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Regularity properties of the generalized Hamiltonian flow. (English) Zbl 0881.58028
Given a symplectic manifold \(S\) with boundary and a smooth function \(p\) with \(dp\) nonvanishing on the boundary of \(S\), under certain nondegeneracy assumptions there is a way of defining what is called the generalized Hamiltonian flow on the zero set of \(p\) [R. Melrose and J. Sjöstrand, Commun. Pure Appl. Math. 31, 593–617 (1978; Zbl 0368.35020)]. This flow generally has discontinuities, but by making some natural identifications of points, one arrives at a continuous flow \(F_{t}\) on a manifold with boundary. The paper investigates the further regularity of \(F_{t}\), and sketches a proof that it is pointwise \(\alpha\)-Hölder for some positive \(\alpha\). Motivation from a problem in scattering theory is described.
MSC:
35P25 Scattering theory for PDEs
35Q40 PDEs in connection with quantum mechanics
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
Citations:
Zbl 0368.35020
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