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Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator with singular potentials. (English) Zbl 1004.34077

The authors consider the one-dimensional Schrödinger operator with singular, complex-valued, 1-periodic potential \(V\), whose Fourier coefficients \(\widehat{V}(k)\) satisfy \((|k|^{-\alpha}|\widehat{V}(k)|)_{k\in {\mathbb Z}}\in l^2\) for some \(\alpha \in [0,1)\). With eigenvalues ordered primarily by their real parts, they derive the asymptotic estimates \[ n^2\pi ^2 +\widehat{V}(0)-\left(\widehat{V}(-2n)+\widehat{V}(2n)\right)/2+ O\left(n^{\varepsilon +2\alpha -1}\right) \] for the \(n\)th Dirichlet eigenvalue and \[ n^2\pi ^2 +\widehat{V}(0)\pm \left(\widehat{V}(-2n)\widehat{V}(2n)\right)^{1/2} +O\left(n^{\varepsilon +(3\alpha -1)/2}\right) \] for the \(n\)th pair of periodic eigenvalues as \(n\to \infty\). Both estimates hold for all \(\varepsilon >0\).

MSC:

34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47E05 General theory of ordinary differential operators
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