Excited states in the infinite quantum lens potential: conformal mapping and moment quantization methods

The conformal mapping method (CMM) and the eigenvalue moment method (EMM) are employed to study the eigenvalue problem defined by a free particle in a quantum lens geometry. The characteristics of the spectrum and the corresponding spatial properties of wavefunctions are studied for varying symmetry quantization numbers and lens parameter values. It is shown that the states belong to two independent Hilbert subspaces corresponding to even and odd azimuthal, m, quantum numbers. The CMM analysis is used to reduce the Dirichlet problem for the Helmholtz's equation into a 2D problem defined by a region with semicircular geometry. This approach allows one to obtain explicit analytical solutions in terms of the lens geometry. In the case of small geometry deformations (relative to the semicircular case), the solutions can be found by a perturbative method. The exact and approximate solutions are compared for different values of the lens parameters. We compare these results with those derived through the EMM approach, which, although more computationally expensive, can yield converging lower and upper bounds. The particular formulation developed here differs significantly from an earlier application of EMM to the ground state case (within each symmetry class). Accordingly, we develop the new formulation and apply it to a limited number of states in order to confirm the results derived by the other, aforementioned, methods.