eigenvalue estimates for schrodinger operators with complex potentials we discuss properties of eigenvalues of non self adjoint schrodinger operators with complex valued potential v among our results are estimates of the sum of powers of imaginary parts of eigenvalue

the low energy scattering for slowly decreasing potentials for the radial schrodinger equation with a potentialq x decreasing at infinity asq0q 0 2 the low energy asymptotics of spectral and scattering data is found in particular it is shown that forq0 0 the spectral function vanishes exponentially as the energyk2 tends to zero on the contrary there is always a zero energy resonance forq0 0 t in the case 1 2 it is shown that for q0 0 the phase shift tends to ask 0 and its asymptotics is evaluated

on the asymptotical behavior of nonautonomous dynamical systems abstract the uniform asymptotical behavior of nonautonomous dynamical systems and their attractors is investigated in particular it is shown that parametrically inflated pullback attractors are uniformly forward attracting and also that under appropriate conditions the component sets of the pullback attractor a of a system form an almost periodic setvalued mapping t a t p when its driving system is almost periodic this together with the attraction properties of a demonstrates that almost periodic systems exhibit global almost periodic asymptotic behavior

bounds on complex eigenvalues and resonances we obtain bounds on the complex eigenvalues of non self adjoint schrodinger operators with complex potentials and also on the complex resonances of self adjoint schrodinger operators our bounds are compared with numerical results and are seen to provide useful information

isotropic hypoellipticity and trend to equilibrium for the fokker planck equation with a high degree potential we consider the fokker planck equation with a confining or anti confining potential which behaves at infinity like a possibly high degree homogeneous function hypoellipticity techniques provide the well posedness of the weak cauchy problem in both cases as well as instantaneous smoothing and exponential trend to equilibrium lower and upper bounds for the rate of convergence to equilibrium are obtained in terms of the lowest positive eigenvalue of the corresponding witten laplacian with detailed applications

on the trend to global equilibrium in spatially inhomogeneous entropy dissipating systems the linear fokker planck equation we study the long time behavior of kinetic equations in which transport and spatial confinement in an exterior potential or in a box are associated with a degenerate collision operator acting only in the velocity variable we expose a general method based on logarithmic sobolev inequalities and the entropy to overcome the well known problem due to the degeneracy in the position variable of the existence of infinitely many local equilibria this method requires that the solution be somewhat smooth in this paper we apply it to the linear fokker planck equation and prove decay to equilibrium faster than o t 1 for all 0 2001 john wiley sons inc

on dynamical properties of general dynamical systems and differential inclusions abstract let f be a general dynamical system defined on a complete locally compact metric space x we give a slightly improved version of the lyapunov characterization of asymptotic stability in one of our previous works and provide a short self contained proof for the existence of arbitrarily small positively invariant neighborhoods of compact asymptotically stable sets in the present context based on these results we prove that the uniform attractors of f are connected and are stable with respect to perturbations under appropriate conditions we are also interested in the dynamical properties of differential inclusion x t f x t on r m first we show that if no solutions of the system blow up in finite future time then its reachable mapping f is a general dynamical system then we discuss some asymptotic stability properties of the system in particular we prove that if there exists a nonempty compact connected subset m of r m such that m attracts a neighborhood of itself then the system has a connected uniform attractor a we also prove that a is stable with respect to both internal and external perturbations more precisely we prove that when 0 is sufficiently small the perturbed system x t con f x t b 1 b 1 has a connected uniform attractor a moreover h a a 0 as 0 where h is the hausdorff distance in r m

a simple criterion for the existence of nonreal eigenvalues for a class of 2d and 3d pauli operators abstract in this work we investigate the discrete spectrum generated by complex matrix valued perturbations for a class of 2d and 3d pauli operators with nonconstant magnetic fields we establish a simple criterion for the potentials to produce discrete spectrum near the low ground energy of the operators moreover in case of creation of nonreal eigenvalues this criterion specifies also their location

pseudospectra of semiclassical pseudo differential operators the pseudo spectra or spectral instability of non selfadjoint operators is a topic of current interest in applied mathematics for example in computational fluid dynamics it affects the study of the stability of laminar flows in fact even for the most basic flows the computations entirely fails to predict what is observed in the experiments the explanation is that for non normal operators the resolvent could be very large far away from the spectrum which makes computation of the eigenvalues impossible the occurence of false eigenvalues is due to the existence of quasi modes i e approximate local solutions to the eigenvalue problem the quasi modes appear since the nirenberg treves condition psi is not satisfied for topological reasons less

eigenvalue bounds for schrodinger operators with complex potentials iii laptev and safronov conjectured that any non positive eigenvalue of a schrodinger operator v in l 2 r with complex potential has absolute value at most a constant times v 2 2 for 0 2 in dimension 2 we prove this conjecture for radial potentials if 0 2 and we almost disprove it for general potentials if 1 2 2 in addition we prove various bounds that hold in particular for positive eigenvalues

eigenvalue bounds for dirac and fractional schrodinger operators with complex potentials abstract we prove lieb thirring type bounds for fractional schrodinger operators and dirac operators with complex valued potentials the main new ingredient is a resolvent bound in schatten spaces for the unperturbed operator in the spirit of frank and sabin 11

when is a fuzzy relation decomposable in two fuzzy sets two necessary and sufficient conditions are given in order to decompose an assigned fuzzy relation in two fuzzy sets

