Scattering by obstacles of finite capacity
Assume that \(X\) is a locally compact, second countable space equipped with a Radon measure \(m\) of full support. Let \(G\subset X\) be an open set and \(B= X\backslash G\) considered as an obstacle. Assume further that a self-adjoint non-negative operator \(H\) on \(L_ 2(X)\) is given such that: (1) the form \(\mathfrak h\) associated to \(H\) is a regular Dirichlet form, (2) for any \(t>0\), \(\|\exp(- tH): L_ 1\to L_ 1\|< \infty\). Another operator \(H_ G\) on \(L_ 2(G)\) is associated to the closure of the form \({\mathfrak h}\mid D({\mathfrak h})\cap C_ c(G)\). Let \(J^*: L_ 2(G)\to L_ 2(X)\) be the embedding and \(J: L_ 2(X)\to L_ 2(G)\) be the orthogonal projection. The paper develops a method to study the difference \[ D(t):= \exp(- tH)- J^*\exp(- tH_ G)J. \] Sufficient conditions are stated for the operator \(D(t)\) to be Hilbert-Schmidt respectively trace class. As a consequence, one can show that the essential respectively absolutely continuous spectra of \(H\) and \(H_ G\) coincide. The method is based on two theoretical results. The first one is the existence of a Markov process \((\Omega,(\mathbb{P}^ x; x\in X)\), \(({\mathcal X}_ t; t>0))\), i.e. for each \(x\in X\), \((\Omega,\mathbb{P}^ x,({\mathcal X}_ t)_{t>0})\) is a stochastic process with the state space \(X\cup\{\infty\}\). Moreover, it holds \[ \exp(- tH)f(x)= \mathbb{E}^ x(f\circ{\mathcal X}_ t) \] for \(t\geq 0\), \(f\in L_ 2\) and a.e. \(x\in X\) [\textit{M. Fukushima}, `Dirichlet forms and Markov processes', Amsterdam (1980; Zbl 0422.31007)]. The second source is a factorization technique based on the `little Grothendieck theorem'' which yields a remarkable criterion for an operator to be Hilbert-Schmidt respectively trace class. The starting point of the method is then an application of the Feynman- Kac formula giving \[ \exp(- tH_ G)f(x)= \mathbb{E}^ x(\mathbf{1}_{\{\tau_ b>t\}}\cdot f\circ{\mathcal X}_ t). \] Here \(\tau_ B= \inf\{s\geq 0;\;{\mathcal X}_ t\in B\}\) is the hitting time of \(B\).
locally compact, second countable space equipped with a Radon measure of full support
scientific article; zbMATH DE number 599788