Qualitative Properties of Eigenfunctions for some Selfadjoint and Non-Selfadjoint PDE

Research project

Description

This NSF proposal concerns research in the deep relationships between solutions to partial
differential equations, differential geometry, dynamical systems, and mathematical physics.
These different areas of mathematics are often tied together by problems in spectral theory
and microlocal analysis; that is, problems concerned with eigenvalues, eigenfunctions, phase
space localization, and the generalizations thereof. The research in this proposal is divided
between selfadjoint and non-selfadjoint eigenfunction problems. Selfadjoint eigenfunction problems include eigenfunction scarring and non-concentration phenomena, quantum chaos, unique continuation properties, distribution of eigenvalues and resonances, and even waves in general relativity. Non-selfadjoint eigenfunction problems include damped waves, heat flow in inhomogeneous materials, statistical physics problems, predictive micro-fluidics, dialysis and many others.
StatusActive
Effective start/end date9/1/158/31/18

Funding

  • National Science Foundation (NSF)

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Qualitative Properties
Eigenfunctions
Eigenvalue
Microlocal Analysis
Quantum Chaos
Unique Continuation
Statistical Physics
Heat Flow
Microfluidics
Differential Geometry
General Relativity
Spectral Analysis
Damped
Phase Space
Dynamical system
Physics