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We review our recent results on pseudo-hermitian random matrix theory which were hitherto presented in various conferences and talks. (Detailed accounts of our work will appear soon in separate publications.) Following an introduction of this new type of random matrices, we focus on two specific models of matrices which are pseudo-hermitian with respect to a given indefinite metric B. Eigenvalues of pseudo-hermitian matrices are either real, or come in complex-conjugate pairs. The diagrammatic method is applied to deriving explicit analytical expressions for the density of eigenvalues in the complex plane and on the real axis, in the large-N, planar limit. In one of the models we discuss, the metric B depends on a certain real parameter t. As t varies, the model exhibits various "phase transitions" associated with eigenvalues flowing from the complex plane onto the real axis, causing disjoint eigenvalue support intervals to merge. Our analytical results agree well with presented numerical simulations., Comment: 18 pages, 9 figures. Minor typos corrected

We study heavy-tailed Hermitian random matrices that are unitarily invariant. The invariance implies that the eigenvalue and eigenvector statistics are decoupled. The motivating question has been whether a freely stable random matrix has stable eigenvalue statistics for the largest eigenvalues in the tail. We investigate this question through the use of both numerical and analytical means, the latter of which makes use of the supersymmetry method. A surprising behaviour is uncovered in that a freely stable random matrix does not necessarily yield stable statistics and if it does then it might exhibit Poisson or Poisson-like statistics. The Poisson statistics have been already observed for heavy-tailed Wigner matrices. We conclude with two conjectures on this peculiar behaviour., Comment: 30 pages, 7 figures

We report an unusual buildup of the quantum coherence in a qubit subjected to non-Hermitian evolution generated by a Parity-Time ($\mathcal{PT}$) symmetric Hamiltonian, which is reinterpreted as a Hermitian system in a higher dimensional space using Naimark dilation. The coherence is found to be maximum about the exceptional points (EPs), i.e., the points of coalescence of the eigenvalues as well as the eigenvectors. The nontrivial physics about EPs has been observed in various systems, particularly in photonic systems. As a consequence of enhancement in coherence, the various formulations of Leggett-Garg inequality tests show maximal violation about the EPs.

We propose time-dependent Darboux (supersymmetric) transformations that provide a scheme for the calculation of explicitly time-dependent solvable non-Hermitian partner Hamiltonians. Together with two Hermitian Hamilitonians the latter form a quadruple of Hamiltonians that are related by two time-dependent Dyson equations and two intertwining relations in form of a commutative diagram. Our construction is extended to the entire hierarchy of Hamiltonians obtained from time dependent Darboux-Crum transformations. As an alternative approach we also discuss the intertwining relations for Lewis-Riesenfeld invariants for Hermitian as well as non-Hermitian Hamiltonians that reduce the time-dependent equations to auxiliary eigenvalue equations. The working of our proposals is discussed for a hierarchy of explicitly time-dependent rational, hyperbolic, Airy function and nonlocal potentials., 22 pages, 4 figures

The non hermiticity of the Hamiltonian is shown to be generated from the q-deformation of the algebra. As an example the q-deformed harmonic oscillator is investigated within the leading order approximation in the pseudo Hermitian perturbation theory. A generalized form of the metric is constructed. The spectrum as well as eigenstates of the system are derived explicitly.. Some applications and illustrations in q-deformed quantum statistical mechanics are also discussed.

The Eigen problem is a problem that exists when there is an operator that is affected by the function, then that function does not change except being multipled by the constant in the form of an eigenvalue. In general, the eigenvalue problem uses the Hermitian operators by using observable variable in the form of mathematical operations. Hermitian operator is an operator that happens when the Hermit conjugation is performed, then the operator will return to its original condition. This research aims to determine the base matrix of Hermitian operators with order n ≤ 4 (size 2 × 2, 3 × 3 and 4 × 4). Entry element of the hermitian matrix operator uses complex numbers of which each order consists of few Hermitian operators. The method used is an analytical method by inserting problems into the eigen equation then normalized to obtain a diagonal matrix, so as to obtain a complete solution of the eigen problem on form of a matrix (including eigenvalues, normalized eigenvectors, and base matrix). Base matrix that has been obtained will be a diagonal matrix with diagonal elements of the eigenvalue. The results of this research indicate that the base matrix obtained in each order forms a square matrix that has entry elements in the form of complex numbers.

