semi classical estimate of the residues of the scattering amplitude for long range potentials in this paper we study the residue of the scattering amplitude for the schrodinger operator with long range perturbation of the laplacian in the case where there are resonances exponentially close to the real axis if the resonances are simple and under a separation condition one proves that the residue of the scattering amplitude associated with a resonance is bounded by c h im here c h denotes an explicit constant depending polynomially on h 1 and the number of resonances in a fixed box this generalizes a recent result of stefanov concerning compactly supported perturbations and isolated resonances

semiclassical analysis of low lying eigenvalues ii tunneling we discuss the leading asymptotics of eigenvalue splittings of 2 x2v in the limit as x o and where v is a non negative potential with several zeros for example if eo x e1 x are the two lowest eigenvalues in a situation where v has precisely two zeros a and b related by a symmetry then limx x lin e1 x eo x is given as the distance from a to b in a certain riemann metric

minimality of descriptor representations under external equivalence abstract necessary and sufficient conditions are derived for the minimality of descriptor representations under external equivalence these conditions are stated completely in terms of the matrices e a b c and d use is made of the close connection between the descriptor representation and the so called pencil representation this connection is further exploited to derive the transformation group for minimal descriptor representations it is shown that the transformations coincide with the operations of strong equivalence as introduced by verghese et al ieee trans aut control ac 26 811 831 1981

real trajectories in complex time method and barrier penetrationlike phenomena various applications of the real trajectories in the complex time method are discussed it is shown that the proper way of summation over semiclassical orbits removes existing doubts and leads to the more general formulas than the standard wkb type one in the energy eigenvalue problem this method is also used in the description of explicitly time dependent phenomena in the problem of escaping from a metastable state the estimation of a lifetime is in a good agreement with the inverse width of a false ground state energy

resolvent conditions for the control of parabolic equations abstract since the seminal work of russell and weiss in 1994 resolvent conditions for various notions of admissibility observability and controllability and for various notions of linear evolution equations have been investigated intensively sometimes under the name of infinite dimensional hautus test this paper sets out resolvent conditions for null controllability in arbitrary time necessary for general semigroups sufficient for analytic normal semigroups for a positive self adjoint operator a it gives a sufficient condition for the null controllability of the semigroup generated by a which is only logarithmically stronger than the usual condition for the unitary group generated by ia this condition is sharp when the observation operator is bounded the proof combines the so called control transmutation method and a new version of the direct lebeau robbiano strategy the improvement of this strategy also yields interior null controllability of new logarithmic anomalous diffusions

sharp upper bounds on the number of resonances near the real axis for trapping systems we study resonances near the real axis jimz jd o h n n 1 and the corresponding resonant states for semiclassical long range operators p h without a priori assumptions on the distribution or on the multiplicities of the resonances we show that the truncated resonant states form a family of quasimode states for p h stable under small perturbations as a consequence they form also a family of quasimode states for any suitably defined selfadjoint reference operator p h therefore those resonances are perturbed eigenvalues of p h next we show that the semiclassical wave front set of the resonant states is contained in the set of trapped directions t we construct a suitable reference operator from p h by imposing a microlocal barrier outside t to show that the counting function for those resonances admits an upper bound of weyl s type connected with the measure of t we give an example of system for which this bound is optimal and also prove similar bound in case of classical scattering by obstacle this paper is devoted to a detailed study of the behavior of the resonances and resonant states near the real axis we work mainly in the semi classical setting but most results can be easily translated into the classical one by resonances near the real axis we mean resonances in a box h d a0 b0 ci s h 0 where 0 s h d o h k k 1 such resonances may exist only for trapping geometries we accept the convention here that resonances lie in the lower half plane

input output and state space representations of finite dimensional linear time invariant systems in this paper we develop various representations for systems described by aset of high order differential equations of the form r0w r1w rsw s 0 with r0 r1 rs not necessarily square matrices the variables w are the external variables particular attention is paid to the problem of obtaining minimal state space realizations and input output or input state output representations of such systems

operateurs conjugues et proprietes de propagation we use an algebraic criteria based on local positivity of a commutator which asserts the existence of a direction of propagation for the flowe iht associated to a self adjoint operatorh

