Abstract: We analyze a certain class of integral equations related to Marchenko equations and Gel’fand-Levitan equations associated with various systems of ordinary differential operators. When the integral operator is perturbed by a finite-rank perturbation, we explicitly evaluate the change in the solution. We show how this result provides a unified approach to Darboux transformations associa...

n |sn−sn+1|, is presented. This Ising-like model is an interesting example of a graduate-level statistical mechanics problem where the eigenfunctions and eigenvalues of an integral operator, the transfer integral, can be determined exactly. This can be contrasted with the determination of eigenfunctions and eigenvalues of differential operators, such as the quantum mechanical finite-depth squar...

The nonexistence of steady, fully-developed solutions for frictionallyheated buoyant flow in vertical channels is established analytically. Explicit bounds on the values of parameters beyond which solutions to this nonlinear problem cannot exist are derived by integral estimation and by a priori comparison methods. The a priori bound shows that no solution exists if a linear combination of the ...

We study the distribution of the smallest eigenvalue for certain classes of positive-definite Hermitian random matrices, in the limit where the size of the matrices becomes large. Their limit distributions can be expressed as Fredholm determinants of integral operators associated to kernels built out of Meijer G-functions or Wright’s generalized Bessel functions. They generalize in a natural wa...

The nonlinear integral equation P (x) = ∫ β α dy w(y)P (y)P (x + y) is investigated. It is shown that for a given function w(x) the equation admits an infinite set of polynomial solutions P (x). For polynomial solutions, this nonlinear integral equation reduces to a finite set of coupled linear algebraic equations for the coefficients of the polynomials. Interestingly, the set of polynomial sol...

The paper deals with dynamomodels in which the induction effects act within a bounded region surrounded by an electrically conducting medium at rest. Instead of the induction equation, an equivalent integral equation is considered, which again poses an eigenvalue problem. The eigenfunctions and eigenvalues represent the magnetic field modes and corresponding dynamo numbers. In the simplest case...

We consider Hermite and Laguerre β-ensembles of large N × N random matrices. For all β even, corrections to the limiting global density are obtained, and the limiting density at the soft edge is evaluated. We use the saddle point method on multidimensional integral representations of the density which are based on special realizations of the generalized (multivariate) classical orthogonal polyn...

This paper discusses the eigenvalue problem of second-order Sturm-Liouville equation. We transform the governing differential equation to the Fredholm-Volterra integral equation with appropriate end supports. By expanding the unknown function into the shifted Chebyshev polynomials, we directly get the corresponding polynomial characteristic equations, where the lower and higher-order eigenvalue...

We develop an optimization technique to compute local solutions to synthesis problems subject to integral quadratic constraints (IQCs). We use the fact that IQCs may be transformed into semi-infinite maximum eigenvalue constraints over the frequency axis and approach them via nonsmooth optimization methods. We develop a suitable spectral bundle method and prove its convergence in the sense that...