In this paper, we present a general framework to solve a fundamental problem in Random Matrix Theory (RMT), i.e., the problem of describing the joint distribution of eigenvalues of the sum $\bsA+\bsB$ of two independent random Hermitian matrices $\bsA$ and $\bsB$. Some considerations about the mixture of quantum states are basically subsumed into the above mathematical problem. Instead, we focus on deriving the spectral density of the mixture of adjoint orbits of quantum states in terms of Duistermaat-Heckman measure, originated from the theory of symplectic geometry. Based on this method, we can obtain the spectral density of the mixture of independent random states. In particular, we obtain explicit formulas for the mixture of random qubits. We also find that, in the two-level quantum system, the average entropy of the equiprobable mixture of $n$ random density matrices chosen from a random state ensemble (specified in the text) increases with the number $n$. Hence, as a physical application, our results quantitatively explain that the quantum coherence of the mixture monotonously decreases statistically as the number of components $n$ in the mixture. Besides, our method may be used to investigate some statistical properties of a special subclass of unital qubit channels., 40 pages, 10 figures, LaTeX, the final version accepted for publication in J. Phys. A

We study state conversion in parity-time (PT) symmetry broken non-Hermitian two level system. We construct a theory and explain underlying mechanism for state conversion and define adiabatic evolutions in non-Hermitian systems. The adiabatic theorem can be used if initial state is an instantaneous eigenstate in Hermitian systems. We show that the system can adiabatically follow the modulationally stable instantaneous eigenstate sooner or later, regardless of initial state if the system parameters are varied slowly in non-Hermitian systems. We discuss the topological feature of dynamical circling in the parameter space., Comment: To Appear in Journal of Physics A: Mathematical and Theoretical

Sparse non-Hermitian random matrices arise in the study of disordered physical systems with asymmetric local interactions, and have applications ranging from neural networks to ecosystem dynamics. The spectral characteristics of these matrices provide crucial information on system stability and susceptibility, however, their study is greatly complicated by the twin challenges of a lack of symmetry and a sparse interaction structure. In this review we provide a concise and systematic introduction to the main tools and results in this field. We show how the spectra of sparse non-Hermitian matrices can be computed via an analogy with infinite dimensional operators obeying certain recursion relations. With reference to three illustrative examples --- adjacency matrices of regular oriented graphs, adjacency matrices of oriented Erd\H{o}s-R\'{e}nyi graphs, and adjacency matrices of weighted oriented Erd\H{o}s-R\'{e}nyi graphs --- we demonstrate the use of these methods to obtain both analytic and numerical results for the spectrum, the spectral distribution, the location of outlier eigenvalues, and the statistical properties of eigenvectors., Comment: 60 pages, 10 figures

The Nearest Neighbour Spacing (NNS) distribution can be computed for generalized symmetric 2x2 matrices having different variances in the diagonal and in the off-diagonal elements. Tuning the relative value of the variances we show that the distributions of the level spacings exhibit a crossover from clustering to repulsion as in GOE. The analysis is extended to 3x3 matrices where distributions of NNS as well as Ratio of Nearest Neighbour Spacing (RNNS) show similar crossovers. We show that it is possible to calculate NNS distributions for Hermitian matrices (N=2, 3) where also crossovers take place between clustering and repulsion as in GUE. For large symmetric and Hermitian matrices we use interpolation between clustered and repulsive regimes and identify phase diagrams with respect to the variances., 15 pages, 7 figures

In this study, we introduce a two dimensional complex harmonic oscillator potential with space and time reflection symmetries. The corresponding time independent Schr\"odinger equation yields real eigenvalues with complex eigenfunctions. We also construct the coherent state of the system by using a superposition of 12 eigenfunctions. Using the complex correspondence principle for the probability density we investigate the possible modifications in the probability densities due to the non-Hermitian aspect of the Hamiltonian., Comment: 11 pages and 5 figures

Canonical quantum mechanics postulates Hermitian Hamiltonians to ensure real eigenvalues. Counterintuitively, a non-Hermitian Hamiltonian, satisfying combined parity-time (PT) symmetry, could display entirely real spectra above some phase-transition threshold. Such a counterintuitive discovery has aroused extensive theoretical interest in extending canonical quantum theory by including non-Hermitian but PT-symmetric operators in the last two decades. Despite much fundamental theoretical success in the development of PT-symmetric quantum mechanics, an experimental observation of pseudo-Hermiticity remains elusive as these systems with a complex potential seem absent in Nature. But nevertheless, the notion of PT symmetry has highly survived in many other branches of physics including optics, photonics, AMO physics, acoustics, electronic circuits, material science over the past ten years, and others, where a judicious balance of gain and loss constitutes a PT-symmetric system. Here, although we concentrate upon reviewing recent progress on PT symmetry in optical microcavity systems, we also wish to present some new results that may help to accelerate the research in the area. Such compound photonic structures with gain and loss provide a powerful platform for testing various theoretical proposals on PT symmetry, and initiate new possibilities for shaping optical beams and pulses beyond conservative structures. Throughout this article there is an effort to clearly present the physical aspects of PT-symmetry in optical microcavity systems, but mathematical formulations are reduced to the indispensable ones. Readers who prefer strict mathematical treatments should resort to the extensive list of references. Despite the rapid progress on the subject, new ideas and applications of PT symmetry using optical microcavities are still expected in the future.

