Interference of non-Hermiticity with Hermiticity at exceptional points

Miloslav Znojil

doi:10.3390/math10203721

paper

Interference of non-Hermiticity with Hermiticity at exceptional points

arXiv:2208.14257 · doi:10.3390/math10203721

Abstract

A family of non-Hermitian but $P T -$ symmetric $2 J$ by $2 J$ toy-model tridiagonal-matrix Hamiltonians $H^{(2 J)} = H^{(2 J)} (t)$ with $J = K + M = 1, 2, \dots$ and $t < J^{2}$ is studied, for which a real but non-Hermitian $2 K$ by $2 K$ tridiagonal-submatrix component $C (t)$ of the Hamiltonian is assumed coupled to its other two complex but Hermitian $M$ by $M$ tridiagonal-submatrix components $A (t)$ and $B (t)$ . By construction, (i) all of the submatrices get decoupled at $t = t_{M} = M (2 J - M)$ with $M = 1, 2, \dots, J$ ; (ii) at all of the parameters $t = t_{M}$ with $M = J - K = 0, 1, \dots, J - 1$ the Hamiltonian ceases to be diagonalizable exhibiting the Kato's exceptional-point degeneracy of order $2 K$ ; (iv) the system's $P T -$ symmetry gets spontaneously broken when $t \leq t_{J - 1} = J^{2} - 1$ .

arXiv comment33 pages, 1 figure

Interference of non-Hermiticity with Hermiticity at exceptional points

Abstract

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