Algebraic Treatment of 𝓟𝓣-Symmetric Coupled Oscillators

Abstract

The purpose of this paper is the discussion of a pair of coupled linear oscillators that has recently been proposed as a model of a system of two optical resonators. By means of an algebraic approach we show that the frequencies of the classical and quantum-mechanical interpretations of the optical phenomenon are exactly the same. Consequently, if the classical frequencies are real, then the quantum-mechanical eigenvalues are also real.

Exceptional point in a coupled Swanson system

“…In connection with the system of two optical resonators [40], an algebraic treatment was developed for PT -symmetric coupled bosonic oscillators to find out the classical and quantum mechanical relevance [41], and stability tests were undertaken with regard to the gain-loss parameter [42]. Here we consider a two-mode interacting system of generalized oscillators that are linked directly with different coupling strengths between them.…”

Symmetric quadratic Hamiltonians with pseudo-Hermitian matrix representation

“…A necessary condition for the spectrum of the symmetric quadratic Hamiltonian (38) to be real is that the two roots of the polynomial (40) are real and positive. A more detailed discussion of this spectrum is given elsewhere [1,4].…”

“…In two recent papers Fernández [4,5] proposed the application of a simple and straightforward algebraic method based on the construction of the adjoint or regular matrix representation of the Hamiltonian operator in a suitable basis set of operators [6,7]. The eigenvalues of such matrix representation are the natural frequencies of the Hamiltonian operator.…”

Algebraic treatment of the Pais–Uhlenbeck oscillator and its PT-variant

“…The algebraic method is extremely useful for the analysis of the mathematical properties of quadratic Hamiltonians [11,16,17]. It consists of associating each quadratic function of K coordinates and their K conjugate momenta with a 2K × 2K matrix.…”