Algebraic treatment of non-Hermitian quadratic Hamiltonians
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Algebraic treatment of non‑Hermitian quadratic Hamiltonians Francisco M. Fernández1 Received: 5 March 2019 / Accepted: 28 August 2020 © Springer Nature Switzerland AG 2020
Abstract We generalize a recently proposed algebraic method in order to treat non-Hermitian Hamiltonians. The approach is applied to several quadratic Hamiltonians studied earlier by other authors. Instead of solving the Schrödinger equation we simply obtain the eigenvalues of a suitable matrix representation of the operator. We take into account the existence of unitary and antiunitary symmetries in the quantummechanical problem. Keywords Quadratic Hamiltonian · Algebraic method · Adjoint matrix · Unitary symmetry · Antiunitary symmetry · Exceptional point
1 Introduction Hamiltonian operators that are quadratic functions of the coordinates and momenta proved to be useful for the study of interesting physical phenomena [1–4]. The eigenvalues of such Hamiltonians may be real or complex. The occurrence of real or complex spectrum depends on the values of the model parameters that determine the experimental setting. The transition from one regime to the other is commonly interpreted as the breaking of PT symmetry. In some cases those PT symmetric Hamiltonians are also Hermitian [5, 6]. In addition to those Hamiltonians directly related to experiment there are other quadratic oscillators that have been used to illustrate physical concepts in a more theoretical setting. They may be Hermitian [7–9] or non-Hermitian [10–14]. The eigenvalue equation for a quadratic Hamiltonian can be solved exactly in several different ways [3, 8, 15]. The algebraic method [5, 6], based on well known properties of Lie algebras [16, 17], is particularly simple and straightforward. It focusses on the natural frequencies of the quantum-mechanical problem and reveals the transition from real to complex spectrum without solving the eigenvalue equation explicitly or * Francisco M. Fernández [email protected] 1
División Química Teórica, INIFTA, Diag. 113 y 64 S/N, 1900 La Plata, Argentina
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Journal of Mathematical Chemistry
writing the Hamiltonian in diagonal form. The whole problem reduces to diagonalizing a 2N × 2N matrix, where N is the number of coordinates. Those earlier applications of the algebraic method focused on Hermitian Hamiltonians [5, 6] but the approach can also be applied to non-Hermitian quadratic ones. The purpose of this paper is to generalize those results and take into account possible unitary and antiunitary symmetries of the quadratic Hamiltonians. In Sect. 2 we briefly address the concepts of unitary and antiunitary symmetries. In Sect. 3 we outline the main ideas of the algebraic method and derive the regular or adjoint matrix representation for non-Hermitian quadratic Hamiltonians. The approach is similar to that in the previous papers [5, 6] but the results are slightly more general and convenient for present aims. In this section we also consider Hermitian quadratic Hamiltonians and illustrate the main r
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