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.
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 in contrast with the common belief that hermiticity is a suffcient condition for real spectrum. The imaginary eigenvalues are derived in algebraic method with imaginary-frequency boson operators for the Hamiltonian of inverted potential well. Dual sets of mutually orthogonal eigenstates are required corresponding respectively to the complex conjugate eigenvalues. Arbitrary order eigenfunctions seen to be the polynomials of imaginary frequency 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 is turnable by the interaction constant. The transformation operator should not be unitary but 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, where all eigenstates are degenerate with zero eigenvalue and transition from imaginary to real spectra appears. The SU(1; 1) generator Sz with real eigenvalues determined by the commutation relation of operators, however, is non-Hermitian in the realization of imaginay-frequency boson operators. The classical counterpart of the quantum Hamiltonian with non-Hermitian interaction is a complex function of the canonical variables. It becomes by the canonical transformation of variables a real function indicating exactly the one to one quantum-classical correspondence of Hamiltonians.
We in this paper study the relation of hermiticity and energy spectrum for Hamiltonians consisting of SU(1,1) generators. In contrast with the common belief, the transition from real to imaginary spectra can appear at an exceptional point for the Hermitian Hamiltonian of a conservative system. The imaginary domain of spectrum resulted from an inverted potential well unbounded from below. An outward force applies on the particle, which moves acceleratingly away from the central point similar to the minisuperspace model of expanding universe. The Hamiltonian remains Hermitian beyond the exceptional point in a price that the boson-operator realization of SU(1,1) generator Sz becomes non-Hermitian. Oppositely a non-Hermitian Hamiltonian (called pseudo-Hermitian) possesses real eigenvalues in the entire region of interaction constant, which increases the gradient of potential well. While the Hermitian interaction decreases the gradient continuously to zero, namely the exceptional point, where eigenstates are degenerate. We extend the non-Hermitian Hamiltonian to the periodically time-dependent and two-dimensional system. The probability density of wave function coincides with classical orbits, which are derived from the corresponding classical-counterpart of the non-Hermitian quantum Hamiltonian.