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## Eigenvalue and level-spacing statistics of random, self-adjoint, non-Hermitian matrices

Yogesh N. Joglekar, William A. Karr

We investigate the eigenvalue distribution $\sigma(x)$ and level-spacing distribution $p(s)$ of random matrices $M=AF\neq M^{\dagger}$ where $F$ is a diagonal inner-product and $A$ is a random, real symmetric or complex Hermitian matrix with independent entries drawn from a probability distribution $q(x)$ with zero mean and finite higher moments. Although not Hermitian, the matrix $M$ is self-adjoint with respect to $F$ and thus has a purely real spectrum. We find that the eigenvalue probability distribution $\sigma_F(x)$ is independent of the underlying distribution $q(x)$, is solely characterized by $F$, and therefore generalizes Wigner’s semicircle distribution $\sigma_W(x)$. We find that the level-spacing distributions $p(s)$ are independent of $q(x)$, are dependent upon the inner-product $F$ and whether $A$ is real or complex, and therefore generalize Wigner’s surmise for level spacing.

http://arxiv.org/abs/1012.1202
Disordered Systems and Neural Networks (cond-mat.dis-nn); Statistical Mechanics (cond-mat.stat-mech)

## PT invariant complex E(8) root spaces

Andreas Fring, Monique Smith

We provide a construction procedure for complex root spaces invariant under antilinear transformations, which may be applied to any Coxeter group. The procedure is based on the factorisation of a chosen element of the Coxeter group into two factors. Each of the factors constitutes an involution and may therefore be deformed in an antilinear fashion. Having the importance of the E(8)-Coxeter group in mind, such as underlying a particular perturbation of the Ising model and the fact that for it no solution could be found previously, we exemplify the procedure for this particular case. As a concrete application of this construction we propose new generalisations of Calogero-Moser Sutherland models and affine Toda field theories based on the invariant complex root spaces and deformed complex simple roots, respectively.

http://arxiv.org/abs/1010.2218
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)

## Constructing Exactly Solvable Pseudo-hermitian Many-particle Quantum Systems by Isospectral Deformation

Pijush K. Ghosh

A class of non-Dirac-hermitian many-particle quantum systems admitting entirely real spectra and unitary time-evolution is presented. These quantum models are isospectral with Dirac-hermitian systems and are exactly solvable. The general method involves a realization of the basic canonical commutation relations defining the quantum system in terms of operators those are hermitian with respect to a pre-determined positive definite metric in the Hilbert space. Appropriate combinations of these operators result in a large number of pseudo-hermitian quantum systems admitting entirely real spectra and unitary time evolution. Examples of a pseudo-hermitian rational Calogero model and XXZ spin-chain are considered.

http://arxiv.org/abs/1012.0907
Quantum Physics (quant-ph); Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)

## Matrix Algebras in Non-Hermitian Quantum Mechanics

Alessandro Sergi

Non-Hermitian quantum dynamics can be defined by giving a more fundamental role either to the Heisenberg’s or to the Schr\”odinger’s picture of quantum dynamics. In both cases, it is shown how to map the algebra of commutators, defining time evolution in terms of a non-Hermitian Hamiltonian, onto a non-Hamiltonian algebra with a Hermitian Hamiltonian. The results and discussions are of interest to methods for simulating open quantum systems dynamics in terms of non-Hermitian time evolution.

http://arxiv.org/abs/1012.0906
Quantum Physics (quant-ph)

## On complexified mechanics and coquaternions

Dorje C Brody, Eva-Maria Graefe

While real Hamiltonian mechanics and Hermitian quantum mechanics can both be cast in the framework of complex canonical equations, their complex generalisations have hitherto been remained tangential. In this paper quaternionic and coquaternionic (split-signature analogue of quaternions) extensions of Hamiltonian mechanics are introduced, and are shown to offer a unifying framework for complexified classical and quantum mechanics. In particular, quantum theories characterised by complex Hamiltonians invariant under space-time reflection are shown to be equivalent to certain coquaternionic extensions of Hermitian quantum theories. One of the interesting consequences is that the space-time dimension of these systems is six, not four, on account of the structures of coquaternionic quantum mechanics.

http://arxiv.org/abs/1012.0757
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)

## Metric operators for non-Hermitian quadratic su(2) Hamiltonians

Paulo E. G. Assis

A class of non-Hermitian quadratic su(2) Hamiltonians that fulfil an anti-linear symmetry is constructed. If unbroken this anti-linear symmetry leads to a purely real spectrum and the Hamiltonian can be mapped to a Hermitian counterpart by, amongst other possibilities, a similarity transformation. Here Lie-algebraic methods which were used to investigate the generalised Swanson Hamiltonian is used to construct a class of quadratic Hamiltonians that allow for such a simple mapping to the Hermitian counterpart. While for the linear su(2) Hamiltonian every Hamiltonian of this type can be mapped to a Hermitian counterpart by a transformation which is itself an exponential of a linear combination of su(2) generators, the situation is more complicated for quadratic Hamiltonians. The existence of finite dimensional representations for the su(2) Hamiltonian, as opposed to the su(1,1) studied before, allows for comparison with explicit diagonalisation results for finite matrices. The possibility of more elaborate similarity transformations, including quadratic exponents, is also discussed. Finally, the similarity transformations are compared with the analogue of Swanson’s method of diagonalising the problem.

http://arxiv.org/abs/1012.0194
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

## Effective Non-Hermitian Hamiltonians for Studying Resonance Statistics in Open Disordered Systems

Joshua Feinberg

We briefly discuss construction of energy-dependent effective non-hermitian hamiltonians for studying resonances in open disordered systems

http://arxiv.org/abs/1011.5932
Disordered Systems and Neural Networks (cond-mat.dis-nn); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)