The probability for the exclusion of eigenvalues from an interval $(-x,x)$ symmetrical about the origin for a scaled ensemble of Hermitian random matrices, where the Fredholm kernel is a type of Bessel kernel with parameter $ a $ (a generalisation of the sine kernel in the bulk scaling case), is considered. It is shown that this probability is the square of a $τ$-function, in the sense of Okamoto, for the Painlevé system \PIII. This then leads to a factorisation of the probability as the product of two $τ$-functions for the Painlevé system \PIIIdash. A previous study has given a formula of this type but involving \PIIIdash systems with different parameters consequently implying an identity between products of $τ$-functions or equivalently sums of Hamiltonians.
One of the simplest non-Hermitian Hamiltonians first proposed by Schwartz (1960 {\it Commun. Pure Appl. Math.} \tb{13} 609) which may possess a spectral singularity is analyzed from the point of view of non-Hermitian generalization of quantum mechanics. It is shown that $η$ operator, being a second order differential operator, has supersymmetric structure. Asymptotic behavior of eigenfunctions of a Hermitian Hamiltonian equivalent to the given non-Hermitian one is found. As a result the corresponding scattering matrix and cross section are given explicitly. It is demonstrated that the possible…
We consider non-Hermitian PT-symmetric deformation of $A_{N-1}$ type Calogero model without confining potential to investigate the possible existence of spectral singularity. By considering the Wronskian between asymptotic incoming and outgoing scattering state wave functions, we found that there exist no spectral singularity in this model. We further explicitly show that the transmission coefficient vanishes and the reflection coefficient becomes unity for all values of the energy in such a momentum dependent non-Hermitian PT-symmetric model.
Using techniques of supersymmetric quantum mechanics, scattering properties of Hermitian Hamiltonians, which are related to non-Hermitian ones by similarity transformations, are studied. We have found that the scattering matrix of the Hermitian Hamiltonian coincides with the phase factor of the non-unitary scattering matrix of the non-Hermitian Hamiltonian. The possible presence of a spectral singularity in a non-Hermitian Hamiltonian translates into a pronounced resonance in the scattering cross section of its Hermitian counterpart. This opens a way for detecting spectral singularities in sca…
We investigate the transition probabilities for the "flavor" eigenstates in the two-level quantum system, which is described by a non-Hermitian Hamiltonian with the parity and time-reversal (PT) symmetry. Particularly, we concentrate on the so-called PT-broken phase, where two eigenvalues of the non-Hermitian Hamiltonian turn out to be a complex conjugate pair. In this case, we find that the transition probabilities will be unbounded in the limit of infinite time $t \to +\infty$. However, after performing a connection between a non-Hermitian system, which exhibits passive PT-symmetry and globa…
We examine the completeness of bi-orthogonal sets of eigenfunctions for non-Hermitian Hamiltonians possessing a spectral singularity. The correct resolutions of identity are constructed for delta like and smooth potentials. Their form and the contribution of a spectral singularity depend on the class of functions employed for physical states. With this specification there is no obstruction to completeness originating from a spectral singularity.
The spectral singularity have been extensively studied over the last one and half decade for different non-Hermitian potentials in non-Hermitian quantum mechanics. The nature of spectral singularities have not been studied for the case of quaternionic potential. In the present work we perform an analytical study for the scattering from a quaternionic point interaction represented by delta function. New features of spectral singularities are observed which are different than the case of complex (non-quaternionic) point interaction. Most notable difference is the occurrence of spectral singulari…
We investigate how to define in a consistent way the probabilities of the transitions between the "flavor" states of the two-level quantum system, which is described by a non-Hermitian but parity and time-reversal (PT) symmetric Hamiltonian. Explicit calculations are carried out to demonstrate the conservation of probability if a proper definition of the final state is adopted. Finally, this formalism is applied to two-flavor neutrino oscillations $ν^{}_μ\to ν^{}_μ$ and $ν^{}_μ\to ν^{}_τ$ in vacuum, where the exact PT symmetry requires the vacuum mixing angle to be maximal, which is compatible…
In this work we construct self-adjoint extensions of the Dirac operator associated to Hermitian matrix potentials with Coulomb decay and prove that the domain is maximal. The result is obtained by means of a Hardy-Dirac type inequality. In particular, we can work with some electromagnetic potentials such that both, the electric potential and the magnetic one, have Coulomb type singularity.
