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New type shift operators for circular well potential in two dimensions. (English) Zbl 1238.81110

Summary: New type shift operators for circular well potential in two dimensions are identified. These so-called shift operators connect those quantum systems with the different potentials but with same energy spectrum. It should be noted that these operators depend on both the radial circular and angular variables \(r\) and \(\varphi \). We find that the operators \(P_{\pm }=P_x\pm P_y\) play the role of the shift operators. The radial linear momentum \(P_r=-i\hbar\frac{\partial}{\partial r}\), the angular momentum \(L_z=-i\hbar\frac{\partial}{\partial \varphi}\) and the Hamiltonian form a complete set of commuting operators with the \(SO(2)\) symmetry.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81R15 Operator algebra methods applied to problems in quantum theory
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