Some results on the quantum dynamics of a particle in a Markovian potential
Let (K,E,Y,X) be a Kato set-up. This means that (i) E is a topological space (ii) Y is a Banach space continuously and densely embedded in X, (iii) -K(e) is, for every e in E, a generator of (strongly continuous contraction) semigroup on X, and at the same time a bounded operator from Y to X. Finally (iv) there is an isometry \(S:Y\to X\) such that \(SK(e)S^{-1}-K(e)\) is a bounded operator on X and strongly continuous on \(e\in E\). Then for an E-valued process \(\xi\) on (\(\Omega\),P) one may consider the evolution equation: \[ \frac{d}{dt}U(t,s)y=-K(\xi (t))U(t,s)y,\quad t>s;\quad U(t,s)y=y,\quad t=s. \] In particular, let (\(\xi\) (t)) be a Markov process with state space E and unique invariant measure \(\mu\). Let \({\mathbb{X}}=L^ 2(E,\mu;X)\) be the \(L^ 2\)-functions on (E,\(\mu)\), valued in a Banach space X, there is an operator \(Q^ t\) on \({\mathbb{X}}\), defined by \[ Q^ tx(e)=conditional\quad \exp ectation\quad of\quad U(t,0)x(\xi (0)),\quad given\quad that\quad \xi (t)=e. \] This is a semigroup with generator log Q. Let the semigroups exp(-\({\mathbb{K}}t)\) and exp(-\({\mathbb{A}}t)\) be defined by \(\exp (-{\mathbb{K}}t)x(e)=e^{-K(e)t}x(e),\) \(\exp (-{\mathbb{A}}t)x(e)=conditional\) expectation of x(\(\xi\) (0)) given that \(\xi (t)=e\). Theorem 3.2. says that log Q\(=(-{\mathbb{A}})+(-{\mathbb{K}}).\) In quantum mechanics, let V be a measurable function \(E\times {\mathbb{R}}^{\nu}\) such that the conditions (B), (C) (on p. 247) and \[ (A):\quad \| V(e,\cdot)\phi \| \leq a\| \Delta \phi \| +b(e)\| \phi \|,\quad \forall \phi \in Dom(\Delta), \] are valid (with \(a<1\) fixed). Then \(K(e)=+i\Delta -iV(e,\cdot)\) satisfies the requisite and generates a (random) unitary propagator u(t,s), which induces \(U^+:\) \(U^+(t,s)C=U(t,s)^*CU(t,s),\) propagating bounded observables C. Theorem 4.1 says that the long-time average \(\lim_{t\to \infty}t^{- 1}\int^{t}_{0}U^+(s,0)C ds=\bar C\) exists (almost surely) in the uniform topology when \(C\in {\mathbb{C}}_{pt}\) \((=\{compact\) operators on \(L^ 2({\mathbb{R}}^{\nu})\})\) and \(C\mapsto \bar C\) is continuous from \({\mathbb{C}}_{pt}\) to \(L^ 1(\Omega,{\mathbb{C}}_{pt}).\) Theorem 4.2 says that under the condition (\(\Sigma)\): zero is not an eigenvalue of log Q, \(\bar C=0\) (a.e.). (A simple criterion of this condition (\(\Sigma)\) is given in the Appendix of the paper.)
scientific article; zbMATH DE number 3949438
scientific article