Based on the PT-symmetric quantum theory, the concepts of PT-frame, PT-symmetric operator and CPT-frame on a Hilbert space K and for an operator on K are proposed. It is proved that the spectrum and point spectrum of a PT-symmetric linear operator are both symmetric with respect to the real axis and the eigenvalues of an unbroken PT-symmetric operator are real. For a linear operator H on Cd, it is proved that H has unbroken PT-symmetry if and only if it has d different eigenvalues and the corresponding eigenstates are eigenstates of PT. Given a CPT-frame on K, a new positive inner product on K is induced and called CPT-inner product. Te relationship between the CPT-adjoint and the Dirac adjoint of a densely defined linear operator is derived, and it is proved that an operator which has a bounded CPT-frame is CPT-Hermitian if and only if it is T-symmetric, in that case, it is similar to a Hermitian operator. The existence of an operator C consisting of a CPT-frame is discussed. These concepts and results will serve a mathematical discussion about PT-symmetric quantum mechanics.

We study the universality of spectral statistics of large random matrices. We consider $N\times N$ symmetric, hermitian or quaternion self-dual random matrices with independent, identically distributed entries (Wigner matrices) where the probability distribution for each matrix element is given by a measure $\nu$ with a subexponential decay. Our main result is that the correlation functions of the local eigenvalue statistics in the bulk of the spectrum coincide with those of the Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble (GUE) and the Gaussian Symplectic Ensemble (GSE), respectively, in the limit $N\to \infty$. Our approach is based on the study of the Dyson Brownian motion via a related new dynamics, the local relaxation flow. As a main input, we establish that the density of eigenvalues converges to the Wigner semicircle law and this holds even down to the smallest possible scale, and, moreover, we show that eigenvectors are fully delocalized. These results hold even without the condition that the matrix elements are identically distributed, only independence is used. In fact, we give strong estimates on the matrix elements of the Green function as well that imply that the local statistics of any two ensembles in the bulk are identical if the first four moments of the matrix elements match. Universality at the spectral edges requires matching only two moments. We also prove a Wegner type estimate and that the eigenvalues repel each other on arbitrarily small scales., Comment: 111 pages

Ensembles of random Hermitian matrices with a distribution measure defined by an anharmonic potential perturbed by an external source are considered. The limiting characteristics of the eigenvalue distribution of the matrices in these ensembles are related to the asymptotic behaviour of a certain system of multiple orthogonal polynomials. Strong asymptotic formulae are derived for this system. As a consequence, for matrices in this ensemble the limit mean eigenvalue density is found, and a variational principle is proposed to characterize this density. Bibliography: 35 titles.

In this paper we extend the orthogonal polynomials approach for extreme value calculations of Hermitian random matrices, developed by Nadal and Majumdar [1102.0738], to normal random matrices and 2D Coulomb gases in general. Firstly, we show that this approach provides an alternative derivation of results in the literature. More precisely, we show convergence of the rescaled eigenvalue with largest modulus of a normal Gaussian ensemble to a Gumbel distribution, as well as universality for an arbitrary radially symmetric potential. Secondly, it is shown that this approach can be generalised to obtain convergence of the eigenvalue with smallest modulus and its universality for ring distributions. Most interestingly, the here presented techniques are used to compute all slowly varying finite N correction of the above distributions, which is important for practical applications, given the slow convergence. Another interesting aspect of this work is the fact that we can use standard techniques from Hermitian random matrices to obtain the extreme value statistics of non-Hermitian random matrices resembling the large N expansion used in context of the double scaling limit of Hermitian matrix models in string theory., Comment: 25 pages, 6 figures, revised and extended version, including new finite N results, change in title to highlight new results, as to be published in JSTAT

An algorithm for finding the bound-state eigenvalues and eigenfunctions of a Hermitian Hamiltonian operator using Green's method, developed by Waxman\cite{W98},has been extended to include non-Hermitian Hamiltonian operators.

