Title:

Topics on Hamiltonian dynamics related to symbols of certain Schrodinger operators associated with generators of Levy processes

L ́evy processes give rise to positivity preserving oneparameter operator semigroups, a source for many studies. They are completely characterised by their characteristic exponent which is continuous and negative definite. Using Fourier analysis it can be found that also characterises the corresponding semigroup and its generator. An interesting case is where e−tψ ∈ L1 (Rn) for all t > 0. This allows us to represent the semigroup as a convolution operator. Through this transition densities are introduced, who are greatly relevant in research associated with probability theory. As a continuous negative definite function, can define a metric dψ on Rn which under natural conditions generates the Euclidean topology. This property also leads to looking at metric measure spaces associated with dψ. A key observation is that the diagonal term of the transition density is entirely controlled by the volume of the balls induced by dψ. An idea was to estimate the offdiagonal transition density by a further metric. This raises questions about the convexities of the balls, the dependence of isotropy on parameters, etc. There is also some interest in the spectral theory of some operators, such as Schr ̈odinger operators associated with . The introduction of microlocal analysis induced some development in the spectral theory of (pseudo) differential operators. From this came an idea of considering the symbol of a (pseudo) differential operator as a function on the cotangent bundle. In the case the operator admits a principal part these studies suggest to look at the symbol of the principal part as a Hamilton function. The study of the corresponding dynamics then gives further information about the original pseudodifferential operator, for example its spectral properties. The aim of this thesis is to start to systematically find out which of the techniques and results from microlocal analysis and the theory of metric measure spaces can be extended to pseudodifferential operators with negative definite symbols generating Feller or L2subMarkovian semigroups. Through this we would like to provide tools for further studies of such pseudodifferential operators. To achieve this we concentrate on operators of the form of a Schr ̈odinger operator. In particular, we look at operators − (D) + V (x), with Hamilton functionH(q, p) = (p) + V (q) for a suitable potential V . Since Hamilton’s principle is a central tool we need also study the Lagrange function corresponding to H. The semiclassical asymptotics of classical Schr ̈odinger operators needs a study of the corresponding action function. Hence introducing the Lagrange function leads smoothly into this interest to find the corresponding action function to H.To accomplish all of this some understanding about convexity as well as negative definite functions is needed and so we collect this required theory in the first two chapters of the thesis. The dynamics associated with H, including the corresponding action function, is then developed in the third chapter. In some cases, it would be useful to have rather explicit solutions which require concrete potentials. Hence in the fourth chapter we also suggest substitutions for wellstudied examples in classical mechanics to provide explicit formulae that could benefit further investigations. The final two chapters extends the dynamics by taking ideas from the HamiltonJacobi theory. This is possible using the action function associated to H. It provides an equivalent way of studying the dynamics and in some cases, an arguably more efficient way of doing so.
