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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 $H = -d^2/dx^2 + Bx^2 + \lambda/x^{\alpha}$ $(B > 0, \lambda > 0)$, for arbitrary $\alpha >0$. A compact topological proof is presented that the set $S = \{\psi _n\} $ of known exact solutions for $\alpha = 2$ constitutes an orthonormal basis of the Hilbert space $L_2(0,\infty)$. 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

81Q05Closed and approximate solutions to quantum-mechanical equations
34L15Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators
34L40Particular ordinary differential operators
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