Solvable rational extensions of the Morse and Kepler-Coulomb potentials

We show that it is possible to generate an infinite set of solvable rational extensions from every exceptional first category translationally shape invariant potential. This is made by using Darboux-B\"acklund transformations based on unphysical regular Riccati-Schr\"odinger functions which are obtained from specific symmetries associated to the considered family of potentials

On Superstring Disk Amplitudes in a Rolling Tachyon Background

We study the tree level scattering or emission of n closed superstrings from a decaying non-BPS brane in Type II superstring theory. We attempt to calculate generic n-point superstring disk amplitudes in the rolling tachyon background. We show that these can be written as infinite power series of Toeplitz determinants, related to expectation values of a periodic function in Circular Unitary Ensembles. Further analytical progress is possible in the special case of bulk-boundary disk amplitudes. These are interpreted as probability amplitudes for emission of a closed string with initial conditions perturbed by the addition of an open string vertex operator. This calculation has been performed previously in bosonic string theory, here we extend the analysis for superstrings. We obtain a result for the average energy of closed superstrings produced in the perturbed background.Comment: 15 pages, LaTeX2e; uses latexsym, amssymb, amsmath, slashed macros; (v2): references added, some typo fixes; (v3): reference adde

Average characteristic polynomials in the two-matrix model

The two-matrix model is defined on pairs of Hermitian matrices $(M_1,M_2)$ of size $n\times n$ by the probability measure $$\frac{1}{Z_n} \exp\left(\textrm{Tr} (-V(M_1)-W(M_2)+\tau M_1M_2)\right)\ dM_1\ dM_2,$$ where $V$ and $W$ are given potential functions and \tau\in\er. We study averages of products and ratios of characteristic polynomials in the two-matrix model, where both matrices $M_1$ and $M_2$ may appear in a combined way in both numerator and denominator. We obtain determinantal expressions for such averages. The determinants are constructed from several building blocks: the biorthogonal polynomials $p_n(x)$ and $q_n(y)$ associated to the two-matrix model; certain transformed functions $\P_n(w)$ and \Q_n(v); and finally Cauchy-type transforms of the four Eynard-Mehta kernels $K_{1,1}$, $K_{1,2}$, $K_{2,1}$ and $K_{2,2}$. In this way we generalize known results for the $1$-matrix model. Our results also imply a new proof of the Eynard-Mehta theorem for correlation functions in the two-matrix model, and they lead to a generating function for averages of products of traces.Comment: 28 pages, references adde

Bound States of the Klein-Gordon Equation for Woods-Saxon Potential With Position Dependent Mass

The effective mass Klein-Gordon equation in one dimension for the Woods-Saxon potential is solved by using the Nikiforov-Uvarov method. Energy eigenvalues and the corresponding eigenfunctions are computed. Results are also given for the constant mass case.Comment: 13 page

High Energy Quark-Antiquark Elastic scattering with Mesonic Exchange

We studies the high energy elastic scattering of quark anti-quark with an exchange of a mesonic state in the $t$ channel with $-t/\Lambda^{2} \gg 1$. Both the normalization factor and the Regge trajectory can be calculated in PQCD in cases of fixed (non-running) and running coupling constant. The dependence of the Regge trajectory on the coupling constant is highly non-linear and the trajectory is of order of $0.2$ in the interesting physical range.Comment: 29 page

Multipartite minimum uncertainty products

In our previous work we have found a lower bound for the multipartite uncertainty product of the position and momentum observables over all separable states. In this work we are trying to minimize this uncertainty product over a broader class of states to find the fundamental limits imposed by nature on the observable quantites. We show that it is necessary to consider pure states only and find the infimum of the uncertainty product over a special class of pure states (states with spherically symmetric wave functions). It is shown that this infimum is not attained. We also explicitly construct a parametrized family of states that approaches the infimum by varying the parameter. Since the constructed states beat the lower bound for separable states, they are entangled. We thus show that there is a gap that separates the values of a simple measurable quantity for separable states from entangled ones and we also try to find the size of this gap.Comment: 18 pages, 5 figure

