I show that the potential V (x,m) = [b 4 −m(1−m)a(a+ 1) ] sn(x,m) dn(x,m) − b(a+ 1 2 ) cn(x,m) dn(x,m) constitutes an unusual QES band-structure problem in one dimension. In particular, I show that for any positive integral or half-integral a, 2a+ 1 band edge eigenvalues and eigenfunctions can be obtained analytically. However, the associated orthogonal polynomials satisfy fourterm (rather than...

This paper deals with spectral type differential equations of the self-adjoint differential operator, 2r order: L(2r)[Y ](x)= 1 (x) d dxr ( (x) r (x) dY (x) dxr ) = rnY (x). If (x) is the weight function and (x) is a second degree polynomial function, then the corresponding classical orthogonal polynomials, {Qn(x)}∞n=0, are shown to satisfy this differential equation when rn is given by rn = r−...

In this paper the linear stability properties of the steady states of a no-slip lubrication equation are studied. The steady states are configurations of droplets and arise during the late-phase dewetting process under the influence of both destabilizing van der Waals and stabilizing Born intermolecular forces, which in turn give rise to the minimum thickness ε of the remaining film connecting ...

A new computational method for determining the eigenvalues and eigenfunctions of the Schrodinger equation is described. Conventional methods for solving this problem rely on diagonalization of a Hamiltonian matrix or iterative numerical solutions of a time independent wave equation. The new method, in contrast, is based on the spectral properties of solutions to the time-dependent Schrodinger e...

We show that in two dimensional flat torus the number of intersections between random eigenfunctions general eigenvalues and a given smooth curve is almost exponentially concentrated around its mean, even when randomness not gaussian.

We report on numerical calculations of nonadiabatic eigenvalues and eigenfunctions for g-modes in ZZ Ceti variables. The spectrum of overstable l \ 1 modes delineates the instability strip. Its blue edge occurs where for the n \ 1 mode. Here u is radian frequency and is about 4 times the uq c B 1 q c thermal time at the bottom of the surface convection zone. As a ZZ Ceti cools, its convection z...

For the case B(u) ≡ I and a Riemann solution consisting of n Lax shocks, I shall explain how geometric singular perturbation theory can be used to construct the u2(x) and to find eigenvalues and eigenfunctions of the linearization of (4) at u2(x). The eigenvalues are related to (i) eigenvalues of the traveling waves that correspond to the individual shock waves, which have been studied by a num...

We develop an efficient spectral-element method for computing the transmission eigenvalues in two-dimensional radially stratified media. Our method is based on a dimension reduction approach which reduces the problem to a sequence of one-dimensional eigenvalue problems that can be efficiently solved by a spectral-element method. We provide an error analysis which shows that the convergence rate...

We study the spectral properties of the transition semigroup of the killed onedimensional Cauchy process on the half-line (0,∞) and the interval (−1, 1). This process is related to the square root of one-dimensional Laplacian A = − √ − d dx with a Dirichlet exterior condition (on a complement of a domain), and to a mixed Steklov problem in the half-plane. For the half-line, an explicit formula ...