Mathematical Sciences: Schrodinger Operators and Elliptic Eigenvalue Problems with an Indefinite Weight Function

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This research treats spectral theory of Schrodinger operators and elliptic boundary value problems with indefinite weight functions. Such eigenvalue problems occur naturally in the study of nonlinear partial differential equations and are of broad interest in mathematical physics and applied mathematics. In physics and engineering they arise, for instance, in transport theory, lasers, crystal coloration, hydrodynamics, and reaction-diffusion. Professor Lapidus will conduct research on semilinear elliptic equations and the related topic of eigenvalue problems with indefinite weight. The elliptic problems are set on bounded domains with Dirichlet or Neumann boundary conditions. He will also consider Schrodinger operators on unbounded domains with discrete spectrum. He will investigate both eigenvalues and eigenfunctions, with special emphasis on extensions of classical asymptotic formulas and remainder estimates, distribution of eigenvalues, inverse problems, and decay and oscillation of the eigenfunctions.