On a Stochastic Trotter Formula with Application to Spontaneous Localization Models

We consider the relation between so called continuous localization models—i.e. non-linear stochastic Schrödinger evolutions—and the discrete GRW-model of wave function collapse. The former can be understood as scaling limit of the GRW process. The proof relies on a stochastic Trotter formula, which is of interest in its own right. Our Trotter formula also allows to complement results on existence theory of stochastic Schrödinger evolutions by Holevo and Mora/Rebolledo.     

Operator-Norm Convergence of the Trotter Product Formula for Sectorial Generators

The operator-norm convergence of the Trotter product formula is known for self-adjoint semigroups with compactness or smallness conditions on the generators involved in this formula. We generalize these two types of results to sectorial generators.     

On Error Estimates for the Trotter–Kato Product Formula

We study the error bound in the operator-norm topology for the Trotter exponential product formula as well as for its generalization à la Kato. Within the framework of an abstract setting, we give a simple proof of error estimates which improve some recent results in this direction.     

A KAM Theorem for Hamiltonian Partial Differential Equations with Unbounded Perturbations

We establish an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field, which could be applied to a large class of Hamiltonian PDEs containing the derivative ? _x in the perturbation. Especially, in this range of application lie a class of derivative nonlinear Schrödinger equations with Dirichlet boundary conditions and perturbed Benjamin-Ono equation with periodic boundary conditions, so KAM tori and thus quasi-periodic solutions are obtained for them.     

Notes for a Quantum Index Theorem

We view DHR superselection sectors with finite statistics as Quantum Field Theory analogs of elliptic operators where KMS functionals play the role of the trace composed with the heat kernel regularization. We extend our local holomorphic dimension formula and prove an analogue of the index theorem in the Quantum Field Theory context. The analytic index is the Jones index, more precisely the minimal dimension, and, on a 4-dimensional spacetime, the DHR theorem gives the integrality of the index. We introduce the notion of holomorphic dimension; the geometric dimension is then defined as the part of the holomorphic dimension which is symmetric under charge conjugation. We apply the AHKT theory of chemical potential and we extend it to the low dimensional case, by using conformal field theory. Concerning Quantum Field Theory on a curved spacetime, the geometry of the manifold enters in the expression for the dimension. If a quantum black hole is described by a spacetime with bifurcate Killing horizon and sectors are localizable on the horizon, the variation of logarithm of the geometric dimension is proportional to the incremental free energy, due to the addition of the charge, and to the inverse temperature, hence to the inverse of the surface gravity in the Hartle–Hawking KMS state. For this analysis we consider a conformal net obtained by restricting the field to the horizon (“holography”). Compared with our previous work on Rindler spacetime, this result differs inasmuch as it concerns true black hole spacetimes, like the Schwarzschild–Kruskal manifold, and pertains to the entropy of the black hole itself, rather than of the outside system. An outlook concerns a possible relation with supersymmetry and noncommutative geometry. Received: 8 March 2000 / Accepted: 17 April 2001     

A Local Index Formula for the Quantum Sphere

For the Dirac operator D on the standard quantum sphere we obtain an asymptotic expansion of the SU_q(2)-equivariant entire cyclic cocycle corresponding to when evaluated on the element The constant term of this expansion is a twisted cyclic cocycle which up to a scalar coincides with the volume form and computes the quantum as well as the classical Fredholm indices.Partially supported by the Norwegian Research Council.Supported by the SUP-program of the Norwegian Research Council.Acknowledgement The preparation of this paper was finished during the authors stay at Institute Mittag-Leffler in September 2003. They would like to express their gratitude to the staff at the institute and to the organizers of the year in Noncommutative Geometry.     

