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Structure of positive solutions to \((-\Delta +V)u=0\) in \(R^ n\). (English) Zbl 0624.35023

The structure of all positive solutions of the Schrödinger equation \((1)\quad (-\Delta +V)u=0\) in \({\mathbb{R}}^ n,\) where the potential V is assumed to be a real-valued function belonging to \(L^ p_{loc}({\mathbb{R}}^ n)\) with \(p>n/2\) for \(n\geq 2\) and p=1 for n=1, is investigated. Three classes of potentials are distinguished: V is called subcritical, if \(-\Delta +V\) has a positive Green’s function, V is called critical, if \(-\Delta +V\) is nonnegative and does not have a positive Green’s function, and V is called supercritical, if \(-\Delta +V\) is not nonnegative. Positive solutions of (1) exist, if and only if \(-\Delta +V\geq 0.\) Then the following results are established
1. A subcritical potential is characterized by stability under small local perturbations, a critical one by unstability.
2. If V is critical then (1) has only one positive solution modulo constant multiple.
3. A potential V, for which (1) has a positive \(L^ 2\)-solution, is critical, whereas the converse is false: the positive solutions for a critical potential can behave very badly at infinity.
4. For subcritical potentials a method of R. S. Martin is applied to show that any positive solution of (1) is represented uniquely by \[ u(x)=\int_{\sigma}P(x,\omega)d\mu (\omega). \] The purpose of the paper is to construct explicitly \(\sigma\) (the minimal Martin boundary), \(P(x,\omega)\) and the measure \(d\mu(\omega)\).
Both, “principally” radially symmetric potentials and potentials, which are nonisotropic in an extreme way, are treated. In the former case the asymptotics of the solutions at infinity are determined explicitly. Also Dirichlet’s problem with boundary conditions at infinity is then solved.
Reviewer: R.Weikard

MSC:

35J10 Schrödinger operator, Schrödinger equation
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35C20 Asymptotic expansions of solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
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