sharp strichartz estimates on nontrapping asymptotically conic manifolds we obtain the strichartz inequalities u l q t l r x 0 1 m c u 0 l 2 m for any smooth n dimensional riemannian manifold m which is asymptotically conic at infinity with either short range or long range metric perturbation and nontrapping where u is a solution to the schrodinger equation iu t 1 2 m u 0 and 2 q r are admissible strichartz exponents 2 q n r n 2 this corresponds with the estimates available for euclidean space except for the endpoint q r 2 2 n n 2 when n 2 these estimates imply existence theorems for semi linear schrodinger equations on m by adapting arguments from cazenave and weissler and kato this result improves on our previous result which was an l 4 t x strichartz estimate in three dimensions it is closely related to results of staffilani tataru burq robbiano zuily and tataru who consider the case of asymptotically flat manifolds

complex scaling and the distribution of scattering poles the purpose of this paper is to establish sharp polynomial bounds on the number of scattering poles for a general class of compactly supported self adjoint perturbations of the laplacian in r n odd we also consider more general types of bounds that give sharper estimates in certain situations the general conclusion can be stated as follows the order of growth of the poles is the same as the order of growth of eigenvalues for corresponding compact problems from the few known cases however the exact asymptotics are expected to be different the scattering poles for compactly supported perturbations were rigorously defined by lax and phillips 13 in a more general setting they correspond to resonances the study of which has a long tradition in mathematical physics in the lax phillips theory they appear as the poles of the meromorphic continuation of the scattering matrix and coincide with the poles of the meromorphic continuation of the resolvent of the perturbed operator because of the latter characterization they can be considered as the analogue of the discrete spectral data for problems on noncompact domains the problem of estimating the counting function

geometric bounds on the density of resonances for semiclassical problems in this paper we shall give upper bounds on the number of resonances in certain regions in the complex plane close to the real axis for semiclassical operators like h 2 v x when h is small

transformations of linear systems under external equivalence abstract we consider systems of linear differential and algebraic equations in which some of the variables are distinguished as external variables two systems are called equivalent if the set of solutions for the external variables is the same for both systems we give an operational form for this definition of equivalence i e we describe a set of system transformations having the property that two systems are equivalent if and only if they can be taken into each other by transformations from that set next an algorithm is described to transform a given system in general form to a system in minimal state space form this algorithm differs from existing methods in that it first takes the equations to first order form so that each subsequent step can be formulated and interpreted in state space terms we also compute the structure indices in terms of a state space description in nonminimal form and use this to prove the minimality of the end result of the algorithm finally an application is shown to the problem of ill posedness of feedback connections

on the location of resonances for schrodinger operators in the semiclassical limit i resonances free domains abstract we show that under suitable non trapping conditions on the potential v at energy e the schrodinger operator h h 2 v exhibits no resonance in a neighbourhood of e for small h we briefly discuss some critical values of e like thresholds or barrier tops

semi classical dispersive estimates we prove dispersive estimates for the wave group e it sqrt p h and the schrodinger group e itp h where p h is a self adjoint elliptic second order differential operator depending on a parameter 0 h le 1 which is supposed to be a short range perturbation of h 2 delta delta being the euclidean laplacian in particular applications are made to non trapping metric perturbations and to perturbations by a magnetic potential

a comparison of the computational efficiency of three averaging procedures in a numerical scheme for the shallow water equations abstract three numerical schemes all based on numerical characteristic decomposition are compared in terms of their computational efficiency when applied to the shallow water equations the schemes differ through the use of alternative averaging procedures

lower bounds for resonance widths in potential and obstacle scattering explicit lower bounds are given for the size of the imaginary parts of resonances for schrodinger operators with non trapping or trapping potentials and for the dirichlet laplacian in the exterior of a star shaped obstacle both acting in three dimensions

resonance free regions in magnetic scattering by two solenoidal fields at large separation abstract we consider the problem of quantum resonances in magnetic scattering by two solenoidal fields at large separation in two dimensions this system has trapped trajectories oscillating between two centers of the fields we give a sharp lower bound on resonance widths when the distance between the two centers goes to infinity the bound is described in terms of backward amplitudes calculated explicitly for scattering by each solenoidal field the study is based on a new type of complex scaling method as an application we also discuss the relation to semiclassical resonances in scattering by two solenoidal fields

on the absence of resonances for schrodinger operators with non trapping potentials in the classical limit we provide bounds on resolvents of dilated schrodinger operators via exterior scaling this depends crucially on a non trapping condition on the potential which has a clear interpretation in classical mechanics these bounds imply absence of resonances due to the tail of the potential in the shape resonance problem

