Geodesic
Oscillator with singular perturbation
Teoretičeskaâ i matematičeskaâ fizika, Tome 70 (1987) no. 3, pp. 384-393
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The Rayleign–Schrödinger perturbation theory is formulated for even states of a one-dimensional oscillator with the singular perturbation $\lambda|x|^{-\nu}(1\leq\nu <2)$. It is shown that the matrix elements of the perturbation and the Rayleigh–Schrödinger series evist for $1\leq\nu <3/2$ if the induced point perturbation $$-2\lambda(\nu-1)^{-1}|x|^{1-\nu}\delta(x) \quad (1<\nu <3/2), \quad 2\lambda\ln |x|\delta(x) \quad (\nu=1).$$ arising as the result of the singular perturbation is taken into account. For $3/2<\nu <2$ the standard perturbation theory cannot be constructed although the energy levels are analytic in $\lambda$. 
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     author = {V. B. Gostev and V. S. Mineev and A. R. Frenkin},
     title = {Oscillator with singular perturbation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     volume = {70},
     number = {3},
     language = {ru},
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}
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V. B. Gostev; V. S. Mineev; A. R. Frenkin. Oscillator with singular perturbation. Teoretičeskaâ i matematičeskaâ fizika, Tome 70 (1987) no. 3, pp. 384-393. http://geodesic.mathdoc.fr/item/TMF_1987_70_3_a5/

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