Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum

Gesztesy, Fritz; Simon, Barry

doi:10.1090/S0002-9947-99-02544-1

Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum
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by Fritz Gesztesy and Barry Simon PDF

Trans. Amer. Math. Soc. 352 (2000), 2765-2787

Abstract:

We discuss results where the discrete spectrum (or partial information on the discrete spectrum) and partial information on the potential $q$ of a one-dimensional Schrödinger operator $H=-\frac {d^{2}}{dx^{2}}+q$ determine the potential completely. Included are theorems for finite intervals and for the whole line. In particular, we pose and solve a new type of inverse spectral problem involving fractions of the eigenvalues of $H$ on a finite interval and knowledge of $q$ over a corresponding fraction of the interval. The methods employed rest on Weyl $m$-function techniques and densities of zeros of a class of entire functions.

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