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

local smoothing for scattering manifolds with hyperbolic trapped sets we prove a resolvent estimate for the laplace beltrami operator on a scattering manifold with a hyperbolic trapped set and as a corollary deduce local smoothing we use a result of nonnenmacher zworski to provide an estimate near the trapped region a result of burq and cardoso vodev to provide an estimate near infinity and the microlocal calculus on scattering manifolds to combine the two

krein s formula and one dimensional multiple well abstract it is shown that all the discrete eigenvalues of a one dimensional schrodinger operator with a multiple well potential possess an asymptotic expansions in power of h 1 2 when h 0 a formula for all the coefficients of these expansions is given the method uses two main tools perturbation by boundary conditions and exponential decay of eigenfunctions which are developed in this article as a by product of this work the exponential localization of eigenvectors when h goes to zero can be proved

application of spectral deformation to the vlasov poisson system ii mathematical results this paper presents a mathematical description of the linearized vlasov poisson operator lk acting on a family of banach spaces xp related to l p r and the application of the method of spectral deformation to this model it is shown that a type a analytic family of operators lk c lk 0 lk can be associated with lk by means of this family the landau damped modes of the plasma are identified as the spectral resonances of lk existence and uniqueness of solutions to the initial value problem for the evolution equation g lk g is proven an expansion of any solution to the initial value problem with sufficiently smooth initial data is obtained in terms of the eigenfunctions of lk and a spectral integral over the essential spectrum this is applied to derive an expansion for solutions to the vlasov equation in which the landau damped portions of the distribution function are manifestly exhibited a self contained discussion of the spectral deformation method and an extension of it to ce

resonances en limite semiclassique et exposants de lyapunov we determine the width of resonance free domains in the complex plane for the semiclassical schrodinger operator h2 v x whenh 0 in terms of lyapunov exponents for the associated classical flow

perturbation theory for shape resonances and large barrier potentials we develop a systematic perturbation and resonance theory for the one dimensional schrodinger equation of the form d 2 dx 2 u x lambda v x e psi x 0 0 leqq x infty where the barrier potentialv x is supported only wherex 1 and is non negative there and is a real parameter tending to infinity we prove that every eigenvalue turns into a resonance or an eigenvalue for finite

lower bounds for shape resonances widths of long range schrodinger operators in this article we give general lower bounds of shape resonance widths in the semi classical limit for schrodinger operators in the exterior of smooth compact obstacles with dirichlet or neuman boundary conditions and with long range dilation analytic potentials these lower bounds are exponentially small with respect to the planck constant

strichartz estimates for long range perturbations we study local in time strichartz estimates for the schroedinger equation associated to long range perturbations of the flat laplacian on the euclidean space we prove that in such a geometric situation outside of a large ball centered at the origin the solutions of the schroedinger equation enjoy the same strichartz estimates as in the non perturbed situation the proof is based on the isozaki kitada parametrix construction if in addition the metric is non trapping we prove that the strichartz estimates hold in the whole space

control from an interior hypersurface we consider a compact riemannian manifold m possibly with boundary and m m an interior hypersurface possibly with boundary we study observation and control from for both the wave and heat equations for the wave equation we prove controllability from in time t under the assumption t gcc that all generalized bicharacteristics intersect transversally in the time interval 0 t for the heat equation we prove unconditional controllability from as a result we obtain uniform lower bounds for the cauchy data of laplace eigenfunctions on under t gcc and unconditional exponential lower bounds on such cauchy data

general lower bounds for resonances in one dimension lower bounds are derived for the magnitude of the imaginary parts of the resonance eigenvalues of a schrodinger operator frac d 2 dx 2 v x on the line depending only on the support and bounds ofv and on the real part of the resonance eigenvalue for example if the resonance eigenvalue is denotede i then there existc and l0 depending only on e ande such that if the support ofv is contained in an interval of length l l0 then left varepsilon right frac m 3 sqrt e m sqrt e 2 exp m ell 1 c ell 1 wherem v x e 1 2

absence of singular continuous spectrum for certain self adjoint operators we give a sufficient condition for a self adjoint operator to have the following properties in a neighborhood of a pointe of its spectrum a its point spectrum is finite b its singular continuous spectrum is empty c its resolvent satisfies a class of a priori estimates

