Mathematical Sciences: Partial Differential Equations of Mathematical Physics

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Work will be done on problems of partial differential equations arising from models of diverse physical phenomena. The work focuses on linear differential operators of three primary types. The first goal will to expand on earler successes in studying isospectral Schrodinger operators with periodic potentials (if you know that two operators have the same spectrum, how are they related?). Techniques should be applicable to general self-adjoint elliptic operators with periodic coefficients. Given that the spectrum is known for an operator acting on smooth functions on a torus which is a fundamental domain for the period lattice, the investigators want to determine characteristics of the operator's coefficients. Two important cases worthy of study will be that of the Laplace-Beltrami operator and the hamiltonian for an electron moving in a magnetic field through a lattice of ions which produce the potential. A second thrust will concentrate on initial-value problems for hyperbolic equations defined in domains with corners. The principal investigators succeeded in providing conditions along the "edge" for smooth solutions of elliptic equations to exist. In the hyperbolic setting singularities not only occur where hypersurfaces meet but they also propagate in some complex manner. In particular, gliding rays are reflected and transmitted at the edge giving rise to cones of rays. One of the long-term goals of this work will be to provide justification to some conjectures from the theory of diffraction. A final area involved concerns approximation in solid state physics. The source of this work lies in the study of the Laplace-Beltrami operator. In a setting of exceeding small length scale one arrives at wave packets satisfying the time-dependent Schrodinger equation up to a precise order. These packets follow trajectories of a hamiltonian. At least that is customarily assumed. Part of this effort will be directed to a careful analysis of the entire model.