Exactly solvable potentials with finite positive position-dependent mass

Complete text of publication follows. Quantum mechanical potential problems with position-dependent mass occur in various branches of physics. There have been considerable efforts recently to obtain the exact analytical solution of the Schroedinger equation with various effective masses, but the results are less numerous and less systematic than in the case of constant-mass potential problems. The quantum mechanically acceptable one-dimensional position-dependent mass problems can be transformed to the form (- d/dx 1/M(x) d/dx + Veff (x)) ψ(x) = Eψ(x). (1) The procedure of solving this equation is usually done by adapting the methods applied in the constant mass case, i.e. Eq. (1) is transformed to the differential equation of some F special function of mathematical physics by substituting ψ(x) = f(x)F(g(x)). The g(x) function maps the domain of definition of ψ to that of F in a monotonous way. The mass function M(x) also has to be included in this procedure, and in the previous works it has been defined in terms of the derivative of g(x). Since g(x) and g'(x) are usually not bound and can be zero, the resulting mass function also has these features typically. The physical interpretation of the zeros and the singularities of M(x) usually need some kind of justification, but it may not be compatible with the actual physical nature of the problem. Recently we generalized the procedure of solving the position-dependent mass Schroedinger equation in a way that guarantees that M(x) remains finite and positive everywhere. We developed the formalism for the generalized Laguerre polynomials F(g) = Lnα (g) and considered the mass function M(g(x)) M0(γ + g(x))(δ + g(x))-1. The g(x) function was defined by g'(x) g(x)[B(δ + g(x)]-1/2, which contains as special cases the g(x) functions generating the harmonic oscillator (δ = 0) and the Morse potential (δ → ∞, Bδ = const.). Integrating g'(x) results in an implicit x(g) function x + K = (Bδ)1/2 ln ((δ + g)1/2 - δ1/2 / (δ + g)1/2 + δ1/2) + 2B1/2(δ + g)1/2. The resulting Veff (g(x)) potential also depends on x in an implicit way, nevertheless, the physically relevant quantities can be obtained exactly. The potential depends on four parameters (B, γ, δ and d) and combines the properties of the harmonic oscillator (x → ∞, small n) and the Morse potential (x → -∞, n ∼ nmax). The bound-state energy eigenvalues are En = 1/B (n + 1/2 - γ/4 + √1/16 - d - γn - γ/2). The constant mass case is obtained for γ = δ. The potential then contains both the harmonic oscillator and the Morse potentials as special cases, similarly to the generalized Coulomb problem, which contains the harmonic oscillator and the Coulomb potential. The method can be generalized in several ways, which may allow tailoring the effective potential to realistic physical situations.