invariant subspaces and invertibility properties for singular systems the general case abstract open loop definitions and properties of several subspaces for general singular systems are characterized by means of a fully algebraic distributional framework simple recursive algorithms for producing these spaces as well as related duality aspects turn out to follow directly from these definitions next we provide definitions and conditions for two notions of left right invertibility of a general singular system in terms of our distributions subspaces and rosenbrock s system matrix and we show which conditions represent the gap between our invertibility concepts finally we prove that in many cases left right invertibility is equivalent to left right invertibility of the system matrix

classical and generalized solutions of time dependent linear differential algebraic equations abstract a coordinate free reduction procedure is developed for linear time dependent differential algebraic equations that transforms their solutions into solutions of smaller systems of ordinary differential equations the procedure applies to classical as well as distribution solutions in the case of analytic coefficients the hypotheses required for the reduction not only are necessary for the validity of the existence and uniqueness results but even allow for the presence of singularities straightforward extensions including undetermined systems and systems with nonanalytic coefficients are also discussed

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

the spectrum of the scattering matrix near resonant energies in the semiclassical limit the object of study in this paper is the on shell scattering matrix s e of the schrodinger operator with the potential satisfying assumptions typical in the theory of shape resonances we study the spectrum of s e in the semiclassical limit when the energy parameter e varies from eres e to eres e where eres is a real part of a resonance and e is sufficiently small the main result of our work describes the spectral flow of the scattering matrix through a given point on the unit circle this result is closely related to the breit wigner effect

spectral distributions for long range perturbations abstract we study distributions which generalize the concept of spectral shift function for pseudo differential operators on r d we call such distributions spectral distributions relations between relative scattering determinants and spectral distributions are established they lead to the definition of regularized scattering phase these relations are analogous to the usual one for the standard spectral shift function we give several asymptotic properties in the high energy and semiclassical limits where both nontrapping and trapping cases are considered in particular we prove breit wigner formulae for the regularized scattering phases for semiclassical schrodinger operators with long range potentials

the shape resonance for a class of schrodinger operatorsh 2 2m v onl 2 n with potentials having minima embedded in the continuum of the spectrum and non trapping tails we show the existence of shape resonances exponentially close to the real axis as 0 the resonant energies are given by a convergent perturbation expansion in powers of a parameter exhibiting the expected exponentially small behaviour for tunneling

the efficient prediction of shallow water flows part 1 theory abstract finite difference schemes are presented for the solution of the one dimensional shallow water equations in open channels and the two dimensional shallow water equations the numerical results are subsequently compared for a range of problems

realization and canonicity for implicit systems this paper studies realization theory and canonicity conditions for implicit systems here an implicit system is any system defined by an input output finite dimensional linear time invariant relation in discrete or continuous time they are also called generalized or descriptor systems in the literature the familiar definitions of canonicity and minimality are extended to such systems and a so called reduced form proves that as for classical systems minimality and canonicity are two equivalent concepts we replace the notion of transfer function by a notion of external form and prove that minimal realization can always be achieved

the problem of two fixed centers bifurcation diagram for positive energies we give a comprehensive analysis of the euler jacobi problem of motion in the field of two fixed centers with arbitrary relative strength and for positive values of the energy these systems represent nontrivial examples of integrable dynamics and are analysed from the point of view of the energy momentum mapping from the phase space to the space of the integration constants in this setting we describe the structure of the scattering trajectories in phase space and derive an explicit description of the bifurcation diagram i e the set of critical value of the energy momentum map

effective limiting absorption principles and applications the limiting absorption principle asserts that if h is a suitable schrodinger operator and f lives in a suitable weighted l2 space namely h 0 1 2 sigma for some sigma 0 then the functions r lambda i varepsilon f h lambda i varepsilon 1 f converge in another weighted l2 space h 0 1 2 sigma to the unique solution u of the helmholtz equation h lambda u f which obeys the sommerfeld outgoing radiation condition in this paper we investigate more quantitative or effective versions of this principle for the schrodinger operator on asymptotically conic manifolds with short range potentials and in particular consider estimates of the form r lambda i varepsilon f h 0 1 2 sigma leq c lambda h f h 0 1 2 sigma we are particularly interested in the exact nature of the dependence of the constants c lambda h on both lambda and h it turns out that the answer to this question is quite subtle with distinctions being made between low energies lambda ll 1 medium energies lambda sim 1 and large energies lambda gg 1 and there is also a non trivial distinction between qualitative estimates on a single operator h possibly obeying some spectral condition such as non resonance or a geometric condition such as non trapping and quantitative estimates which hold uniformly for all operators h in a certain class using elementary methods integration by parts and ode techniques we give some sharp answers to these questions as applications of these estimates we present a global in time local smoothing estimate and pointwise decay estimates for the associated time dependent schrodinger equation as well as an integrated local energy decay estimate and pointwise decay estimates for solutions of the corresponding wave equation under some additional assumptions on the operator h

