## Deformed algebras, position-dependent effective masses and curved spaces: an exactly solvable Coulomb problem.(English)Zbl 1063.81070

The authors first show that 3 current unconvential Schrödinger equations (with deformed canonical commutation relation resp. position-dependent effective mass resp. curved space) are equivalent if their relevant functions (deforming function resp. position-dependent mass resp. diagonal metric) satisfy a simple relation. Then they study a Coulomb problem with standard potential and a deforming function linear in $$r$$. They determine spectrum and wavefunctions by using all 3 different interpretations. They get only a finite number of bound states (in correspondence to the old result for the Coulomb potential in a space with constant negative curvature).

### MSC:

 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 81U15 Exactly and quasi-solvable systems arising in quantum theory 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations 81Q99 General mathematical topics and methods in quantum theory

### Keywords:

unconventional Schrödinger equations; Coulomb problem
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