UC Berkeley

Zeroth order pseudodifferential operators with Morse--Smale dynamical assumption have been used as a model for the dynamics of internal waves. The operators provide a novel mathematical explanation to the formation of attractors for two dimensional internal waves in a bounded aquarium. In addition, the spectral properties are somewhat surprisingly related to phenomena in quantum scattering.

In this thesis we introduce an analogue of the scattering matrix and describe its geometric structure. We also study the dynamics of resonances and answer a question posed by Colin de Verdi ere by proving the generic absence of embedded eigenvalues. The definition of the scattering matrix, although based on analogues with Euclidean scattering, is more involved as we need to deal with the principal symbols of Lagrangian distributions. After being conjugated by certain natural unitary operators, our scattering matrix turns out to be a Fourier integral operator and its canonical relation is related to the bicharacteristics of the operator. Resonances for 0th order pseudodifferential operators are defined as limits of eigenvalues for the operator with viscosity. We show that the convergences rate is linear for simple embedded eigenvalues of the operator. We also consider 0th order perturbations of 0th pseudodifferential operators. A series expansion of resonances for the perturbed operators near embedded eigenvalues for the original operator is proved. A special case of this expansion gives the Fermi golden rule. Using the formulae for the coefficients in the expansion, we prove the absence of embedded eigenvalues for generic perturbation.As a by-product of the study of 0th order pseudodifferential operators, we state and prove the sharp radial estimates in Besov spaces, which generalize the corresponding estimates in Sobolev spaces."/> ere by proving the generic absence of embedded eigenvalues. The definition of the scattering matrix, although based on analogues with Euclidean scattering, is more involved as we need to deal with the principal symbols of Lagrangian distributions. After being conjugated by certain natural unitary operators, our scattering matrix turns out to be a Fourier integral operator and its canonical relation is related to the bicharacteristics of the operator. Resonances for 0th order pseudodifferential operators are defined as limits of eigenvalues for the operator with viscosity. We show that the convergences rate is linear for simple embedded eigenvalues of the operator. We also consider 0th order perturbations of 0th pseudodifferential operators. A series expansion of resonances for the perturbed operators near embedded eigenvalues for the original operator is proved. A special case of this expansion gives the Fermi golden rule. Using the formulae for the coefficients in the expansion, we prove the absence of embedded eigenvalues for generic perturbation.As a by-product of the study of 0th order pseudodifferential operators, we state and prove the sharp radial estimates in Besov spaces, which generalize the corresponding estimates in Sobolev spaces."/> ere by proving the generic absence of embedded eigenvalues. The definition of the scattering matrix, although based on analogues with Euclidean scattering, is more involved as we need to deal with the principal symbols of Lagrangian distributions. After being conjugated by certain natural unitary operators, our scattering matrix turns out to be a Fourier integral operator and its canonical relation is related to the bicharacteristics of the operator. Resonances for 0th order pseudodifferential operators are defined as limits of eigenvalues for the operator with viscosity. We show that the convergences rate is linear for simple embedded eigenvalues of the operator. We also consider 0th order perturbations of 0th pseudodifferential operators. A series expansion of resonances for the perturbed operators near embedded eigenvalues for the original operator is proved. A special case of this expansion gives the Fermi golden rule. Using the formulae for the coefficients in the expansion, we prove the absence of embedded eigenvalues for generic perturbation.As a by-product of the study of 0th order pseudodifferential operators, we state and prove the sharp radial estimates in Besov spaces, which generalize the corresponding estimates in Sobolev spaces."/> ere by proving the generic absence of embedded eigenvalues.

The definition of the scattering matrix, although based on analogues with Euclidean scattering, is more involved as we need to deal with the principal symbols of Lagrangian distributions. After being conjugated by certain natural unitary operators, our scattering matrix turns out to be a Fourier integral operator and its canonical relation is related to the bicharacteristics of the operator.

Resonances for 0th order pseudodifferential operators are defined as limits of eigenvalues for the operator with viscosity. We show that the convergences rate is linear for simple embedded eigenvalues of the operator. We also consider 0th order perturbations of 0th pseudodifferential operators. A series expansion of resonances for the perturbed operators near embedded eigenvalues for the original operator is proved. A special case of this expansion gives the Fermi golden rule. Using the formulae for the coefficients in the expansion, we prove the absence of embedded eigenvalues for generic perturbation.

As a by-product of the study of 0th order pseudodifferential operators, we state and prove the sharp radial estimates in Besov spaces, which generalize the corresponding estimates in Sobolev spaces.