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Spiked harmonic oscillators. (English) Zbl 1059.81044
Summary: A complete variational treatment is provided for a family of spiked-harmonic oscillator Hamiltonians H=-d 2 /dx 2 +Bx 2 +λ/x α (B>0,λ>0), for arbitrary α>0. A compact topological proof is presented that the set S={ψ n } of known exact solutions for α=2 constitutes an orthonormal basis of the Hilbert space L 2 (0,). Closed-form expressions are derived for the matrix elements of H with respect to S. These analytical results, and the inclusion of a further free parameter, facilitate optimized variational estimation of the eigenvalues of H to high accuracy
MSC:
81Q05Closed and approximate solutions to quantum-mechanical equations
34L15Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators
34L40Particular ordinary differential operators