We compute the gap probability that a circle of radius r around the origin contains exactly k complex eigenvalues. Four different ensembles of random matrices are considered: the Ginibre ensembles and their chiral complex counterparts, with both complex (beta=2) or quaternion real (beta=4) matrix elements. For general non-Gaussian weights we give a Fredholm determinant or Pfaffian representation respectively, depending on the non-Hermiticity parameter. At maximal non-Hermiticity, that is for rotationally invariant weights, the product of Fredholm eigenvalues for beta=4 follows from beta=2 by s…
We construct a Hermitian matrix model for the total descendant potential of a simple singularity of type D similar to the Kontsevich matrix model for the generating function of intersection numbers on the Deligne--Mumford moduli spaces $\overline{\mathcal{M}}_{g,n}$.
Using techniques of supersymmetric gauge theories, we present the Ricci-flat metrics on non-compact Kahler manifolds whose conical singularity is repaired by the Hermitian symmetric space. These manifolds can be identified as the complex line bundles over the Hermitian symmetric spaces. Each of the metrics contains a resolution parameter which controls the size of these base manifolds, and the conical singularity appears when the parameter vanishes.
A background-independent quantization of the Universe near its Big Bang singularity is considered using a drastically simplified toy model. Several conceptual issues are addressed. (1) The observable spatial-geometry characteristics of our empty-space expanding Universe is sampled by the time-dependent operator $Q=Q(t)$ of the distance between two space-attached observers (``Alice and Bob''). (2) For any pre-selected guess of the simple, non-covariant time-dependent observable $Q(t)$ one of the Kato's exceptional points (viz., $t=τ_{(EP)}$) is postulated {\em real-valued}. This enables us to t…
Boundary conditions in quantum graph vertices are generally given in terms of a unitary matrix $U$. Observing that if $U$ has at most two eigenvalues, then the scattering matrix $\mathcal{S}(k)$ of the vertex is a linear combination of the identity matrix and a fixed Hermitian unitary matrix, we construct vertex couplings with this property: For all momenta $k$, the transmission probability from the $j$-th edge to $\ell$-th edge is independent of $(j,\ell)$, and all the reflection probabilities are equal. We classify these couplings according to their scattering properties, which leads to the …
Our goal is to show that the one-interval gap probability for the q-Hahn orthogonal polynomial ensemble can be expressed through a solution of the asymmetric q-Painleve V equation. The case of the q-Hahn ensemble we consider is the most general case of the orthogonal polynomial ensembles that have been studied. Our approach is based on the analysis of q-connections on the Riemann sphere with a particular singularity structure. It requires a new derivation of a q-difference equation of Sakai's hierarchy of type A_{2}^{(1)}. We also calculate its Lax pair.
The probabilities for gaps in the eigenvalue spectrum of finite $ N\times N $ random unitary ensembles on the unit circle with a singular weight, and the related hermitian ensembles on the line with Cauchy weight, are found exactly. The finite cases for exclusion from single and double intervals are given in terms of second order second degree ODEs which are related to certain \mbox{Painlevé-VI} transcendents. The scaled cases in the thermodynamic limit are again second degree and second order, this time related to \mbox{Painlevé-V} transcendents. Using transformations relating the second degr…
Versions of Tracy-Widom (TW) and Adler-Shiota-van Moerbeke (ASvM) approaches are applied to derive various partial differential equations (PDE) satisfied by joint eigenvalue spacing probabilities of two coupled Gaussian Hermitian matrices (coupled GUE). All the lowest (third) order PDE satisfied by the probabilities for the largest eigenvalues of two coupled GUE are found, and the results of both approaches are compared. The TW approach allows to derive all PDE at once, while in the ASvM one starting with different bilinear identities leads to different subsets of the full set of equations. An…
We consider eigenvalues of a product of n non-Hermitian, independent random matrices. Each matrix in this product is of size N\times N with independent standard complex Gaussian variables. The eigenvalues of such a product form a determinantal point process on the complex plane (Akemann and Burda J. Phys A: Math. Theor. 45 (2012) 465201), which can be understood as a generalization of the finite Ginibre ensemble. As N\rightarrow\infty, a generalized infinite Ginibre ensemble arises. We show that the set of absolute values of the points of this determinantal process has the same distribution as…
The unitarily invariant probability measures on infinite Hermitian matrices have been classified by Pickrell, and by Olshanski and Vershik. This classification is equivalent to determining the boundary of a certain inhomogeneous Markov chain with given transition probabilities. This formulation of the problem makes sense for general $β$-ensembles when one takes as the transition probabilities the Dixon-Anderson conditional probability distribution. In this paper we determine the boundary of this Markov chain for any $β\in (0,\infty]$, also giving in this way a new proof of the classical $β=2$ …
