Asymptotic Conditional Distribution of Exceedance Counts: Fragility Index with Different Margins

Let $\bm X=(X_1,...,X_d)$ be a random vector, whose components are not necessarily independent nor are they required to have identical distribution functions $F_1,...,F_d$. Denote by $N_s$ the number of exceedances among $X_1,...,X_d$ above a high threshold $s$. The fragility index, defined by $FI=\lim_{s\nearrow}E(N_s\mid N_s>0)$ if this limit exists, measures the asymptotic stability of the stochastic system $\bm X$ as the threshold increases. The system is called stable if $FI=1$ and fragile otherwise. In this paper we show that the asymptotic conditional distribution of exceedance counts (ACDEC) $p_k=\lim_{s\nearrow}P(N_s=k\mid N_s>0)$, $1\le k\le d$, exists, if the copula of $\bm X$ is in the domain of attraction of a multivariate extreme value distribution, and if $\lim_{s\nearrow}(1-F_i(s))/(1-F_\kappa(s))=\gamma_i\in[0,\infty)$ exists for $1\le i\le d$ and some $\kappa\in{1,...,d}$. This enables the computation of the FI corresponding to $\bm X$ and of the extended FI as well as of the asymptotic distribution of the exceedance cluster length also in that case, where the components of $\bm X$ are not identically distributed

Quantum toboggans: models exhibiting a multisheeted PT symmetry

A generalization of the concept of PT-symmetric Hamiltonians H=p^2+V(x) is described. It uses analytic potentials V(x) (with singularities) and a generalized concept of PT-symmetric asymptotic boundary conditions. Nontrivial toboggans are defined as integrated along topologically nontrivial paths of coordinates running over several Riemann sheets of wave functions.Comment: 16 pp, 5 figs. Written version of the talk given during 5th International Symposium on Quantum Theory and Symmetries, University of Valladolid, Spain, July 22 - 28 2007, webpage http://tristan.fam.uva.es/~qts

Eigenvalues of PT-symmetric oscillators with polynomial potentials

We study the eigenvalue problem $-u^{\prime\prime}(z)-[(iz)^m+P_{m-1}(iz)]u(z)=\lambda u(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the rays $\arg z=-\frac{\pi}{2}\pm \frac{2\pi}{m+2}$, where $P_{m-1}(z)=a_1 z^{m-1}+a_2 z^{m-2}+...+a_{m-1} z$ is a polynomial and integers $m\geq 3$. We provide an asymptotic expansion of the eigenvalues $\lambda_n$ as $n\to+\infty$, and prove that for each {\it real} polynomial $P_{m-1}$, the eigenvalues are all real and positive, with only finitely many exceptions.Comment: 23 pages, 1 figure. v2: equation (14) as well as a few subsequent equations has been changed. v3: typos correcte

Identification of observables in quantum toboggans

Quantum systems with real energies generated by an apparently non-Hermitian Hamiltonian may re-acquire the consistent probabilistic interpretation via an ad hoc metric which specifies the set of observables in the updated Hilbert space of states. The recipe is extended here to quantum toboggans. In the first step the tobogganic integration path is rectified and the Schroedinger equation is given the generalized eigenvalue-problem form. In the second step the general double-series representation of the eligible metric operators is derived.Comment: 25 p

PT-Symmetric Quantum Theory Defined in a Krein Space

We provide a mathematical framework for PT-symmetric quantum theory, which is applicable irrespective of whether a system is defined on R or a complex contour, whether PT symmetry is unbroken, and so on. The linear space in which PT-symmetric quantum theory is naturally defined is a Krein space constructed by introducing an indefinite metric into a Hilbert space composed of square integrable complex functions in a complex contour. We show that in this Krein space every PT-symmetric operator is P-Hermitian if and only if it has transposition symmetry as well, from which the characteristic properties of the PT-symmetric Hamiltonians found in the literature follow. Some possible ways to construct physical theories are discussed within the restriction to the class K(H).Comment: 8 pages, no figures; Refs. added, minor revisio

Existence and Uniqueness of Tri-tronqu\'ee Solutions of the second Painlev\'e hierarchy

The first five classical Painlev\'e equations are known to have solutions described by divergent asymptotic power series near infinity. Here we prove that such solutions also exist for the infinite hierarchy of equations associated with the second Painlev\'e equation. Moreover we prove that these are unique in certain sectors near infinity.Comment: 13 pages, Late

Quasi-exactly solvable quartic: elementary integrals and asymptotics

We study elementary eigenfunctions y=p exp(h) of operators L(y)=y"+Py, where p, h and P are polynomials in one variable. For the case when h is an odd cubic polynomial, we found an interesting identity which is used to describe the spectral locus. We also establish some asymptotic properties of the QES spectral locus.Comment: 20 pages, 1 figure. Added Introduction and several references, corrected misprint

Joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles

The density function for the joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles is found in terms of a Painlev\'e II transcendent and its associated isomonodromic system. As a corollary, the density function for the spacing between these two eigenvalues is similarly characterized.The particular solution of Painlev\'e II that arises is a double shifted B\"acklund transformation of the Hasting-McLeod solution, which applies in the case of the distribution of the largest eigenvalue at the soft edge. Our deductions are made by employing the hard-to-soft edge transitions to existing results for the joint distribution of the first and second eigenvalue at the hard edge \cite{FW_2007}. In addition recursions under $a \mapsto a+1$ of quantities specifying the latter are obtained. A Fredholm determinant type characterisation is used to provide accurate numerics for the distribution of the spacing between the two largest eigenvalues.Comment: 26 pages, 1 Figure, 2 Table

Rational Solutions of the Painleve' VI Equation

In this paper, we classify all values of the parameters $\alpha$, $\beta$, $\gamma$ and $\delta$ of the Painlev\'e VI equation such that there are rational solutions. We give a formula for them up to the birational canonical transformations and the symmetries of the Painlev\'e VI equation.Comment: 13 pages, 1 Postscript figure Typos fixe

Spectral zeta functions of a 1D Schr\"odinger problem

We study the spectral zeta functions associated to the radial Schr\"odinger problem with potential V(x)=x^{2M}+alpha x^{M-1}+(lambda^2-1/4)/x^2. Using the quantum Wronskian equation, we provide results such as closed-form evaluations for some of the second zeta functions i.e. the sum over the inverse eigenvalues squared. Also we discuss how our results can be used to derive relationships and identities involving special functions, using a particular 5F_4 hypergeometric series as an example. Our work is then extended to a class of related PT-symmetric eigenvalue problems. Using the fused quantum Wronskian we give a simple method for calculating the related spectral zeta functions. This method has a number of applications including the use of the ODE/IM correspondence to compute the (vacuum) nonlocal integrals of motion G_n which appear in an associated integrable quantum field theory.Comment: 15 pages, version