Continued fraction solution of Krein's inverse problem

The spectral data of a vibrating string are encoded in its so-called characteristic function. We consider the problem of recovering the distribution of mass along the string from its characteristic function. It is well-known that Stieltjes' continued fraction provides a solution of this inverse problem in the particular case where the distribution of mass is purely discrete. We show how to adapt Stieltjes' method to solve the inverse problem for a related class of strings. An application to the excursion theory of diffusion processes is presented.Comment: 18 pages, 2 figure

Small-Energy Analysis for the Selfadjoint Matrix Schroedinger Operator on the Half Line

The matrix Schroedinger equation with a selfadjoint matrix potential is considered on the half line with the most general selfadjoint boundary condition at the origin. When the matrix potential is integrable and has a first moment, it is shown that the corresponding scattering matrix is continuous at zero energy. An explicit formula is provided for the scattering matrix at zero energy. The small-energy asymptotics are established also for the corresponding Jost matrix, its inverse, and various other quantities relevant to the corresponding direct and inverse scattering problems.Comment: This published version has been edited to improve the presentation of the result

Gauss Sums and Quantum Mechanics

By adapting Feynman's sum over paths method to a quantum mechanical system whose phase space is a torus, a new proof of the Landsberg-Schaar identity for quadratic Gauss sums is given. In contrast to existing non-elementary proofs, which use infinite sums and a limiting process or contour integration, only finite sums are involved. The toroidal nature of the classical phase space leads to discrete position and momentum, and hence discrete time. The corresponding `path integrals' are finite sums whose normalisations are derived and which are shown to intertwine cyclicity and discreteness to give a finite version of Kelvin's method of images.Comment: 14 pages, LaTe

Ferromagnetic Ordering of Energy Levels for Uq(sl2)U_q(\mathfrak{sl}_2) Symmetric Spin Chains

We consider the class of quantum spin chains with arbitrary $U_q(\mathfrak{sl}_2)$-invariant nearest neighbor interactions, sometimes called $\textrm{SU}_q(2)$ for the quantum deformation of $\textrm{SU}(2)$, for $q>0$. We derive sufficient conditions for the Hamiltonian to satisfy the property we call {\em Ferromagnetic Ordering of Energy Levels}. This is the property that the ground state energy restricted to a fixed total spin subspace is a decreasing function of the total spin. Using the Perron-Frobenius theorem, we show sufficient conditions are positivity of all interactions in the dual canonical basis of Lusztig. We characterize the cone of positive interactions, showing that it is a simplicial cone consisting of all non-positive linear combinations of "cascade operators," a special new basis of $U_q(\mathfrak{sl}_2)$ intertwiners we define. We also state applications to interacting particle processes.Comment: 23 page

A local-global principle for linear dependence of noncommutative polynomials

A set of polynomials in noncommuting variables is called locally linearly dependent if their evaluations at tuples of matrices are always linearly dependent. By a theorem of Camino, Helton, Skelton and Ye, a finite locally linearly dependent set of polynomials is linearly dependent. In this short note an alternative proof based on the theory of polynomial identities is given. The method of the proof yields generalizations to directional local linear dependence and evaluations in general algebras over fields of arbitrary characteristic. A main feature of the proof is that it makes it possible to deduce bounds on the size of the matrices where the (directional) local linear dependence needs to be tested in order to establish linear dependence.Comment: 8 page

Asymptotic analysis for the generalized langevin equation

Various qualitative properties of solutions to the generalized Langevin equation (GLE) in a periodic or a confining potential are studied in this paper. We consider a class of quasi-Markovian GLEs, similar to the model that was introduced in \cite{EPR99}. Geometric ergodicity, a homogenization theorem (invariance principle), short time asymptotics and the white noise limit are studied. Our proofs are based on a careful analysis of a hypoelliptic operator which is the generator of an auxiliary Markov process. Systematic use of the recently developed theory of hypocoercivity \cite{Vil04HPI} is made.Comment: 27 pages, no figures. Submitted to Nonlinearity

Translation Representations and Scattering By Two Intervals

Studying unitary one-parameter groups in Hilbert space (U(t),H), we show that a model for obstacle scattering can be built, up to unitary equivalence, with the use of translation representations for L2-functions in the complement of two finite and disjoint intervals. The model encompasses a family of systems (U (t), H). For each, we obtain a detailed spectral representation, and we compute the scattering operator, and scattering matrix. We illustrate our results in the Lax-Phillips model where (U (t), H) represents an acoustic wave equation in an exterior domain; and in quantum tunneling for dynamics of quantum states

Nonprofit governance: Improving performance in troubled economic times

Nonprofit management is currently pressured to perform effectively in a weak economy. Yet, nonprofit governance continues to suffer from unclear conceptions of the division of labor between board of directors and executive directors. This online survey of 114 executive directors aims to provide clarification and recommendations for social administration

AR and MA representation of partial autocorrelation functions, with applications

We prove a representation of the partial autocorrelation function (PACF), or the Verblunsky coefficients, of a stationary process in terms of the AR and MA coefficients. We apply it to show the asymptotic behaviour of the PACF. We also propose a new definition of short and long memory in terms of the PACF.Comment: Published in Probability Theory and Related Field

De Branges spaces and Krein's theory of entire operators

This work presents a contemporary treatment of Krein's entire operators with deficiency indices $(1,1)$ and de Branges' Hilbert spaces of entire functions. Each of these theories played a central role in the research of both renown mathematicians. Remarkably, entire operators and de Branges spaces are intimately connected and the interplay between them has had an impact in both spectral theory and the theory of functions. This work exhibits the interrelation between Krein's and de Branges' theories by means of a functional model and discusses recent developments, giving illustrations of the main objects and applications to the spectral theory of difference and differential operators.Comment: 37 pages, no figures. The abstract was extended. Typographical errors were corrected. The bibliography style was change