Based on the approach of biorthogonal basis, we carry out the quasinormal modes (QNMs) expansions for a class of open systems described by the wave equation with outgoing wave boundary conditions. For such a non-Hermitian system, the eigenfunction perturbation expansions and Green function method, which are based on the orthogonal eigenvectors of the Hermitian Hamiltonian for the closed quantum system, can be generalized in terms of the biorthogonal basis, the two sets of eigenfunctions of H and its adjointness H†. The time-independent perturbation theory for the complex frequencies can be also developed.

We simply use the relation between the asymptotic iteration method and the Nikiforov–Uvarov method for the analytical solution of the second order linear ordinary differential equations. We apply this relation to study the Schrodinger equation with potentials admitting quasinormal modes. Non-Hermitian PT symmetric potentials have also been studied. Energy eigenvalues in all the cases by the relation are found to be consistent with exact results.

A spectral average which generalises the local spacing distribution of the eigenvalues of random $ N\times N $ hermitian matrices in the bulk of their spectrum as $ N\to\infty $ is known to be a $\tau$-function of the fifth Painlev\'e system. This $\tau$-function, $ \tau(s) $, has generic parameters and is transcendental but is characterised by particular boundary conditions about the singular point $s=0$, which we determine here. When the average reduces to the local spacing distribution we find that $\tau$-function is of the separatrix, or partially truncated type., Comment: 23 pages, 2 figures, to appear in proceedings of Symmetries and Integrability of Difference Equations VII, Melbourne 2006

The iterative algorithm recently proposed by Waxman for solving eigenvalue problems, which relies on the method of moments, has been modified to improve its convergence considerably without sacrificing its benefits or elegance. The suggested modification is based on methods to calculate low-lying eigenpairs of large bounded hermitian operators or matrices.

The tangent formulation for photonic bandgap materials is used to derive explicit expressions for the energy eigenvalues at the centre and the edge of the Brillouin zone of multi-layer photonic bandgap materials with integer ratios of optical path lengths (e.g. any combination of quarter-wave, half-wave, etc., stacks). Unlike the transfer matrix formalism?in which such solutions are implicit functions of photon energy and require a root search?here, all the eigenfrequencies and wave amplitudes are found through a single diagonalization of a Hermitian eigenvalue problem. By reformulating the general nonlinear problem as an algebraic eigenvalue problem, the present solution replaces the search for resonance frequencies with a single eigenproblem, which considerably reduces the computational cost and provides a more accurate solution and a better insight for structure resonances.

We introduce creation and annihilation operators of pseudo-Hermitian fermions for two-level systems described by pseudo-Hermitian Hamiltonian with real eigenvalues. This allows the generalization of the fermionic coherent states approach to such systems. Pseudo-fermionic coherent states are constructed as eigenstates of two pseudo-fermion annihilation operators. These coherent states form a bi-normal and bi-overcomplete system, and their evolution governed by the pseudo-Hermitian Hamiltonian is temporally stable. In terms of the introduced pseudo-fermion operators the two-level system' Hamiltonian takes a factorized form similar to that of a harmonic oscillator., 13 pages (Latex, article class), no figures; v2: some amendments in section 2, seven new refs added

Recently much effort has been made towards the introduction of non-Hermitian random matrix models respecting $PT$-symmetry. Here we show that there is a one-to-one correspondence between complex $PT$-symmetric matrices and split-complex and split-quaternionic versions of Hermitian matrices. We introduce two new random matrix ensembles of (a) Gaussian split-complex Hermitian, and (b) Gaussian split-quaternionic Hermitian matrices, of arbitrary sizes. They are related to the split signature versions of the complex and the quaternionic numbers, respectively. We conjecture that these ensembles represent universality classes for $PT$-symmetric matrices. For the case of $2\times2$ matrices we derive analytic expressions for the joint probability distributions of the eigenvalues, the one-level densities and the level spacings in the case of real eigenvalues., Comment: 9 pages, 3 figures, typos corrected, small changes, accepted for publication in Journal of Physics A

Two of the Hermitian eigenvalue equations resulting from the separation of the three-dimensional Schroedinger equation for H2+ in spheroidals are solved perturbatively for the ground state by expanding the action in positive powers of the internuclear distance R near the united atom He+. The dispersion relations between the separation constants A and Ee are seen to have rigorous analytic solutions, the third-order equation leading to an exact expansion for the inner determinantal equation up to R10. The explicit form for the expansion coefficients is determined up to n = 10, and is seen to contain up to the third power of (γ + ln 4R) logarithmic terms. Even if the general range of validity of the short-range Rn-expansion is expected to be smaller than the corresponding long-range R−n-expansion, it is important to stress that such higher expansion coefficients are calculated exactly for the first time. These formulae give extremely accurate numerical results up to R 0.3a0.