realization of autoregressive equations in pencil and descriptor form a linear system described by autoregressive equations with a given input output structure cannot be transformed to standard state space form if the implied input output relation is nonproper instead a realization in descriptor form must be used in this paper it is shown how to obtain minimal descriptor realizations from autoregressive equations without separating finite and infinite frequencies and without going through a reduction process external equivalence is used so that even situations in which there is no transfer matrix can be considered the approach is based on the so called pencil representation of linear systems and it is shown that there is a natural realization of autoregressive equations in pencil form in this way the link between the realization theories of willems and fuhrmann can also be clarified

a bound for the eigenvalue counting function for krein von neumann and friedrichs extensions for an arbitrary open nonempty bounded set rn n n and sufficiently smooth coefficients a b q we consider the closed strictly positive higher order differential operator a 2m a b q in l2 defined on w02m 2 associated with the differential expression 2m a b q j k 1n i j bj aj k i k bk q m m n and its krein von neumann extension ak 2m a b q in l2 denoting by n ak 2m a b q 0 the eigenvalue counting function corresponding to the strictly positive eigenvalues of ak 2m a b q we derive the bound n ak 2m a b q cvn 2 n 1 2m2m n n 2m n 2m 0 where c c a b q 0 with c in 0 0 is connected to the eigenfunction expansion of the self adjoint operator a 2m a b q in l2 rn defined on w2m 2 rn corresponding to 2m a b q here vn n 2 n 2 2 denotes the euclidean volume of the unit ball in rn our method of proof relies on variational considerations exploiting the fundamental link between the krein von neumann extension and an underlying abstract buckling problem and on the distorted fourier transform defined in terms of the eigenfunction transform of a 2 a b q in l2 rn we also consider the analogous bound for the eigenvalue counting function for the friedrichs extension af 2m a b q in l2 of a 2m a b q no assumptions on the boundary of are made

semiclassical asymptotics for the spectral function of long range schrodinger operators we obtain semiclassical asymptotics for the spectral function of schrodinger operators with long range potentials we study the case of non trapping energy levels and also the case of a potential well inside an island where resonances due to trapped trajectories appear

representability of impulsive input output behaviors in descriptor form abstract it is known that the descriptor representation of linear smooth behaviors imposes no constraints on the partitioning of external variables into inputs and outputs the issue is reconsidered here in the setting of impulsive behaviors in this context there are two different descriptor representations that may be used the conventional form and the unconventional one it is shown that the conventional descriptor form again does not constrain the choice of inputs and outputs but the unconventional form does the latter representation only allows input output partitionings in which inputs are dirac free this notion is newly defined here

lower bounds on the width of stark wannier type resonances we prove that the schrodinger operator d2 dx2 fx w x onl2 r withw bounded and analytic in a strip has no resonances in a region ime exp c f

the efficient prediction of shallow water flows part ii application abstract numerical results are presented and compared for the application of finite difference schemes for the shallow water equations on a range of problems

resonance expansions of propagators in the presence of potential barriers abstract we show that the schrodinger propagator can be expanded in terms of resonances at energy levels at which a barrier separates the interaction region from infinity the expansions hold for all times with errors small in the semi classical parameter as a byproduct we obtain a result on the approximation of clusters of resonant states by clusters of eigenfunctions of a self adjoint reference operator

the equivalence structure of descriptor representations of systems with possibly inconsistent initial conditions abstract necessary and sufficient conditions for minimality of descriptor representations of impulsive smooth behaviors are derived we obtain a complete set of transformations by which minimal descriptor representations that give rise to the same behavior can be transformed into each other in particular this leads to a jump behavioral interpretation of the notion of strong equivalence of descriptor representations

impulsive smooth behavior in multimode systems part ii minimality and equivalence this is the second part of a two part study of linear multimode systems in the first part it was argued that the behavior of such a system on an interval between switches should be described in a framework that allows for impulses at the switching instant and both first order and polynomial representations were introduced that satisfy this requirement here we determine the conditions under which first order representations are minimal we also show how two minimal representations of the same behavior are related this leads in particular to an appropriate state space isomorphism theorem the minimality conditions are given a dynamic interpretation

