The existence of countably many positive solutions for nonlinear singular m-point boundary value problems on the half-line

AbstractIn this paper, by introducing a new operator, improving and generating a p-Laplace operator for some p>1, we study the existence of countably many positive solutions for nonlinear boundary value problems on the half-line (φ(u′))′+a(t)f(u(t))=0,0<t<+∞,u(0)=∑i=1m−2αiu(ξi),u′(∞)=0, where φ:R→R is the increasing homeomorphism and positive homomorphism and φ(0)=0. We show the sufficient conditions for the existence of countably many positive solutions by using the fixed-point index theory and a new fixed-point theorem in cones

On Weyl-Titchmarsh Theory for Singular Finite Difference Hamiltonian Systems

We develop the basic theory of matrix-valued Weyl-Titchmarsh M-functions and the associated Green's matrices for whole-line and half-line self-adjoint Hamiltonian finite difference systems with separated boundary conditions.Comment: 30 pages, to appear in J. Comput. Appl. Mat

Singular perturbation of polynomial potentials in the complex domain with applications to PT-symmetric families

In the first part of the paper, we discuss eigenvalue problems of the form -w"+Pw=Ew with complex potential P and zero boundary conditions at infinity on two rays in the complex plane. We give sufficient conditions for continuity of the spectrum when the leading coefficient of P tends to 0. In the second part, we apply these results to the study of topology and geometry of the real spectral loci of PT-symmetric families with P of degree 3 and 4, and prove several related results on the location of zeros of their eigenfunctions.Comment: The main result on singular perturbation is substantially improved, generalized, and the proof is simplified. 37 pages, 16 figure

Classical backgrounds and scattering for affine Toda theory on a half-line

We find classical solutions to the simply-laced affine Toda equations which satisfy integrable boundary conditions using solitons which are analytically continued from imaginary coupling theories. Both static `vacuum' configurations and the time-dependent perturbations about them which correspond to classical vacua and particle scattering solutions respectively are considered. A large class of classical scattering matrices are calculated and found to satisfy the reflection bootstrap equation.Comment: Latex document, 28 pages, 3 figures include