Algebraic calculation of the resolvent of a generalized quantum oscillator in a space of dimension $D$

K. S. Karpov^ab, Yu. M. Pis'mak<sup>b

a</sup> Mendeleyev Institute for Metrology (VNIIM), St. Petersburg, Russia
^b St. Petersburg State University, St. Petersburg, Russia

Abstract: 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, spectral decomposition.

DOI: 10.4213/tmf8923

 English version: Theoretical and Mathematical Physics, 2015, 185:1, 1454–1461

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