The complex eigenvalues of some non-Hermitian Hamiltonians, e.g., parity-time-symmetric Hamiltonians, come in complex-conjugate pairs. We show that for non-Hermitian scattering Hamiltonians (of a structureless particle in one dimension) possessing one of four certain symmetries, the poles of the $S$ -matrix eigenvalues in the complex momentum plane are symmetric about the imaginary axis, i.e., they are complex-conjugate pairs on the complex-energy plane. This applies even to states which are not bounded eigenstates of the system, i.e., antibound or virtual states, resonances, and antiresonances. The four Hamiltonian symmetries are formulated as the commutation of the Hamiltonian with specific antilinear operators. Example potentials with such symmetries are constructed and their pole structures and scattering properties are calculated.The complex eigenvalues of some non-Hermitian Hamiltonians, e.g., parity-time-symmetric Hamiltonians, come in complex-conjugate pairs. We show that for non-Hermitian scattering Hamiltonians (of a structureless particle in one dimension) possessing one of four certain symmetries, the poles of the $S$-matrix eigenvalues in the complex momentum plane are symmetric about the imaginary axis, i.e., they are complex-conjugate pairs on the complex-energy plane. This applies even to states which are not bounded eigenstates of the system, i.e., antibound or virtual states, resonances, and antiresonances. The four Hamiltonian symmetries are formulated as the commutation of the Hamiltonian with specific antilinear operators. Example potentials with such symmetries are constructed and their pole structures and scattering properties are calculated.

arXiv:1811.06270Physical Review AMay 13, 201917 citations

THE CONNECTION BETWEEN THE R-MATRIX AND THE S-MATRIX

N. G. van Kampen

Revista Mexicana de FísicaOct 1, 19535 citations

Complex Eigenvalues in a pseudo-Hermitian \b{eta}-Laguerre ensemble

Cleverson Andrade Goulart, Gleb Oshanin, Mauricio Porto Pato

Non-Hermitian PT-symmetric models have been extensively studied in recent years. Following the seminal work that reduced classical random matrix ensembles to a tridiagonal form, several efforts have aimed to generalize this framework to non-Hermitian extensions of the so-called \b{eta}-ensembles. In particular, while the transition of eigenvalues from the real axis to the complex plane has been well characterized for the \b{eta}-Hermite ensemble under symmetry breaking, the behavior of the \b{eta}-Laguerre ensemble in a similar non-Hermitian setting remains less understood. In this work, we investigate an ensemble of unstable matrices isospectral to the \b{eta}-Laguerre ensemble. Introducing a small non-Hermitian perturbation breaks the symmetry and drives the eigenvalues into the complex plane. We derive analytical expressions for the loci of complex-conjugate eigenvalue pairs, which organize into a balloon-like structure in the complex plane, followed by a discrete finite line of real eigenvalues. The asymptotic behavior of these eigenvalues is analyzed in the large matrix-size limit, and our theoretical predictions are supported by numerical simulations.Non-Hermitian PT-symmetric models have been extensively studied in recent years. Following the seminal work that reduced classical random matrix ensembles to a tridiagonal form, several efforts have aimed to generalize this framework to non-Hermitian extensions of the so-called \b{eta}-ensembles. In particular, while the transition of eigenvalues from the real axis to the complex plane has been well characterized for the \b{eta}-Hermite ensemble under symmetry breaking, the behavior of the \b{eta}-Laguerre ensemble in a similar non-Hermitian setting remains less understood. In this work, we investigate an ensemble of unstable matrices isospectral to the \b{eta}-Laguerre ensemble. Introducing a small non-Hermitian perturbation breaks the symmetry and drives the eigenvalues into the complex plane. We derive analytical expressions for the loci of complex-conjugate eigenvalue pairs, which organize into a balloon-like structure in the complex plane, followed by a discrete finite line of real eigenvalues. The asymptotic behavior of these eigenvalues is analyzed in the large matrix-size limit, and our theoretical predictions are supported by numerical simulations.

cond-mat.stat-mechCross-listedarXiv:2511.08857Nov 13, 2025

Evidence of virtual dipole-supported states in electron scattering by methyl chloride

I. I. Fabrikant, R. S. Wilde

A previously detected negative-ion dipole-supported state of is identified as a virtual state whose energy is about eV at J=K=0 , where J and K are the rotational angular momentum and its component about the symmetry axis. For higher J and low K the S -matrix poles in the complex energy plane move off the real axis and typically do not correspond to quasibound states, i.e. there is no time delay in scattering. If K is close to J , the S -matrix poles correspond to quasibound states and may appear as Feshbach resonances.A previously detected negative-ion dipole-supported state of is identified as a virtual state whose energy is about eV at J=K=0 , where J and K are the rotational angular momentum and its component about the symmetry axis. For higher J and low K the S -matrix poles in the complex energy plane move off the real axis and typically do not correspond to quasibound states, i.e. there is no time delay in scattering. If K is close to J , the S -matrix poles correspond to quasibound states and may appear as Feshbach resonances.

Journal of Physics B Atomic Molecular and Optical PhysicsJan 1, 199914 citations

Symmetry of quantum matrix models

C.-W. H. Lee, S. G. Rajeev

AIP conference proceedingsJan 1, 19980 citations

S

-matrix eigenvalues in the complex momentum plane are symmetric about the imaginary axis, i.e., they are complex-conjugate pairs on the complex-energy plane. This applies even to states which are not bounded eigenstates of the system, i.e., antibound or virtual states, resonances, and antiresonances. The four Hamiltonian symmetries are formulated as the commutation of the Hamiltonian with specific antilinear operators. Example potentials with such symmetries are constructed and their pole structures and scattering properties are calculated.