minoru Murata
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  2. Tetsuo Tsuchida
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Correspondence to Tetsuo Tsuchida.

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Murata, M., Tsuchida, T. Positive Solutions of Schrödinger Equations in Product form and Martin Compactifications of the Plane, II. Potential Anal 59, 519–563 (2023). https://doi.org/10.1007/s11118-021-09977-2

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