2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/162559In 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.Revtex, 9 pages, 1 eps figure, claim on continuous bound state spectra removed, more discussions on self-adjoint extensions addedQuantum PhysicsHigh Energy Physics - Theorytext