the eigenvalue distribution of degenerate dirichlet forms we obtain a positive lower bound to the spectrum of certain second order elliptic operators h on l2 where rn has a lipschitz boundary and the coefficients of h become singular as one approaches the boundary we also find a general formula for the order of the asymptotic eigenvalue distribution of h in some situations where the classical limit formula of weyl courant titchmarsh and others is not applicable

nonclassical eigenvalue asymptotics abstract the leading asymptotics for the growth of the number of eigenvalues of the two dimensional dirichlet laplacian in the regions x y x y 1 and for x y all of which are non weyl because of infinite phase space volumes are computed along the way a general inequality on quantum partition functions computed in a kind of born oppenheimer approximation is proved

finite controllability of infinite dimensional quantum systems quantum phenomena of interest in connection with applications to computation and communication often involve generating specific transfers between eigenstates and their linear superpositions for some quantum systems such as spin systems the quantum evolution equation the schrodinger equation is finite dimensional and old results on controllability of systems defined on on lie groups and quotient spaces provide most of what is needed insofar as controllability of non dissipative systems is concerned however in an infinite dimensional setting controlling the evolution of quantum systems often presents difficulties both conceptual and technical in this paper we present a systematic approach to a class of such problems for which it is possible to avoid some of the technical issues in particular we analyze controllability for infinite dimensional bilinear systems under assumptions that make controllability possible using trajectories lying in a nested family of pre defined subspaces this result which we call the finite controllability theorem provides a set of sufficient conditions for controllability in an infinite dimensional setting we consider specific physical systems that are of interest for quantum computing and provide insights into the types of quantum operations gates that may be developed

the hardy rellich inequality for polyharmonic operators the hardy rellich inequality given here generalizes a hardy inequality of davies from the case of the dirichlet laplacian of a region n to that of the higher order polyharmonic operators with dirichlet boundary conditions the inequality yields some immediate spectral information for the polyharmonic operators and also bounds on the trace of the associated semigroups and resolvents

on the spectrum of the dirichlet laplacian for horn shaped regions in rn with infinite volume abstract an inequality for trace et d is proven where d is the dirichlet laplacian for horn shaped regions d in rn the results of rozenbljum and simon for the leading asymptotics for the growth of the number of eigenvalues of the two dimensional dirichlet laplacian in the regions x y x y 1 0 are easily recovered an example of a horn shaped region in r2 where that asymptotics is exponential is given

trace properties of the dirichlet laplacian on obtient des bornes superieures et inferieures pour tr exp ht en termes de la fonction r x

on the relaxation of fluctuations in the steady state of the stratonovich model a detailed investigation has been carried out of the relaxation of fluctuations in the steady state of the stratonovich model also known as the random growing rate model the autocorrelation time of this system driven by parametric white noise in the linear term has been measured for an electronic circuit model computed by digital simulation and calculated by use of a continued fraction expansion method the results are all consistent with each other and with those obtained from the matrix continued fraction method of jung and risken 1985 provided that explicit account is taken of the ways in which any macroscopic real physical system is bound to differ from the idealisation represented by the original form of the stratonovich model in particular it is necessary to recognise the existence of weak additive noise and a small additive constant the physical origins of some earlier seemingly discrepant calculations and experimental data are discussed

eigenvalue asymptotics of the neumann laplacian of regions and manifolds with cusps abstract we study the eigenvalue asymptotics of a neumann laplacian n in unbounded regions of r2 with cusps at infinity a typical example is x y r 2 x 1 y x 2 and prove that n e n n e h v e 2 vol where hv is the canonical one dimensional schrodinger operator associated to the problem we establish a similar formula for manifolds with cusps and derive the eigenvalue asymptotics of a dirichlet laplacian d for a class of cusp type regions of infinite volume

non linear dynamics of systems coupled with external noise some exact results abstract we solve exactly a class of non linear stochastic differential equations describing a variety of physical and chemical relevant systems coupled to external sources of noise the static and dynamic solutions are valid in the whole space of parameters including the critical points of non equilibrium instabilities critical slowing down is found and an unexpected long time tail evolution is predicted near and at the critical point this last effect is completely due to the coupling with the external noise

on the asymptotics of the heat equation and bounds on traces associated with the dirichlet laplacian we use the feynman kac formula and a decomposition of the brownian bridge to obtain pointwise estimates on the diagonal elements of the heat kernel we also compute bounds on trace et d where d is the dirichlet laplacian for i a bounded region d in rn with smooth boundary or ii an unbounded region d in r2 with finite area

controllability of quantum harmonic oscillators it is proven in a previous paper that any modal approximation of the one dimensional quantum harmonic oscillator is controllable we prove here that contrary to such finite dimensional approximations the original infinite dimensional system is not controllable its controllable part is of dimension 2 and corresponds to the dynamics of the average position more generally we prove that for the quantum harmonic oscillator of any dimension similar lacks of controllability occur whatever the number of control is the controllable part still corresponds to the average position dynamics we show with the quantum particle in a moving quadratic potential that some physically interesting motion planning questions can be however solved

