Voznyak O. Supersymmetry and quasi-exactly solvable potentials for the particle with a position-dependent mass

In the thesis, the supersymmetric method of construction of quasi-exactly solvable potentials for the case of periodic and random potentials and systems with a position-dependent mass has been generalized. In the case of a periodic generating function, new periodic potentials with two and three known eigenstates have been found. The effective low-energy Hamiltonian for the tight-binding model with the hopping integral slowly varying along the chain has been derived from the basic principles, the problem of ordering for mass and momentum in the kinetic energy has been solved and new quasi-exactly solvable potentials with one and two known eigenstates have been found.