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Register a new userApproximation and limit theorems for quantum stochastic models with unbounded coefficients
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We prove a limit theorem for quantum stochastic differential equations with unbounded coefficients which extends the Trotter-Kato theorem for contraction semigroups. From this theorem, general results on the convergence of approximations and singular perturbations are obtained. The results are illustrated in several examples of physical interest.
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We develop a general technique for proving convergence of repeated quantum interactions to the solution of a quantum stochastic differential equation. The wide applicability of the method is illustrated in a variety of examples. Our main theorem, which is based on the Trotter-Kato theorem, is not restricted to a specific noise model and does not require boundedness of the limit coefficients.
In this paper we study quantum stochastic differential equations (QSDEs) that are driven by strongly squeezed vacuum noise. We show that for strong squeezing such a QSDE can be approximated (via a limit in the strong sense) by a QSDE that is driven by a single commuting noise process. We find that the approximation has an additional Hamiltonian term.
After a brief review of stochastic limit approximation with spin-boson system from physical points of view, amplification phenomenon-stochastic resonance phenomenon-in driven spin-boson system is observed which is helped by the quantum white noise introduced through the stochastic limit approximation. The shift in frequency of the system due to the interaction with the environment-Lamb shift-has an important role in these phenomena.
A new characterization of the singular packing subspaces of general bounded self-adjoint operators is presented, which is used to show that the set of operators whose spectral measures have upper packing dimension equal to one is a $G_delta$ (in suitable metric spaces). As an application, it is proven that, generically (in space of continuous sampling functions), spectral measures of the limit-periodic Schrodinger operators have upper packing dimensions equal to one. Consequently, in a generic set, these operators are quasiballistic.
In this paper we consider Schr{o}dinger operators on $M times mathbb{Z}^{d_2}$, with $M={M_{1}, ldots, M_{2}}^{d_1}$ (`quantum wave guides) with a `$Gamma$-trimmed random potential, namely a potential which vanishes outside a subset $Gamma$ which is periodic with respect to a sub lattice. We prove that (under appropriate assumptions) for strong disorder these operators have emph{pure point spectrum } outside the set $Sigma_{0}=sigma(H_{0,Gamma^{c}})$ where $H_{0,Gamma^{c}} $ is the free (discrete) Laplacian on the complement $Gamma^{c} $ of $Gamma $. We also prove that the operators have some emph{absolutely continuous spectrum} in an energy region $mathcal{E}subsetSigma_{0}$. Consequently, there is a mobility edge for such models. We also consider the case $-M_{1}=M_{2}=infty$, i.~e.~ $Gamma $-trimmed operators on $mathbb{Z}^{d}=mathbb{Z}^{d_1}timesmathbb{Z}^{d_2}$. Again, we prove localisation outside $Sigma_{0} $ by showing exponential decay of the Green function $G_{E+ieta}(x,y) $ uniformly in $eta>0 $. For emph{all} energies $Einmathcal{E}$ we prove that the Greens function $G_{E+ieta} $ is emph{not} (uniformly) in $ell^{1}$ as $eta$ approaches $0$. This implies that neither the fractional moment method nor multi scale analysis emph{can} be applied here.
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