the effect of multiplicative noise on the relaxation time of a real non linear physical system a comparison of experiment and theory for the random growing rate model rgrm it is found that the theoretical behaviour of the relaxation time t for the random growing rate model rgrm under the influence of multiplicative noise of intensity q a monotonic decrease of t 1 towards a limiting value as q to infinity differs markedly from that actually measured in electronic simulators of that system a new electronic analogue experiment is described in which a distinct minimum in t 1 q has been observed for the first time with clear evidence for an increase of t 1 q with q at large q in good qualitative agreement with an earlier analogue experiment the discrepancy between experiment and the theoretical solutions of the idealised equation is attributed to the profound influence exerted by the very weak additive noise which must also in some measure always be present in a real physical system

a new approximate stationary probability distribution for processes driven by ornstein uhlenbeck noise a novel approximate probability distribution for a class of nonlinear processes driven by multiplicative linear ornstein uhlenbeck noise is presented a comparison is made with other approximate distributions reported in the literature

dynamics of the stochastic duffing oscillator in gaussian approximation abstract the previously developed general concept of the gaussian approximation allows the description of the driven stochastic duffing oscillator by ordinary differential equations these equations are analyzed analytically as well as numerically to investigate the modification of simple bifurcations and of the feigenbaum cascade by stochastic forces furthermore the method allows a qualitative discussion of the stochastic system in the deterministic irregular region

on the controllability of quantum mechanical systems the systems theoretic concept of controllability is elaborated for quantum mechanical systems sufficient conditions being sought under which the state vector can be guided in time to a chosen point in the hilbert space h of the system the schrodinger equation for a quantum object influenced by adjustable external fields provides a state evolution equation which is linear in and linear in the external controls thus a bilinear control system for such systems the existence of a dense analytic domain d in the sense of nelson together with the assumption that the lie algebra associated with the system dynamics gives rise to a tangent space of constant finite dimension permits the adaptation of the geometric approach developed for finite dimensional bilinear and nonlinear control systems conditions are derived for global controllability on the intersection of d with a suitably defined finite dimensional submanifold of the unit sphere sh in h several soluble examples are presented to illuminate th

refined asymptotic expansion for the partition function of unbounded quantum billiards this paper presents a refined asymptotic expansion for the partition function t tr et of quantum billiards in the unbounded regions 0 x 0 yx 1 0 and 0 ye x 1 r2 where is the dirichlet laplacian simon ann phys 146 209 1983 j funct anal 53 84 1983 determined the leading divergence of the trace of the heat kernel for the first class of systems standard techniques are combined for the evaluation of for bounded region billiards with results by van den berg j funct anal 71 279 1987 for horn shaped regions using an optimized way of dividing the region into narrow and wide parts to determine the first three terms in the asymptotic expansion of results are also stated for bounded regions with cusps that can be obtained by the same method as an application the spectral staircase of the strongly chaotic billiard system defined in the region 0 xy 2 x 0 which has been discussed in connection with the riemann function and the search for quantum chaos is c

exact solution for the diffusion in bistable potentials we solve analytically the fokker planck equation for a one parameter family of symmetric attractive nonharmonic potentials which include double well situations the exact knowledge of the eigenfunctions and eigenvalues allows us to fully discuss the transient behavior of the probability density in particular for the bistable potentials we can give analytical expressions for the probability current over the working barrier and for the onset time which characterizes the transition from uni to bimodal probability densities

wavefunction controllability for finite dimensional bilinear quantum systems we present controllability results for quantum systems interacting with lasers exact controllability for the wavefunction in these bilinear systems is proved in the finite dimensional case under very natural hypotheses

method of divergent series summation in the problem of particle diffusion in a bistable potential an infinite system for the moments of a particle moving in a potential with two minima under the action of fluctuations is considered a class of solutions is found for which a divergent series can be summed up to a simple transcendent function it is shown numerically that the true solution belongs to this class and conditions are set to define the solution uniquely

localization in infinite billiards a comparison between quantum and classical ergodicity consider the non compact billiard in the first quandrant bounded by the positive x semiaxis the positive y semiaxis and the graph of f x x 1 1 2 although the schnirelman theorem holds the quantum average of the position xis finite on any eigenstate while classical ergodicity entails that the classical time average of xis unbounded

generating function for a multiplicative noise with group analysis the authors find the differential equation for the generating function or a multiplicative stochastic process and they apply to it the group analysis they give the general form of the lie generators and find the conditions for the existence of similarity solutions two classes of similarity solutions are presented and the analytic expression of the generating function is given