Quantum defect theory (QDT) is widely employed to describe Rydberg states. We show that the effective Hamiltonian employed by QDT is intrinsically non-Hermitian and examine the consequences arising in practical calculations. In particular if the eigenstates are treated in the 'standard' manner, i.e. as if the Hamiltonian were Hermitian, unphysical errors may appear, such as the non-unitarity of the time evolution. Although in most cases the degree of non-Hermiticity is small, we give examples involving the autocorrelation function and the time dependence of entanglement generation where non-Hermiticity must be explicitly accounted for to avoid such unphysical errors. This is done by introducing a second basis set forming with the QDT eigenstates a biorthogonal basis. We give practical schemes to construct this basis and discuss how to express physical quantities. We illustrate the formalism by computing the time evolution of the autocorrelation function and of the linear entropy for a model Rydberg system and compare with the results obtained in the 'standard' manner.

There has been renewed interest in the exploitation of Barta's configuration space theorem (BCST) (Barta 1937 C. R. Acad. Sci. Paris 204 472) which bounds the ground-state energy, , by using any Φ lying within the space of positive, bounded, and sufficiently smooth functions, . Mouchet's (Mouchet 2005 J. Phys. A: Math. Gen. 38 1039) BCST analysis is based on gradient optimization (GO). However, it overlooks significant difficulties: (i) appearance of multi-extrema; (ii) inefficiency of GO for stiff (singular perturbation/strong coupling) problems; (iii) the nonexistence of a systematic procedure for arbitrarily improving the bounds within . These deficiencies can be corrected by transforming BCST into a moments' representation equivalent, and exploiting a generalization of the eigenvalue moment method (EMM), within the context of the well-known generalized eigenvalue problem (GEP), as developed here. EMM is an alternative eigenenergy bounding, variational procedure, overlooked by Mouchet, which also exploits the positivity of the desired physical solution. Furthermore, it is applicable to Hermitian and non-Hermitian systems with complex-number quantization parameters (Handy and Bessis 1985 Phys. Rev. Lett. 55 931, Handy et al 1988 Phys. Rev. Lett. 60 253, Handy 2001 J. Phys. A: Math. Gen. 34 5065, Handy et al 2002 J. Phys. A: Math. Gen. 35 6359). Our analysis exploits various quasi-convexity/concavity theorems common to the GEP representation. We outline the general theory, and present some illustrative examples.

We show that squeezed spin states achieved through nonlinear interactions are eigenstates of a non-Hermitian operator, which however enjoys the property of pseudo-Hermiticity such that the resulting eigenvalues are real. We represent the squeezed states in terms of Wigner d-matrices, making the evaluation of squeezing straightforward. We show that the spin squeezing can go up to 70%.

The correspondence between quantum level spacing distributions and classical motion of 1-D PT symmetric non-Hermitian systems is investigated using two PT symmetric complex potentials: complex rational power potential V1(x) = (ix)(2n + 1)/m and general polynomial potential V2(x) = x2M + ib1 x2M − 1 + b2 x2M − 2 + ... + ib2M − 1x. The level spacing distribution of V1 has two forms. When 2n + 1 − 2m is positive, the level spacing distribution of real eigen values assumes a decreasing power function, while it behaves as an increasing power function when 2n + 1 − 2m is negative. The PT symmetry of this system is spontaneously broken as 2n + 1 − 2m becomes negative. This change manifests itself in classical mechanics as it is found by Bender et al. However, it was found that the change in the form of level spacing distribution mentioned above is not due to the spontaneous breaking down of PT symmetry. Level spacing distribution of V2 assumes an increasing power function when order of the polynomial is greater than two.

For the Friedrichs model, obtained as a one-dimensional ?-perturbation of the orthogonal sum of the momentum and a finite Hermitian matrix: , the scattering matrix is presented as S?(p) = [I + i?M(p)]?1[I ? i?M(p)]. The rational Nevanlinna-class Krein?Weyl function M is associated with the operator A and has poles at the eigenvalues of A. It is proven that for any eigenvalue ?0 of A there exists an intermediate operator , which is constructed as a one-dimensional perturbation of the orthogonal sum of the momentum and an appropriate one-dimensional operator?a solvable model, which plays the role of an intermediate operator in the scattering problem to the pair . The scattering matrix S0? to the pair is an analytic function of ? and the total scattering matrix to the pair can be factorized as a product where S?0 is the scattering matrix to the pair . It is represented by a single Blaschke factor with a pole and zero approaching ?0 when ? ? 0. The non-analytic factor describes creation of the resonance from the eigenvalue ?0 of the operator A.

