Quasi-exactly solvable extended trigonometric Pöschl-Teller potentials with position-dependent mass

Infinite families of quasi-exactly solvable position-dependent mass Schrödinger equations with known ground and first excited states are constructed in a deformed supersymmetric background. The starting points consist in one- and two-parameter trigonometric Pöschl-Teller potentials endowed with a deformed shape invariance property and, therefore, exactly solvable. Some extensions of them are considered with the same position-dependent mass and dealt with by a generating function method. The latter enables to construct the first two superpotentials of a deformed supersymmetric hierarchy, as well as the first two partner potentials and the first two eigenstates of the first potential from some generating function W+(x) (and its accompanying function W-(x) . The generalized trigonometric Pöschl-Teller potentials so obtained are thought to have interesting applications in molecular and solid state physics.