Inverse Spectral Problem with Partial Information¶on the Potential: The Case of the Whole Real Line
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The Schrödinger operator \(\) is considered on the real axis. We discuss the inverse spectral problem where discrete spectrum and the potential on the positive half-axis determine the potential completely. We do not impose any restrictions on the growth of the potential but only assume that the operator is bounded from below, has discrete spectrum, and the potential obeys \(\). Under these assertions we prove that the potential for x≥ 0 and the spectrum of the problem uniquely determine the potential on the whole real axis. Also, we study the uniqueness under slightly different conditions on the potential. The method employed uses Weyl m-function techniques and asymptotic behavior of the Herglotz functions.
KeywordsAsymptotic Behavior Real Axis Discrete Spectrum Spectral Problem Partial Information
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