A position dependent mass Hamiltonian and abstract ladder operators

Abstract

We consider the Hamiltonian $H$ of a particle in one dimension with a position dependent mass for which we apply the recent strategy of the so-called {\em abstract ladder operators}, in the attempt to find its eigenvalues and eigenvectors. We don't assume that $H$ is self-adjoint, while we focus on the case of a factorizable operator. We show then that pseudo-bosonic operators play a relevant role in this analysis, and we construct bi-coherent states attached to these operators. Explicit examples are discussed.

The paper asks when a one-dimensional position-dependent-mass Hamiltonian with a possibly complex potential admits an explicit ladder-operator solution. It studies $H=-\hbar^2\partial_x^2/[2m(x)]+\hbar^2m'(x)\partial_x/[2m(x)^2]+V(x)$, with smooth positive mass, imposing a first-order operator $A$ satisfying $[H,A]=\lambda A$, $\lambda\ne0$, and a factorization $H-E_0=BA$. Solving the resulting differential constraints fixes the potential in terms of $F(x)=\int\sqrt{m(x)}\,dx$ and complex parameters $\lambda,\gamma$. The factors satisfy $[A,B]=-\lambda I$, yielding a pseudo-bosonic algebra after rescaling. The constructed eigenfunctions are Hermite polynomials in an affine function of $F$, multiplied by $m^{1/4}\exp[\lambda F^2/(2\hbar^2)-\gamma F]$, with eigenvalues $E_n=-\hbar^2\gamma^2/2-(n+1/2)\lambda$. A coordinate and multiplication transformation also connects the problem to a quadratic oscillator. Square-integrability reduces to an exponential integral over the range of $F$: bounded ranges allow either sign of $\Re\lambda$, whereas the full real line requires $\Re\lambda<0$. Three mass profiles illustrate bounded, full-line, and half-line ranges. Associated bi-coherent states are constructed both as lowering-operator eigenstates and as series, with their equivalence established using the Hermite generating function. The imposed commutator and factorization restrict the solvable potentials and enforce equally spaced levels. Totality of the eigenfunction families is established under the stated decay conditions, but whether they form bases remains unresolved. Some displacement-operator calculations are explicitly formal; quasi-basis and identity-resolution statements require a suitable dense subspace.