Symmetric quadratic Hamiltonians with pseudo-Hermitian matrix representation

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.

References in corpus (7)

• Twofold Transition in PT-Symmetric Coupled Oscillators
• Bypassing the bandwidth theorem with PT symmetry
• Complex modes in unstable quadratic bosonic forms
• Dynamics of entanglement between two harmonic modes in stable and unstable regimes
• Algebraic treatment of $\mathcal{P}\mathcal{T}$-symmetric coupled oscillators
• PT-symmetric interpretation of the electromagnetic self-force
• Algebraic treatment of a simple model for the electromagnetic self-force

Cited by in corpus (5)

• Spectrum and normal modes of non-hermitian quadratic boson operators
• A chain of solvable non-Hermitian Hamiltonians constructed by a series of metric operators
• Abstract ladder operators and their applications
• Algebraic analysis of non-Hermitian quadratic Hamiltonians
• Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. I. Two-Dimensional Model