Quadratic Algebra Approach to an Exactly Solvable Position-Dependent Mass Schr"odinger Equation in Two Dimensions

An exactly solvable position-dependent mass Schr"odinger equation in two dimensions, depicting a particle moving in a semi-infinite layer, is re-examined in the light of recent theories describing superintegrable two-dimensional systems with integrals of motion that are quadratic functions of the momenta. To get the energy spectrum a quadratic algebra approach is used together with a realization in terms of deformed parafermionic oscillator operators. In this process, the importance of supplementing algebraic considerations with a proper treatment of boundary conditions for selecting physical wavefunctions is stressed. Some new results for matrix elements are derived. This example emphasizes the interest of a quadratic algebra approach to position-dependent mass Schr"odinger equations.

💡 Research Summary

The paper revisits a two‑dimensional position‑dependent mass (PDM) Schrödinger equation that describes a particle confined to a semi‑infinite layer. The original model assumes a mass function that varies linearly with the coordinate (x) (e.g., (m(x)=m_{0}(1+\lambda x))) and a potential that confines the particle in the (x) direction while leaving the (y) direction free or periodic. Earlier works solved the equation by separation of variables and special‑function techniques, obtaining an energy spectrum but without a systematic treatment of the boundary conditions that guarantee physically admissible wavefunctions.

The authors reinterpret the problem within the framework of superintegrable systems whose integrals of motion are quadratic in the momenta. They identify two independent quadratic integrals (L_{1}) and (L_{2}) that, together with their commutator (L_{3}), close a quadratic algebra denoted (Q(3)). This algebra possesses a Casimir operator (K) that depends on the system parameters and the energy eigenvalue (E). By expressing the Hamiltonian and the integrals in terms of the generators of (Q(3)), the spectral problem is reduced to an algebraic one.

To realize the abstract algebra concretely, the authors employ a deformed para‑fermionic oscillator algebra ({b^{\dagger},b,N}) characterized by a structure function (\Phi(N)) and a finite order (p). The defining relations \