We describe a method that allows for a practical application of the theory of pseudo-Hermitian operators to PT-symmetric systems defined on a complex contour. We apply this method to study the Hamiltonians H = p2 + x2(ix)ν with ν (−2, ∞) that are defined along the corresponding anti-Stokes lines. In particular, we reveal the intrinsic non-Hermiticity of H for the cases that ν is an even integer, so that H = p2 ± x2+ν, and give a proof of the discreteness of the spectrum of H for all ν (−2, ∞). Furthermore, we study the consequences of defining a square-well Hamiltonian on a wedge-shaped complex contour. This yields a PT-symmetric system with a finite number of real eigenvalues. We present a comprehensive analysis of this system within the framework of pseudo-Hermitian quantum mechanics. We also outline a direct pseudo-Hermitian treatment of PT-symmetric systems defined on a complex contour which clarifies the underlying mathematical structure of the formulation of PT-symmetric quantum mechanics based on the charge-conjugation operator. Our results provide conclusive evidence that pseudo-Hermitian quantum mechanics provides a complete description of general PT-symmetric systems regardless of whether they are defined along the real line or a complex contour.

The pseudo Hermiticity with respect to the exchange operators of N-D complex Hamiltonians is investigated. It is shown that if an N-D Hamiltonian is pseudo Hermitian and any eigenfunction of it retains παT symmetry then the corresponding eigen value is real, where πα is an exchange operator with respect to the permutation α of coordinates and T is the time reversal operator. We construct a special class of N-D pseudo Hermitian Hamiltonians with respect to exchange operators from both N/2-D and N-D general complex Hamiltonians. Examples are presented for Hamiltonians with πT symmetry (π:x↔y, px↔py) and the reality of these systems are investigated.

The general expression with the physical significance and positive definite condition of the eigenvalues of $4\times 4$ Hermitian and trace-one matrix are obtained. This implies that the eigenvalue problem of the $4\times 4$ density matrix is generally solved. The obvious expression of Peres' separability condition for an arbitrary state of two qubits is then given out and it is very easy to use. Furthermore, we discuss some applications to the calculation of the entanglement, the upper bound of the entanglement, and a model of the transfer of entanglement in a qubit chain through a noisy channel., Comment: 12 pages (Revtex); Revised Version; The example of transfer of entanglement is rewritten

In this paper we calculate, in the large N limit, the eigenvalue density of an infinite product of random unitary matrices, each of them generated by a random hermitian matrix. This is equivalent to solving unitary diffusion generated by a hamiltonian random in time. We find that the result is universal and depends only on the second moment of the generator of the stochastic evolution. We find indications of critical behavior (eigenvalue spacing scaling like $1/N^{3/4}$) close to $\theta=\pi$ for a specific critical evolution time $t_c$., Comment: 12 pages, 2 figures

We formulate an algebraic approach to quantum mechanics in fractional dimensions in which the momentum and position operators P, Q satisfy the R-deformed Heisenberg relations, which depend on an operator ν. We find representations of P, Q in which the dimension d and angular momentum l appear as parameters related to the eigenvalues of ν. We analyse the domain of P and find conditions which ensure that P is Hermitian. We investigate plane wave solutions and also free particle wavefunctions in fractional dimensions, and show that as a consequence of wavefunction continuity l is quantized. The representations of P, Q also lead to the corresponding representations of paraboson operators which are used to solve the harmonic oscillator in dimension d, both algebraically and analytically. We demonstrate that the formalism extends also to time-dependent Hamiltonians by solving the time-dependent harmonic oscillator in any dimension d > 0 using the method of Lewis and Riesenfeld.

We define pseudo-reality and pseudo-adjointness of a Hamiltonian, $H$, as $\rho H \rho^{-1}=H^\ast$ and $\mu H \mu^{-1}=H^\prime$, respectively. We prove that the former yields the {\it necessary} condition for spectrum to be real whereas the latter helps in fixing a definition for inner-product of the eigenstates. Here we separate out adjointness of an operator from its Hermitian-adjointness. It turns out that a Hamiltonian possessing real spectrum is first pseudo-real, further it could be Hermitian, PT-symmetric or pseudo-Hermitian., Comment: No Figures, 11 pages

We propose construction of a unique and definite metric ($\eta_+$), time-reversal operator (T) and an inner product such that the pseudo-Hermitian matrix Hamiltonians are C, PT, and CPT invariant and PT(CPT)-norm is indefinite (definite). Here, P and C denote the generalized symmetries : parity and charge-conjugation respectively. The limitations of the other current approaches have been brought out., Comment: No Fig, 13 pages, E-mail: zahmed@apsara.barc.ernet.in

The delocalization phenomenon was discovered by Hatano et al. and Miller et al. for a class of non-Hermitian quantum-mechanical problems. We show that the delocalization is only one example of many possible critical phenomena which are associated with the self-orthogonality of an eigenstate of the non-Hermitian Hamiltonian. It is shown that in this class of problems the self-orthogonality occurs at the series of branch points in the complex energy plane that serve as gates for the particle to hop from one Bloch energy band to another one.

