The non-Hermitian quadratic Hamiltonian H=ωa†a+αa2+βa†2 is analyzed, where a† and a are harmonic oscillator creation and annihilation operators and ω, α, and β are real constants. For the case that ω2−4αβ⩾0, it is shown using operator techniques that the Hamiltonian possesses real and positive eigenvalues. A generalized Bogoliubov transformation allows the energy eigenstates to be constructed from the algebra and states of the harmonic oscillator. The eigenstates are shown to possess an imaginary norm for a large range of the parameter space. Finding the orthonormal dual space allows the inner product to be redefined using the complexification procedure of Bender et al. for non-Hermitian Hamiltonians. Transition probabilities governed by H are shown to be manifestly unitary when the complexification procedure is followed. A specific transition element between harmonic oscillator states is evaluated for both the Hermitian and non-Hermitian cases to identify the differences in time evolution.
The non-Hermitian quadratic Hamiltonian H=ωa†a+αa2+βa†2 is analyzed, where a† and a are harmonic oscillator creation and annihilation operators and ω, α, and β are real constants. For the case that ω2−4αβ⩾0, it is shown using operator techniques that the Hamiltonian possesses real and positive eigenvalues. A generalized Bogoliubov transformation allows the energy eigenstates to be constructed from the algebra and states of the harmonic oscillator. The eigenstates are shown to possess an imaginary norm for a large range of the parameter space. Finding the orthonormal dual space allows the inner product to be redefined using the complexification procedure of Bender et al. for non-Hermitian Hamiltonians. Transition probabilities governed by H are shown to be manifestly unitary when the complexification procedure is followed. A specific transition element between harmonic oscillator states is evaluated for both the Hermitian and non-Hermitian cases to identify the differences in time evolution.
A possible method to investigate non-Hermitian Hamiltonians is suggested through finding a Hermitian operator η+ and defining the annihilation and creation operators to be η+ -pseudo-Hermitian adjoint to each other. The operator η+ represents the η+ -pseudo-Hermiticity of Hamiltonians. As an example, a non-Hermitian and non-PT-symmetric Hamiltonian with imaginary linear coordinate and linear momentum terms is constructed and analyzed in detail. The operator η+ is found, based on which, a real spectrum and a positive-definite inner product, together with the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution, are obtained for the non-Hermitian and non-PT-symmetric Hamiltonian. Moreover, this Hamiltonian turns out to be coupled when it is extended to the canonical noncommutative space with noncommutative spatial coordinate operators and noncommutative momentum operators as well. Our method is applicable to the coupled Hamiltonian. Then the first and second order noncommutative corrections of energy levels are calculated, and in particular the reality of energy spectra, the positive-definiteness of inner products, and the related properties (the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution) are found not to be altered by the noncommutativity.
In the present paper we study the deformed harmonic oscillator for the non-Hermitian operator [Formula: see text] where λ,θ are real positive parameters, since the parameters α,β,m are for the general case complex. For the case α=1,β=1 and mass m real, we find the eigenfunctions and eigenvalues of energy, the coherent states, the time evolution of the operators [Formula: see text] in the Heisenberg picture and the uncertainty relations. In this case the operator ℋ is Hermitian and PT-symmetric. Also for the case m complex α=1,β=1, the operator ℋ is non-Hermitian and no more PT symmetric, but CPT symmetric with real discrete positive spectrum and the CPT symmetry is preserved. In the general case α,β,m complex, for the non-Hermitian operator ℋ, we obtain complex spectrum and for the special values of the complex parameters α,β the spectrum is real discrete and positive and the CPT symmetry is preserved. The general problem of deformed oscillator for non hermitian operators can be applied to the Solid State Physics.
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.
The requirement of Hermiticity of a quantum mechanical Hamiltonian, for the description of physical processes with real eigenvalues which has been challenged notably by Carl Bender, is examined for the case of a Fock space Hamiltonian which is bilinear in two creation and annihilation operators. An interpretation of this model as a Schrödinger operator leads to an identification of the Hermitian form of the Hamiltonian as the Landau model of a charged particle in a plane, interacting with a constant magnetic field at right angles to the plane. When the parameters of the Hamiltonian are suitably adjusted to make it non-Hermitian, the model represents two harmonic oscillators at right angles interacting with a constant magnetic field in the third direction, but with a pure imaginary coupling, and real energy eigenvalues. It is now symmetric. Multiparticle states are investigated.
The unitary operator V ( t ) that diagonalizes the time-dependent quadratic Hamiltonian (TDQH) into a time-dependent harmonic oscillator (TDHO) is obtained using a Lie algebra. The method involves a factorization of the TDQH into a TDHO through a unitary Bogoliubov transformation in terms of creation and annihilation operators with time-dependent coefficients. It is shown that this operator can be easily achieved by means of the factorization, together with the commonly known Wei–Norman theorem. We discuss the conditions under which this unitary operator converges to the evolution operator U ( t ) of the Schrödinger equation for the TDQH, giving then a straightforward calculation of the evolution operator with respect to the procedures published in the literature.
We analyze a class of non-Hermitian quadratic Hamiltonians, which are of the form H = A † A + αA 2 + βA † 2, where α, β are real constants, with α ̸ = β, and A † and A are generalized creation and annihilation operators. Thus these Hamiltonians may be classified as generalized Swanson models. It is shown that the eigenenergies are real for a certain range of values of the parameters. A similarity transformation ρ, mapping the non-Hermitian Hamiltonian H to a Hermitian one h, is also obtained. It is shown that H and h share identical energies. As explicit examples, the solutions of a couple of models based on the trigonometric Rosen-Morse I and the hyperbolic Rosen-Morse II type potentials are obtained. We also study the case when the non-Hermitian Hamiltonian is PT symmetric. PACS: 03.65-w