pseudomodes for schrodinger operators with complex potentials abstract for one dimensional schrodinger operators with complex valued potentials we construct pseudomodes corresponding to large pseudoeigenvalues we develop a first systematic non semi classical approach which results in a substantial progress in achieving optimal conditions and conclusions as well as in covering a wide class of previously inaccessible potentials including discontinuous ones applications of the present results to higher dimensional schrodinger operators are also discussed

accumulation of complex eigenvalues of a class of analytic operator functions for analytic operator functions we prove accumulation of branches of complex eigenvalues to the essential spectrum moreover we show minimality and completeness of the corresponding system of eig

uniform resolvent estimates and absence of eigenvalues for lame operators with complex potentials abstract we consider the 0 order perturbed lame operator v x it is well known that if one considers the free case namely v 0 the spectrum of is purely continuous and coincides with the non negative semi axis the first purpose of the paper is to show that at least in part this spectral property is preserved in the perturbed setting precisely developing a suitable multipliers technique we will prove the absence of point spectrum for lame operator with potentials which satisfy a variational inequality with suitable small constant we stress that our result also covers complex valued perturbation terms moreover the techniques used to prove the absence of eigenvalues enable us to provide uniform resolvent estimates for the perturbed operator under the same assumptions about v

on the schein rank of matrices over linear lattices abstract we extend the classical concept of schein rank of a boolean matrix to the case of a matrix a defined on a linear lattice and we prove that it coincides with the smallest regular integer for a in other words we show that there exists a partition of the set of the elements of a into classes such that it is possible to find suitable cross vectors whose sum is a

observer based controllers for max plus linear systems max plus algebra is a suitable algebraic setting to model discrete event systems involving synchronization and delay phenomena which are often found in transportation networks communications systems and manufacturing systems one way of controlling this kind of systems consists in choosing the dates of input events in order to achieve the desired performances e g to obtain output events in order to respect given dates this kind of control is optimal according to a just in time criterion if the input event dates are delayed as much as possible while ensuring the output events to occur before a desired reference date this paper presents the observer based controller for max plus linear systems where only estimations of system states are available for the controller as in the classical sense this is a state feedback control problem which is solved in two steps first an observer computes an estimation of the state by using the input and the output measurements then this estimated state is used to compute the state feedback control action as a main result it is shown that the optimal solution of this observer based control problem leads to a greater control input than the one obtained with the output feedback strategy a high throughput screening example in drug discovery illustrates this main result by showing that the scheduling obtained from the observer based controller is better than the scheduling obtained from the output feedback controller

schrodinger operators with slowly decaying potentials several recent papers have obtained bounds on the distribution of eigenvalues of non self adjoint schrodinger operators and resonances of self adjoint operators in this paper we describe two new methods of obtaining such bounds when the potential decays more slowly than previously permitted

resolvent estimates and bounds on eigenvalues for dirac operators on the half line estimates for the eigenvalues of non self adjoint 1d dirac operators considered on the half line are obtained in terms of the l p norms of the potentials the proofs are based on the resolvent est

large time asymptotics of solutions to the kramers fokker planck equation with a short range potential in this work we use a scattering method to study the kramers fokker planck equation with a potential whose gradient tends to zero at the infinity for short range potentials in dimension three we show that complex eigenvalues do not accumulate at low energies and obtain the low energy resolvent asymptotics this combined with high energy pseudospectral estimates valid in more general situations gives the large time asymptotics of solutions in weighted l 2 spaces

time decay of semigroups generated by dissipative schrodinger operators abstract we establish a representation formula for semigroups of contractions in terms of a global limiting absorption principle as applications we prove long time asymptotics and dispersive estimate of the semigroup generated by i h where h is a dissipative schrodinger operator

low energy asymptotics for schrodinger operators with slowly decreasing potentials low energy behavior of schrodinger operators with potentials which decay slowly at infinity is studied it is shown that if the potential is positive then the zero energy is very regular and the resolvent is smooth near 0 this implies rapid local decay for the solutions of the schrodinger equation on the other hand if the potential is negative then the resolvent has discontinuity at zero energy thus one cannot expect local decay faster than ordert 1 ast

improved eigenvalue bounds for schrodinger operators with slowly decaying potentials we extend a result of davies and nath j comput appl math 148 1 1 28 2002 on the location of eigenvalues of schrodinger operators with slowly decaying complex valued potentials to higher dimensions in this context we also discuss various examples related to the laptev and safronov conjecture laptev and safronov in commun math phys 292 1 29 54 2009

non equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures we study the statistical mechanics of a finite dimensional non linear hamiltonian system a chain of anharmonic oscillators coupled to two heat baths described by wave equations assuming that the initial conditions of the heat baths are distributed according to the gibbs measures at two different temperatures we study the dynamics of the oscillators under suitable assumptions on the potential and on the coupling between the chain and the heat baths we prove the existence of an invariant measure for any temperature difference i e we prove the existence of steady states furthermore if the temperature difference is sufficiently small we prove that the invariant measure is unique and mixing in particular we develop new techniques for proving the existence of invariant measures for random processes on a non compact phase space these techniques are based on an extension of the commutator method of hormander used in the study of hypoelliptic differential operators

on dynamical stability in general dynamical systems this paper is concerned with dynamical stability of general dynamical systems we discuss invariance properties of some limit sets investigate connections between various notions and definitions related to stability and attraction properties and establish existence results for invariant uniform attractors we also give a characterization of asymptotic stability of general dynamical systems via uniformly unbounded lyapunov functions