Topics in Spectral Theory and Nonlinear Equations

Northeastern University, Boston MA

Investigators

Abstract

Topics in Spectral Theory and Nonlinear Equations. Abstract of Proposed Research Mikhail Shubin This project will study topics in the spectral theory of the Laplace and Schroedinger operators in n-dimensions, on subdomains and on Riemannian manifolds with boundary, with Dirichlet boundary conditions. We plan to obtain new and precise criteria for discreteness of spectra and strict positivity for the magnetic Schroedinger operators, with general scalar and vector potentials. We will use a gauge optimization to reduce many issues to those for a family of the usual Schroedinger operators with a potential. We wish to establish new and more precise two-sided estimates for the bottom of the spectrum and essential spectrum of (magnetic) Schroedinger operators. We plan to study applications of the spectral theory of one-dimensional Schroedinger operators to KdV and mKdV equations. In particular the construction of solutions of mKdV in classes of functions which may grow at infinity with respect to the space variable will be investigated using a first order evolution PDE satisfied by eigenfunctions of the one-dimensional Schroedinger operators whose time-dependent potentials satisfy KdV. We also propose to investigate Arnold's problem of finding the asymptotic growth rate of the distribution function of the Dirichlet eigenvalues corresponding to a fixed irreducible representation of a finite group G for an elliptic self-adjoint operator A on a compact manifold with boundary and with a G-action commuting with A. This will be done using the method of approximate spectral projection, which should also provide an estimate of remainder in the asymptotic expansion. The spectrum of a Schroedinger operator has long been interpreted in terms of the energy levels of a quantum particle in the electric and magnetic fields associated with the coefficients of the operator. The results of the proposed research may be interpreted in terms of the localization and stability of the quantum particle at various energy levels. We plan to study influence of the symmetries of different states on the asymptotic distribution of the corresponding energy levels. The Korteweg - de Vries (KdV) and modified Korteweg - de Vries (mKdV) equations are important model equations of various phenomena in non-linear dynamics and we expect to obtain new classes of solutions by using spectral methods.