Imaginary eigenvalues of Hermitian Hamiltonian with an inverted potential well and transition to the real spectrum at exceptional point by a non-Hermitian interaction

Abstract We in this paper study the hermiticity of Hamiltonian and energy spectrum for the SU (1, 1) systems. The Hermitian Hamiltonian can possess imaginary eigenvalues if the potential is unbounded below like the Gamow state in nuclear physics. We provide a rigorous quantization for the Hermitian Hamiltonian of inverted harmonic oscillator with the imaginary eigenvalues, which are derived in algebraic method with imaginary frequency boson operators. Dual sets of mutually orthogonal eigenstates are required corresponding respectively to the complex conjugate eigenvalues. Arbitrary order eigenfunctions are generated from the normalized ground state wave functions, which are spatially non-localized. The Hamiltonian including a non-Hermitian interaction term can be converted by similarity transformation to the Hermitian one with an effective potential of reduced slope, which can be manipulated by the interaction constant. The transformation operator is Hermitian different from the unitary transformation in ordinary quantum mechanics. The effective potential vanishes at a critical value of coupling strength called the exceptional point (EP), where all eigenstates are degenerate with zero eigenvalue. The spectrum becomes real beyond the EP just opposite to the usual transition direction. The classical counterpart of the quantum Hamiltonian with non-Hermitian interaction is a complex function of canonical variables. It becomes by the canonical transformation of variables a real function. The quantum–classical correspondence exists precisely for the system with imaginary eigenvalues.