Geodesic
Algebraic calculation of the~resolvent of a~generalized quantum oscillator in a~space of dimension $D$
Teoretičeskaâ i matematičeskaâ fizika, Tome 185 (2015) no. 1, pp. 109-117

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We consider the formalism based on using the $sl(2)$ algebra instead of the conventional Heisenberg algebra for isotropic models of quantum mechanics. The operators of the squared momentum $p^2$ and squared coordinates $q^2$ and also the dilation operator $H=i(pq+qp)$ are used as its generators. This allows calculating with the space dimension $D$ as an arbitrary, not necessarily integer parameter. We obtain integral representations for the resolvent and its trace for a generalized harmonic oscillator with the Hamiltonian $H(a,b,c)=ap^2+bq^2+cH$ and any $D$ and study their analytic properties for different model parameter values.
Keywords: generalized quantum oscillator, $sl(2)$ algebra, isotropic model of quantum mechanics, resolvent
Mots-clés : spectral decomposition. 
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     title = {Algebraic calculation of the~resolvent of a~generalized quantum oscillator in a~space of dimension $D$},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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K. S. Karpov; Yu. M. Pis'mak. Algebraic calculation of the~resolvent of a~generalized quantum oscillator in a~space of dimension $D$. Teoretičeskaâ i matematičeskaâ fizika, Tome 185 (2015) no. 1, pp. 109-117. http://geodesic.mathdoc.fr/item/TMF_2015_185_1_a9/