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Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class \((V)_ a\). (English) Zbl 0702.35228

The Schrödinger equation with cubic nonlinearity \[ i\hslash \cdot \partial \psi /\partial t=-(\hslash^ 2/2m)\partial^ 2\psi /\partial x^ 2+V\psi -\gamma | \psi^ 2| \psi \] has been studied by A. Floer and A. Weinstein [J. Funct. Anal. 69, 397-408 (1986; Zbl 0613.35076)]. However their proof contains an error and the existence of bound semiclassical states requires further investigation. The author makes use of a class of potentials (class “V”) introduced by T. Kato in 1984 which ensure the Schrödinger operator to be selfadjoint. Defining \[ S_{\hslash}(u)=-(1/2)d^ 2u/dy^ 2+(V_{\hslash}-E)-u^ 3;\quad S_{\hslash}: H_{\hslash}\to L^ 2, \] and Sobolev embedding \(D(H_{\hslash})\hookrightarrow H^ 2\hookrightarrow L^ 6\) he shows that \(S_{\hslash}(u)\) is Fréchet differentiable. This permits the author to deduce elaborate estimates of \(S_{\hslash}(u)\) in terms of \(\hslash\) and the estimates of the Fredholm inverse of \(S'_{\hslash}\). This in turn leads to a proof of local existence of solutions for “small” \(\hslash\). Some of these results are not valid for rapidly oscillating potentials.
Reviewer: V.Komkov

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)

Citations:

Zbl 0613.35076
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References:

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