microlocalization of resonant states and estimates of the residue of the scattering amplitude we obtain some microlocal estimates of the resonant states associated to a resonance z0 of an h differential operator more precisely we show that the normalized resonant states are outside the set of trapped trajectories and are in the incoming area of the phase space as an application we show that the residue of the scattering amplitude of a schrodinger operator is small in some directions under an estimate of the norm of the spectral projector finally we prove such a bound in some examples

impulsive smooth behavior in multimode systems part i state space and polynomial representations abstract a switched or multimode system is one that can switch between various modes of operation we consider here switched systems in which the modes of operation are characterized as linear finite dimensional systems not necessarily all of the same mcmillan degree when a switch occurs from one of the modes to another of lower mcmillan degree the state space collapses and an impulse may result followed by a smooth evolution under the new regime this paper is concerned with the description of such impulsive smooth behavior on a typical interval we propose an algebraic framework modeled on the class of impulsive smooth distributions as defined by hautus both state space and polynomial representations are considered and we discuss transformations between the two forms

space time picture of semiclassical resonances this paper addresses the theory of quasiclassical resonances for schrodinger operators with potentials smooth outside some possible local singularities we introduce the notion of quasiresonance similar to that of quasimode but incorporating a condition revealing its scattering nature and describe its space time behaviour the definition is given in terms of the original schrodinger operator and uses a description of its frequency set the result on the space time behaviour justifies the intuitive picture of resonances as metastable states or bound states with finite life times we demonstrate how quasiresonances arise in several natural situations

trapped rays in cylindrically symmetric media and poles of the analytically continued resolvent abstract the acoustic stratified propagator is defined as the positive self adjoint operatork c2 y y z 1 y zon l 2 rn c2 1dnz wherez x y x rk y rm bounds on the weighted resolvent ofk defined as z k 1 z for 1 2 and z 1 z 2 1 2 are obtained at high energies i e in the semi classical limit re 0 using the virial conditionc y c ec 0 this implies energy decay estimates for trapped waves in fiber optics with further smoothness assumptions oncand the resolvent can be analytically extended onto the second riemann sheet the failure of the virial condition on some radial interval form 2 implies the existence ofresonant traveling modesfor the wave equation t2 k 0 in particular it is shown that ifn m 2 thenkhas resonance states at high energies no such states exist ifm 1 this is explained using a shape resonance picture

stark ladder resonances for small electric fields we prove the existence of resonances in the semi classical regime of smallh for stark ladder hamiltonians h h f equiv h 2 frac d 2 dx 2 v fx in one dimension the potentialv is a real periodic function with period which is the restriction to of a function analytic in a strip about the electric field strengthf satisfies the bounds v f 0 in general the imaginary part of the resonances are bounded above by open image in new window for some 0 0 independent ofh andf forh sufficiently small this is an extension of the oppenheimer formula for the stark effect to the case of periodic potentials

traceformula for resonances in small domains abstract traceformulas are useful in the study of resonances in this article we work in a semi classical black box framework and obtain a traceformula of the same type as that of the second author now for the resonances in h dependent domains of size h and under somewhat more restrictive assumptions as an application we extend to a long range situation a result by v petkov and m zworski on breit wigner peaks in the scattering phase

resonances in scattering by two magnetic fields at large separation and a complex scaling method abstract we study the quantum resonances in magnetic scattering in two dimensions the scattering system consists of two obstacles by which the magnetic fields are completely shielded the trajectories trapped between the two obstacles are shown to generate the resonances near the positive real axis when the distance between the obstacles goes to infinity the location is described in terms of the backward amplitudes for scattering by each obstacle a difficulty arises from the fact that even if the supports of the magnetic fields are largely separated from each other the corresponding vector potentials are not expected to be well separated to overcome this we make use of a gauge transformation and develop a new type of complex scaling method we can cover the scattering by two solenoids at large separation as a special case the obtained result heavily depends on the magnetic fluxes of the solenoids this indicates that the aharonov bohm effect influences the location of resonances

convergent expansions for tunneling a new method to compute effects of tunneling in one dimensional multiple well is developed a tunneling parameter built with physical quantities is introduced to measure the splitting between eigenvalues due to tunneling these splittings are given by convergent series in term of this tunneling parameter for a wide class of double well