The study of the edge behaviour in the classical ensembles of Gaussian Hermitian matrices has led to the celebrated distributions of Tracy–Widom. Here we take up a similar line of inquiry in the non-Hermitian setting. We focus on the family of N × N random matrices with all entries independent and distributed as complex Gaussian of mean zero and variance 1/N. This is a fundamental non-Hermitian ensemble for which the eigenvalue density is known. Using this density, our main result is a limit law for the (scaled) spectral radius as N ↑ ∞. As a corollary, we get the analogous statement for the case where the complex Gaussians are replaced by quaternion Gaussians.

We compute the large scale (macroscopic) correlations in ensembles of normal random matrices with an arbitrary measure and in ensembles of general non-Hermition matrices with a class of non-Gaussian measures. In both cases the eigenvalues are complex and in the large $N$ limit they occupy a domain in the complex plane. For the case when the support of eigenvalues is a connected compact domain, we compute two-, three- and four-point connected correlation functions in the first non-vanishing order in 1/N in a manner that the algorithm of computing higher correlations becomes clear. The correlation functions are expressed through the solution of the Dirichlet boundary problem in the domain complementary to the support of eigenvalues. The two-point correlation functions are shown to be universal in the sense that they depend only on the support of eigenvalues and are expressed through the Dirichlet Green function of its complement.

The paper discusses progress in understanding statistical properties of complex eigenvalues (and corresponding eigenvectors) of weakly non-unitary and non-Hermitian random matrices. Ensembles of this type emerge in various physical contexts, most importantly in random matrix description of quantum chaotic scattering as well as in the context of QCD-inspired random matrix models., Published version, with a few more misprints corrected

In the present paper, we enlarge the method of Debergh, Beckers and Szafraniec for non-Hermitian Hamiltonians with complex parameters and obtain real eigenvalues. For the case of non-unitary squeeze operators with two complex parameters of non-Hermitian harmonic oscillator, we obtain discrete complex spectrum and for special values of the complex parameters, the spectrum is discrete real and positive even though the corresponding operators are not PT symmetric.

The typical avoided crossings for Hermitian quantum systems depending on parameters, the diabolic crossing scenario, are generalized to the non-Hermitian case, e.g. for resonances. Two types of crossings appear: for type I, the real parts show an avoided and the imaginary parts a true crossing of the eigenenergies, and for type II the opposite is found. A simple symmetric non-Hermitian two-state matrix Hamiltonian is analysed in detail. The diabolic point bifurcates into two exceptional ones on exceptional lines where the matrices are defective. The adiabatic transport of eigenvectors and eigenstates in parameter space is discussed in this generalized diabolic crossing scenario, in particular the geometric Berry phases for a cyclic variation of system parameters, depending on the topology of the closed curves with respect to the exceptional lines.

In this paper, the inverse eigenvalue problem of Hermitian-generalized Hamiltonian matrices and their optimal approximation problem are considered. Some sufficient and necessary conditions of the solvability for the inverse eigenvalue problem are given. A general representation of the solution is presented for a solvable case. Furthermore, an expression of the solution for the optimal approximation problem is presented.

We analyze the spectral properties of large, time-lagged correlation matrices using the tools of random matrix theory. We compare predictions of the one-dimensional spectra, based on approaches already proposed in the literature. Employing the methods of free random variables and diagrammatic techniques, we solve a general random matrix problem, namely the spectrum of a matrix $\frac{1}{T}XAX^{\dagger}$, where $X$ is an $N\times T$ Gaussian random matrix and $A$ is \textit{any} $T\times T$, not necessarily symmetric (Hermitian) matrix. As a particular application, we present the spectral features of the large lagged correlation matrices as a function of the depth of the time-lag. We also analyze the properties of left and right eigenvector correlations for the time-lagged matrices. We positively verify our results by the numerical simulations., Comment: 44 pages, 11 figures; v2 typos corrected, final version

The eigenvalue moment method (EMM) developed by Handy (J. Phys. A: Math. Gen. 2001 34 L271, J. Phys. A: Math. Gen. 2001 34 5065). Handy et al (J. Phys. A: Math. Gen. 2001 34 5593), and Handy and Xiao-Qian Wang (J. Phys. A: Math. Gen. 2001 34 8297), which generates converging lower and upper bounds to the (complex) discrete state energies, is extended to the case of discrete states with non-Real support. In particular, Bender and Boettcher (Phys. Rev. Lett. 1998 80 5243) have argued on the reality of the discrete state spectrum for the −(iX)N potential. For N (integer) ≥ 4, such PT-invariant solutions can only exist on appropriate complex contours. We develop and apply the necessary EMM formalism to such cases. In particular, the restriction of EMM to the anti-Stokes angles significantly increases the convergence rate of the bounds.

