Related papers: R-transforms for non-Hermitian matrices: a spheric…
A transforation between a hierarchy of integrable equations arising from the standard $R$-matrix construction on the algebra of differential operators and a hierarchy of integrable equations arising from a deformation of the standard…
The diagonalization of Hermitian supermatrices is studied. Such a change of coordinates is inevitable to find certain structures in random matrix theory. However it still poses serious problems since up to now the calculation of all…
In this paper, we explore the non-Hermitian transition matrix and its gravity dual. States in quantum field theories or gravity theories are typically prepared using Euclidean path integrals. We demonstrate that it is both natural and…
Reciprocal transformations of Hamiltonian operators of hydrodynamic type are investigated. The transformed operators are generally nonlocal, possessing a number of remarkable algebraic and differential-geometric properties. We apply our…
The non-Hermitian formalism is used at present in many papers for the description of open quantum systems. A special language developed in this field of physics which makes it difficult for many physicists to follow and to understand the…
We introduce a nonsymmetric real matrix which contains all the information that the usual Hermitian density matrix does, and which has exactly the same tensor product structure. The properties of this matrix are analyzed in detail in the…
I review aspects of work done in collaboration with A. Zee and R. Scalettar \cite{fz1,fz2,fsz} on complex non-hermitean random matrices. I open by explaining why the bag of tools used regularly in analyzing hermitean random matrices cannot…
In this work, a classical-quantum correspondence for two-level pseudo-Hermitian systems is proposed and analyzed. We show that the presence of a complex external field can be described by a pseudo-Hermitian Hamiltonian if there is a…
We calculate the Fourier transform of a spherically symmetric exponential function. Our evaluation is much simpler than the known one. We use the polar coordinates and reduce the Fourier transform to the integral of a rational function of…
In these lectures three different methods of computing the asymptotic expansion of a Hermitian matrix integral is presented. The first one is a combinatorial method using Feynman diagrams. This leads us to the generating function of the…
Non-Hermitian systems display remarkable response effects that reflect a variety of distinct spectral scenarios, such as exceptional points where the eigensystem becomes defective. However, present frameworks treat the different scenarios…
We consider "spectral" matrix-functions for Hermitian matrices, where the novelty is that the function applied to the spectrum is allowed to be a vector-field rather than a scalar function (a.k.a isotropic matrix functions). We prove first…
We sketch our recent application of a non-commutative version of the Cartan `moving-frame' formalism to the quantum Euclidean space $R^N_q$, the space which is covariant under the action of the quantum group $SO_q(N)$. For each of the two…
The motivation for studying non-Hermitian systems and the role of $\mathcal{PT}$-symmetry is discussed. We investigate the use of a quantum algorithm to find the eigenvalues and eigenvectors of non-Hermitian Hamiltonians, with applications…
In this paper, we describe a Bayesian nonparametric approach to make inference for a bivariate spherically symmetric distribution. We consider a Dirichlet invariant process prior on the set of all bivariate spherically symmetric…
A non-Hermitean extension of paradigmatic Wishart random matrices is introduced to set up a theoretical framework for statistical analysis of (real, complex and real quaternion) stochastic time series representing two "remote" complex…
Let g be a (say, sufficiently differentiable) function on the reals. One knows how to apply g to Hermitian elements A of a C* algebra. Yet the question of differentiability of the mapping A to g(A) is not trivial, since in general "A and dA…
Non-Hermitian systems with specific forms of Hamiltonians can exhibit novel phenomena. However, it is difficult to study their quantum thermodynamical properties. In particular, the calculation of work statistics can be challenging in…
Models based on non-Hermitian Hamiltonians can exhibit a range of surprising and potentially useful phenomena. Physical realizations typically involve couplings to sources of incoherent gain and loss; this is problematic in quantum…
Products and sums of random matrices have seen a rapid development in the past decade due to various analytical techniques available. Two of these are the harmonic analysis approach and the concept of polynomial ensembles. Very recently, it…