We consider quasinormal modes with complex energies from the point of view of the theory of quasi-exactly solvable (QES) models. We demonstrate that it is possible to find new potentials which admit exactly solvable or QES quasinormal modes by suitable complexification of parameters defining the QES potentials. Particularly, we obtain one QES and four exactly solvable potentials out of the five one-dimensional QES systems based on the sl(2) algebra.
In this paper we present a novel quasi-exactly solvable model with symmetric inverted potentials which are unbounded from below. The quasi-exactly solvable states are shown to be total transmission (or reflectionless) modes. From these modes even and odd wavefunctions can be constructed which are normalizable and flux-zero. Under the procedure of self-adjoint extension, a discrete spectrum of bound states can be obtained for these inverted potentials and the solvable part of the spectrum is the quasi-exactly solvable states we have discovered.
We study the self-adjoint extensions of the Hamiltonian operator with symmetric potentials which go to −∞ faster than −|x| 2p with p > 1 as x → ±∞. In this extension procedure, one requires the Wronskian between any states in the spectrum to approach to the same limit asx → ±∞. Then the boundary terms cancel and the Hamiltonian operator can be shown to be hermitian. Discrete bound states with even and odd parities are obtained. Since the Wronskian is not required to vanish asymptotically, the energy eigenstates could be degenerate. Some explicit examples are given and analyzed.
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