Symmetric quadratic Hamiltonians with pseudo-Hermitian matrix representation

TL;DR

This paper demonstrates that symmetric quadratic Hamiltonians can be represented by pseudo-Hermitian matrices, linking their eigenvalues to physical frequencies, with implications for systems like coupled resonators and circuits.

Contribution

It establishes that symmetric quadratic Hamiltonians have pseudo-Hermitian matrix representations, connecting eigenvalues to physical frequencies and extending understanding of their spectral properties.

Findings

Eigenvalues of the matrix correspond to natural frequencies.

Real eigenvalues imply the Hamiltonian is Hermitian.

Illustrative examples include coupled resonators and circuits.

Abstract

We prove that any symmetric Hamiltonian that is a quadratic function of the coordinates and momenta has a pseudo-Hermitian adjoint or regular matrix representation. The eigenvalues of the latter matrix are the natural frequencies of the Hamiltonian operator. When all the eigenvalues of the matrix are real, then the spectrum of the symmetric Hamiltonian is real and the operator is Hermitian. As illustrative examples we choose the quadratic Hamiltonians that model a pair of coupled resonators with balanced gain and loss, the electromagnetic self-force on an oscillating charged particle and an active LRC circuit.