We obtain exact functional integrals expressions for many-body matrix elements of the type ψ,J,π,N,Z|e-β|ψ,J,π,N,Z, between states having good angular-momentum, parity and particle number, in the case of particle-number-preserving functional integrals expressions for e-β are used. Compact expressions are obtained without using angular-momentum projectors. In the case of an even number of particles, a new proof of positivity for the pairing plus quadrupole model is given, which is angular-momentum and parity projected to Jπ = 0+. It is shown that the propagator in the functional integral is reduced to a Hermitian positive-definite propagator by the use of angular-momentum eigenstates.

We present a $\mathbf{k} \cdot \mathbf{p}$ theory of photonic crystals containing gain and loss in which the gain and loss are added to separate primitive cells of the underlying Hermitian system, thereby creating a supercell photonic crystal. We show that the supercell bands of this system can merge outward from the degenerate contour formed from folding the bands of the underlying Hermitian system into the supercell Brillouin zone, but that other accidental degeneracies in the band structure of the underlying Hermitian system do not yield band merging behavior. Finally, we show that the modal coupling matrix in PhCs with balanced gain and loss is trace-less, and thus the imaginary components of the eigenvalues can only move relative to one another as the strength of the gain and loss is varied, without any collective motion., 13 pages, 5 figures

Matrix integrals used in random matrix theory for the study of eigenvalues of Hermitian ensembles have been shown to provide $\tau$-functions for several hierarchies of integrable equations. In this article, we extend this relation by showing that such integrals can also provide $\tau$-functions for the discrete KP hierarchy and a coupled version of the same hierarchy obtained through the process of Pfaffianization. To do so, we consider the first equation of the discrete KP hierarchy, the Hirota-Miwa equation. We write the Wronskian determinant solutions to the Hirota-Miwa equation and consider a particular form of matrix integrals, which we show is an example of those Wronskian solutions. The argument is then generalized to the whole hierarchy. A similar strategy is used for the Pfaffianized version of the hierarchy except that in that case, the solutions are written in terms of Pfaffians rather than determinants.

The problem of computing the energy shifts and widths of ground or excited N-electron atomic states perturbed by weak or strong static electric fields is dealt with by formulating a state-specific complex eigenvalue Schrodinger equation (CESE), where the complex energy contains the field-induced shift and width. The CESE is solved to all orders nonperturbatively, by using separately optimized N-electron function spaces, composed of real and complex one-electron functions, the latter being functions of a complex coordinate. The use of such spaces is a salient characteristic of the theory, leading to economy and manageability of calculation in terms of a two-step computational procedure. The first step involves only Hermitian matrices. The second adds complex functions and the overall computation becomes non-Hermitian. Aspects of the formalism and of computational strategy are compared with those of the complex absorption potential (CAP) method, which was recently applied for the calculation of field-induced complex energies in H and Li. Also compared are the numerical results of the two methods, and the questions of accuracy and convergence that were posed by Sahoo and Ho (Sahoo S and Ho Y K 2000 J. Phys. B: At. Mol. Opt. Phys. 33 2195) are explored further. We draw attention to the fact that, because in the region where the field strength is weak the tunnelling rate (imaginary part of the complex eigenvalue) diminishes exponentially, it is possible for even large-scale nonperturbative complex eigenvalue calculations either to fail completely or to produce seemingly stable results which, however, are wrong. It is in this context that the discrepancy in the width of Li 1s22s 2S between results obtained by the CAP method and those obtained by the CESE method is interpreted. We suggest that the very-weak-field regime must be computed by the golden rule, provided the continuum is represented accurately. In this respect, existing one-particle semiclassical formulae seem to be sufficient. In addition to the aforementioned comparisons and conclusions, we present a number of new results from the application of the state-specific CESE theory to the calculation of field-induced shifts and widths of the H n = 3 levels and of the prototypical Be 1s22s2 1S state, for a range of field strengths. Using the H n = 3 manifold as the example, it is shown how errors may occur for small values of the field, unless the function spaces are optimized carefully for each level.