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Solving fractional Schrödinger-type spectral problems: Cauchy oscillator and Cauchy well. (English) Zbl 1304.81069
The authors present the results of a computer-assisted route to obtain the eigenvalues and the eigenfunctions of the \(1D\) Cauchy-Schrödinger operator \(H=(-\Delta)^{1/2} + V\), \(V\) being a local potential. The Cauchy oscillator (which also has an analytical solution) and Cauchy finite well spectral problems are mainly envisaged. The algorithms employed are a non-local version of Strang’s splitting method, which is based on Trotter product formula.

MSC:
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35S05 Pseudodifferential operators as generalizations of partial differential operators
35R11 Fractional partial differential equations
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