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Nonlinear Schrödinger/Gross–Pitaevskii equations play a central role in the understanding of nonlinear optical and macroscopic quantum systems. The large time dynamics of such systems is governed by interactions of the nonlinear ground state manifold, discrete neutral modes (“excited states”) and dispersive radiation. Systems with symmetry, in spatial dimensions larger than one, typically have degenerate neutral modes. Thus, we study the large time dynamics of systems with degenerate neutral modes. This requires a new normal form (nonlinear matrix Fermi Golden Rule) governing the system’s large time asymptotic relaxation to the ground state (soliton) manifold.
The pseudospectrum (or spectral instability) of non-self-adjoint operators is a topic of current interest in applied mathematics. In fact, for non-self-adjoint operators the resolvent could be very large outside the spectrum, making numerical computation of the complex eigenvalues very hard. This has importance, for example, in quantum mechanics, random matrix theory and fluid dynamics.
The occurrence of false eigenvalues (or pseudospectrum) of non-self-adjoint semiclassical differential operators is due to the existence of quasimodes, that is, approximate local solutions to the eigenvalue problem. For scalar operators, the quasimodes appear generically since the bracket condition on the principal symbol is not satisfied for topological reasons.
In this paper we shall investigate how these results can be generalized to square systems of semiclassical differential operators of principal type. These are the systems whose principal symbol vanishes of first order on its kernel. We show that the resolvent blows up as in the scalar case, except in a nowhere dense set of degenerate values. We also define quasisymmetrizable systems and systems of subelliptic type, for which we prove estimates on the resolvent.
In a recent breakthrough, Dvir showed that every Kakeya set in must have cardinality at least , where . We improve this lower bound to for a constant . This pins down the correct growth of the constant as a function of (up to the determination of ).