Impact of localization on Dyson's circular ensemble

A wide variety of complex physical systems described by unitary matrices have been shown numerically to satisfy level statistics predicted by Dyson's circular ensemble. We argue that the impact of localization in such systems is to provide certain restrictions on the eigenvalues. We consider a solvable model which takes into account such restrictions qualitatively and find that within the model a gap is created in the spectrum, and there is a transition from the universal Wigner distribution towards a Poisson distribution with increasing localization.Comment: To be published in J. Phys.

Orthonormal Polynomials on the Unit Circle and Spatially Discrete Painlev\'e II Equation

We consider the polynomials $\phi_n(z)= \kappa_n (z^n+ b_{n-1} z^{n-1}+ >...)$ orthonormal with respect to the weight $\exp(\sqrt{\lambda} (z+ 1/z)) dz/2 \pi i z$ on the unit circle in the complex plane. The leading coefficient $\kappa_n$ is found to satisfy a difference-differential (spatially discrete) equation which is further proved to approach a third order differential equation by double scaling. The third order differential equation is equivalent to the Painlev\'e II equation. The leading coefficient and second leading coefficient of $\phi_n(z)$ can be expressed asymptotically in terms of the Painlev\'e II function.Comment: 16 page

Universality of a family of Random Matrix Ensembles with logarithmic soft-confinement potentials

Recently we introduced a family of $U(N)$ invariant Random Matrix Ensembles which is characterized by a parameter $\lambda$ describing logarithmic soft-confinement potentials $V(H) \sim [\ln H]^{(1+\lambda)} \:(\lambda>0$). We showed that we can study eigenvalue correlations of these "$\lambda$-ensembles" based on the numerical construction of the corresponding orthogonal polynomials with respect to the weight function $\exp[- (\ln x)^{1+\lambda}]$. In this work, we expand our previous work and show that: i) the eigenvalue density is given by a power-law of the form $\rho(x) \propto [\ln x]^{\lambda-1}/x$ and ii) the two-level kernel has an anomalous structure, which is characteristic of the critical ensembles. We further show that the anomalous part, or the so-called "ghost-correlation peak", is controlled by the parameter $\lambda$; decreasing $\lambda$ increases the anomaly. We also identify the two-level kernel of the $\lambda$-ensembles in the semiclassical regime, which can be written in a sinh-kernel form with more general argument that reduces to that of the critical ensembles for $\lambda=1$. Finally, we discuss the universality of the $\lambda$-ensembles, which includes Wigner-Dyson universality ($\lambda \to \infty$ limit), the uncorrelated Poisson-like behavior ($\lambda \to 0$ limit), and a critical behavior for all the intermediate $\lambda$ ($0<\lambda<\infty$) in the semiclassical regime. We also comment on the implications of our results in the context of the localization-delocalization problems as well as the $N$ dependence of the two-level kernel of the fat-tail random matrices.Comment: 10 pages, 13 figure

Edgeworth Expansion of the Largest Eigenvalue Distribution Function of GUE Revisited

We derive expansions of the resolvent Rn(x;y;t)=(Qn(x;t)Pn(y;t)-Qn(y;t)Pn(x;t))/(x-y) of the Hermite kernel Kn at the edge of the spectrum of the finite n Gaussian Unitary Ensemble (GUEn) and the finite n expansion of Qn(x;t) and Pn(x;t). Using these large n expansions, we give another proof of the derivation of an Edgeworth type theorem for the largest eigenvalue distribution function of GUEn. We conclude with a brief discussion on the derivation of the probability distribution function of the corresponding largest eigenvalue in the Gaussian Orthogonal Ensemble (GOEn) and Gaussian Symplectic Ensembles (GSEn)