We propose a unified approach to exactly diagonalize generalized non-Hermitian so(3,2) models. This approach is a series of similarity transformations, which is constructed by some similarity transformation operators associated with su(1,1) and su(2) subalgebras of so(3,2) Lie algebra. During this diagonalization, it is worth noting that a key step is to get rid of the terms Eˆ± and Fˆ± together via the proper similarity transformations first. In this way, exact solutions of the non-Hermitian so(3,2) models are obtained. Meanwhile we give the corresponding eigenstates, which are regarded as Lie algebra so(3,2) coherent-like number states. The results can cover the generic form of the eigenvalues and eigenstates to the generalized non-Hermitian two-mode boson systems with the discrete spectrum, including 2D PT-symmetric and non-PT-symmetric oscillators as the special cases. Also they are true for the Hermitian case.
We suggest a generic algebraic method to solve non-Hermitian Hamiltonian systems with Lie algebraic linear structure. Such method can not only unify the non-Hermitian Hamiltonian and the Hermitian Hamiltonian with the same structure but also keep self-consistent between similarity transformation and unitary transformation. To clearly reveal the correctness and physical meaning of such algebraic method, we illustrate our method with two different types of non-Hermitian Hamiltonians: the non-Hermitian Hamiltonian with Heisenberg algebraic linear structure and the non-Hermitian Hamiltonian with su(1, 1) algebraic linear structure. We obtain energy eigenvalues and the corresponding eigenstates of non-Hermitian forced harmonic oscillator with Heisenberg algebra structure via a proper similarity transformation. We also calculate the eigen-problems of general non-Hermitian Hamiltonian with su(1, 1) structure in terms of the similarity transformation. As an application, we focus on studying the non-Hermitian single-mode squeezed and coherent harmonic oscillator system and find that such similarity transformation associated with this model is in fact gauge-like transformation for simple harmonic oscillator.
A class of non-Hermitian quadratic su (2) Hamiltonians having an anti-linear symmetry is constructed. This is achieved by analysing the possible symmetries of such systems in terms of automorphisms of the algebra. In fact, different realizations for this type of symmetry are obtained, including the natural occurrence of charge conjugation together with parity and time reversal. Once specified, the underlying anti-linear symmetry of the Hamiltonian, the former, if unbroken, leads to a purely real spectrum and the latter can be mapped to a Hermitian counterpart by, amongst other possibilities, a similarity transformation. Here, Lie-algebraic methods which were used to investigate the generalized Swanson Hamiltonian ( Assis and Fring 2009 J. Phys. A: Math. Theor . 42 015203 ) are employed to identify the class of quadratic Hamiltonians that allow for such a mapping to the Hermitian counterpart. Whereas for the linear su (2) system, every Hamiltonian of this type can be mapped to a Hermitian counterpart by a transformation which is itself an exponential of a linear combination of su (2) generators, the situation is more complicated for quadratic Hamiltonians. Therefore, the possibility of more elaborate similarity transformations, including quadratic exponents, is also explored in detail. The existence of finite-dimensional representations for the su (2) Hamiltonian, as opposed to the su (1, 1) studied before, allows for comparison with explicit diagonalization results for finite matrices. Finally, the similarity transformations constructed are compared with the analogue of Swanson's method for exact diagonalization of the problem, establishing a simple relation between both approaches.
We analyse a class of non-Hermitian Hamiltonians, which can be expressed in terms of bilinear combinations of generators in the -Lie algebra or their isomorphic su (1, 1)-counterparts. The Hamiltonians are prototypes for solvable models of Lie algebraic type. Demanding a real spectrum and the existence of a well-defined metric, we systematically investigate the constraints these requirements impose on the coupling constants of the model and the parameters in the metric operator. We compute isospectral Hermitian counterparts for some of the original non-Hermitian Hamiltonians. Alternatively, we employ a generalized Bogoliubov transformation, which allows us to compute explicitly real-energy eigenvalue spectra for these type of Hamiltonians, together with their eigenstates. We compare the two approaches.
The family of metric operators, constructed by Musumbu et al (2007 J. Phys. A: Math. Theor. 40 F75), for a harmonic oscillator Hamiltonian augmented by a non-Hermitian PT-symmetric part, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized non-Hermitian (not necessarily PT-symmetric) oscillator Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed. Short title: Non-Hermitian oscillator Hamiltonian and su(1,1)
We study, classify, and explore the mathematical properties of finite-dimensional Lie algebras occurring in the quantum dynamics of single-mode and self-interacting bosonic systems. These Lie algebras are contained in the real skew-hermitian Weyl algebra
A^1, defined as the real subalgebra of the Weyl algebra
A1 consisting of all skew-hermitian polynomials. A central aspect of our analysis is the choice of basis for
A^1, which is composed of skew-symmetric combinations of two elements of the Weyl algebra called monomials, namely strings of creation and annihilation operators combined with their hermitian conjugate. Motivated by the quest for analytical solutions in quantum optimal control and dynamics, we aim at answering the following three fundamental questions: (i) What are the finite-dimensional Lie subalgebras in
A^1 generated by monomials alone? (ii)~What are the finite-dimensional Lie subalgebras in
A^1 that contain the free Hamiltonian? (iii) What are the non-abelian and finite-dimensional Lie subalgebras that can be faithfully realized in
A^1? We answer the first question by providing all possible realizations of all finite-dimensional non-abelian Lie algebras that are generated by monomials alone. We answer the second question by proving that any non-abelian and finite-dimensional subalgebra of
A^1 that contains a free Hamiltonian term must be a subalgebra of the Schrödinger algebra. We partially answer the third question by classifying all nilpotent and non-solvable Lie algebras that can be realized in
A^1, and comment on the remaining cases. Finally, we also discuss the implications of our results for quantum control theory. Our work constitutes an important stepping stone to understanding quantum dynamics of bosonic systems in full generality.
Abstract From the definition of the standard Perelomov coherent states we introduce the Perelomov number coherent states for any su(2) Lie algebra. With the displacement operator we apply a similarity transformation to the su(2) generators and construct a new set of operators which also close the su(2) Lie algebra, being the Perelomov number coherent states the new basis for its unitary irreducible representation. We apply our results to obtain the energy spectrum, the eigenstates and the partition function of two coupled oscillators. We show that the eigenstates of two coupled oscillators are the SU(2) Perelomov number coherent states of the two-dimensional harmonic oscillator with an appropriate choice of the coherent state parameters.