Kochen-Specker Theorem as a Precondition for Quantum Computing

We study the relation between the Kochen-Specker theorem (the KS theorem) and quantum computing. The KS theorem rules out a realistic theory of the KS type. We consider the realistic theory of the KS type that the results of measurements are either +1 or ?1. We discuss an inconsistency between the realistic theory of the KS type and the controllability of quantum computing. We have to give up the controllability if we accept the realistic theory of the KS type. We discuss an inconsistency between the realistic theory of the KS type and the observability of quantum computing. We discuss the inconsistency by using the double-slit experiment as the most basic experiment in quantum mechanics. This experiment can be for an easy detector to a Pauli observable. We cannot accept the realistic theory of the KS type to simulate the double-slit experiment in a significant specific case. The realistic theory of the KS type can not depicture quantum detector. In short, we have to give up both the observability and the controllability if we accept the realistic theory of the KS type. Therefore, the KS theorem is a precondition for quantum computing, i.e., the realistic theory of the KS type should be ruled out.     

Statistical Properties for Flows with Unbounded Roof Function,Including the Lorenz Attractor

For geometric Lorenz attractors (including the classical Lorenz attractor) we obtain a greatly simplified proof of the central limit theorem which applies also to the more general class of codimension two singular hyperbolic attractors. We also obtain the functional central limit theorem and moment estimates, as well as iterated versions of these results. A consequence is deterministic homogenisation (convergence to a stochastic differential equation) for fast-slow dynamical systems whenever the fast dynamics is singularly hyperbolic of codimension two.     

Small-Mass Behavior of Quantum Gibbs States for Lattice Models with Unbounded Spins

We construct the distribution of the infinite-dimensional Markov process associated with a finite-temperature Gibbs state for a quantum mechanical anharmonic crystal. The corresponding state is constructed via a cluster expansion technique for an arbitrary fixed temperature and, correspondingly, small enough masses of particles.     

Sharp Spectral Asymptotics and Weyl Formula for Elliptic Operators with Non-smooth Coefficients

The aim of this paper is to give the Weyl formula for eigenvalues of self-adjoint elliptic operators, assuming that first-order derivatives of the coefficients are Lipschitz continuous. The approach is based on the asymptotic formula of Hörmander"s type for the spectral function of pseudodifferential operators having Lipschitz continuous Hamiltonian flow and obtained via a regularization procedure of nonsmooth coefficients.     

A Semiclassical Egorov Theorem and Quantum Ergodicity for Matrix Valued Operators

We study the semiclassical time evolution of observables given by matrix valued pseudodifferential operators and construct a decomposition of the Hilbert space L²(^d)ⁿ into a finite number of almost invariant subspaces. For a certain class of observables, that is preserved by the time evolution, we prove an Egorov theorem. We then associate with each almost invariant subspace of L²(^d)ⁿ a classical system on a product phase space T^*d×, where is a compact symplectic manifold on which the classical counterpart of the matrix degrees of freedom is represented. For the projections of eigenvectors of the quantum Hamiltonian to the almost invariant subspaces we finally prove quantum ergodicity to hold, if the associated classical systems are ergodic.     

Long-Time Behavior for the 1-D Stochastic Ising Model with Unbounded Random Couplings

We consider the ferromagnetic Ising model with Glauber spin flip dynamics in one dimension. The external magnetic field vanishes and the couplings are i.i.d. random variables. If their distribution has compact support, the disorder averaged spin auto-correlation function has an exponential decay in time. We prove that, if the couplings are unbounded, the decay switches to either a power law or a stretched exponential, in general.     

General Solution for the Complete Separability of a Pure Quantum State with Real Coefficients for a Quantum Network of Any Nodes

By means of the criterion of entanglement in terms of the covariance correlation tensor in quantum network theory, this article discusses the general solution for the complete separability of the pure quantum state with real coefficients for a quantum network of any nodes.     

Measurements of Entanglement and a Quantum de Finetti Theorem

The problem of defining a natural measure of entanglement of mixed states on tensor products is considered from the point of view of a quantum de Finetti theorem for Bosons.     

Peierls Argument and Long-Range Order Behavior of Quantum Lattice Systems with Unbounded Spins

Results on long-range order behavior are obtained for systems in arbitrary dimension (v2) with a wide class of spin–spin long-range interactions, without assuming the reflection positivity property.