Representation reduction and solution space contraction in quasi-exactly solvable systems

In quasi-exactly solvable problems partial analytic solutions (energy spectrum and associated wavefunctions) are obtained if some potential parameters are assigned specific values. We introduce a new class in which exact solutions are obtained at a given energy for a special set of values of the potential parameters. To obtain a larger solution space one varies the energy over a discrete set (the spectrum) by simply changing the value of a given integer. A unified treatment that includes the standard as well as the new class of quasi-exactly solvable problems is presented and a few examples are given. The solution space is spanned by discrete square integrable basis functions in which the matrix representation of the Hamiltonian is tridiagonal. Consequently, the wave equation gives a three-term recursion relation for the expansion coefficients of the wavefunction. Imposing quasi-exact solvability constraints results in a complete reduction of the representation to the direct sum of a finite and an infinite component. The finite is real and exactly solvable, whereas the infinite is complex and associated with zero norm states. Consequently, the whole physical space contracts to a finite-dimensional subspace with normalizable states.