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On the absolutely continuous and negative discrete spectra of Schrödinger operators on the line with locally integrable globally square summable potentials. (English) Zbl 1068.81024

Summary: For one-dimensional Schrödinger operators with potentials \(q\) subject to \[ \sum_{n=-\infty}^{\infty}\left(\int_{n}^{n+1}\bigl| q(x)\bigr|\text{d}x\right)^2< \infty, \] we prove that the absolutely continuous spectrum is \([0,\infty)\), extending the 1999 result due to Dieft–Killip. As a by-product we show that under the same condition the sequence of the negative eigenvalues is \(3/2\)-summable improving the relevant result by Lieb–Thirring.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47E05 General theory of ordinary differential operators
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