phase space tunneling for operators with symbols in a gevrey class phase space tunneling and exponential decay of eigenfunctions in phase space are well known for operators with symbols which are analytic in some neighborhood of the real axis this can be used to prove an adiabatic theorem of exponential order if one assumes the hamiltonian to depend analytically on time however to study compactly supported switching processes one has to weaken the analyticity assumptions here we examine nonanalytic symbols with gevrey class regularity and show that we get an exponential decay of the corresponding eigenfunctions with respect to 1 a as 0 where a 1 the loss of regularity causes a slower decay in the analysis is done using the methods of martinez and its generalization by nakamura an upper bound for the rate of decay is given

bifurcations of relative periodic orbits in nls gp with a triple well potential abstract the nonlinear schrodinger gross pitaevskii nls gp equation is considered in the presence of three equally spaced potentials the problem is reduced to a finite dimensional hamiltonian system by a galerkin truncation families of oscillatory orbits are sought in the neighborhoods of the system s nine branches of standing wave solutions normal forms are computed in the neighborhood of these branches various hamiltonian hopf and saddle node bifurcations showing how the oscillatory orbits change as a parameter is increased numerical experiments show agreement between normal form theory and numerical solutions to the reduced system and nls gp near the hamiltonian hopf bifurcations and some subtle disagreements near the saddle node bifurcations due to exponentially small terms in the asymptotics

multiscale simulation of transport in an open quantum system resonances and wkb interpolation a numerical scheme for the one dimensional stationary schrodinger poisson model is described the scheme is used to simulate a resonant tunneling diode and provides an important reduction of the simulation time the improvement is twofold first the grid spacing in the position variable is made coarser by using oscillating interpolation functions derived from the wkb asymptotics then the discretization of the energy variable which is a parameter for the schrodinger equation is improved by approaching the wavefunctions in the double barrier region by its projection on the resonant states following the work of presilla sjostrand and jona lasinio on schrodinger equations with concentrated non linearities ann phys 240 1995 1 21

nekhoroshev theorem for small amplitude solutions in nonlinear schrodinger equations we prove a nekhoroshev type result 26 27 for the nonlinear schrodinger equation with vanishing or periodic boundary conditions on 0 pi here m in mathbb r is a parameter and varphi mathbb r to mathbb r is a function analytic in a neighborhood of the origin and such that varphi 0 0 varphi 0 not 0 more precisely we consider the cauchy problem for 0 1 with initial data which extend to analytic entire functions of finite order and prove that all the actions of the linearized system are approximate constants of motion up to times growing faster than any negative power of the size of the initial datum the proof is obtained by a method which applies to hamiltonian perturbations of linear pde s with the following properties i the linear dynamics is periodic ii there exists a finite order birkhoff normal form which is integrable and quasi convex as a function of the action variables eq 0 1 satisfies i and ii when restricted to a level surface of left vert u l 2 right vert which is an integral of motion the main technical tool used in the proof is a normal form lemma for systems with symmetry which is also proved here

schrodinger poisson systems in dimension d 3 the whole space case after a study made for bounded domains and in the periodic case we investigate the variational formulation of schrodinger poisson systems set on the whole space d d 3 this variational formulation leads to a uniqueness result while the existence of a solution is proved only for small data because of the lack of coerciveness the end of this paper briefly presents the extension of this formalism to a physically relevant problem where the potential is periodic in one direction

instability of bound states for abstract nonlinear schrodinger equations abstract we study the instability of bound states for abstract nonlinear schrodinger equations we prove a new instability result for a borderline case between stability and instability we also reprove some known results in a unified way

elementary exponential error estimates for the adiabatic approximation we present an elementary proof that the quantum adiabatic approximation is correct up to exponentially small errors for hamiltonians that depend analytically on the time variable our proof uses optimal truncation of a straightforward asymptotic expansion we estimate the terms of the expansion with standard cauchy estimates

nonlinear double well schrodinger equations in the semiclassical limit we consider time dependent schrodinger equations with a double well potential and an external nonlinear both local and non local perturbation in the semiclassical limit the finite dimensional eigenspace associated to the lowest eigenvalues of the linear operator is almost invariant for times of the order of the beating period and the dominant term of the wavefunction is given by means of the solutions of a finite dimensional dynamical system in the case of local nonlinear perturbation we assume the spatial dimension d 1 or d 2

quantum and fisher information from the husimi and related distributions the two principal immediate influences which we seek to interrelate here upon the undertaking of this study are papers of yczkowski and slomczynski j phys a 34 6689 2001 and of petz and sudar j math phys 37 2262 1996 in the former work a metric the monge one specifically over generalized husimi distributions was employed to define a distance between two arbitrary density matrices in the petz sudar work completing a program of chentsov the quantum analog of the classically unique fisher information monotone metric of a probability simplex was extended to define an uncountable infinitude of riemannian also monotone metrics on the set of positive definite density matrices we pose here the questions of what is the specific unique fisher information metric for the classically defined husimi distributions and how does it relate to the infinitude of quantum metrics over the density matrices of petz and sudar we find a highly proximate small relative entropy relationship bet

the quantum beating and its numerical simulation abstract we examine the suppression of quantum beating in a one dimensional non linear double well potential made up of two focusing nonlinear point interactions the investigation of the schrodinger dynamics is reduced to the study of a system of coupled nonlinear volterra integral equations for various values of the geometric and dynamical parameters of the model we give analytical and numerical results on the way states which are initially confined in one well evolve we show that already for a nonlinearity exponent well below the critical value there is complete suppression of the typical beating behavior characterizing the linear quantum case

asymptotic first eigenvalue estimates for the biharmonic operator on a rectangle abstract we find an asymptotic expression for the first eigenvalue of the biharmonic operator on a long thin rectangle this is done by finding lower and upper bounds which become increasingly accurate with increasing length the lower bound is found by algebraic manipulation of the operator and the upper bound is found by minimising the quadratic form for the operator over a test space consisting of separable functions these bounds can be used to show that the negative part of the groundstate is small

a nonlinear schrodinger equation with two symmetric point interactions in one dimension we consider a time dependent one dimensional nonlinear schrodinger equation with a symmetric double well potential represented by two dirac s among our results we give an explicit formula for the integral kernel of the unitary semigroup associated with the linear part of the hamiltonian then we establish the corresponding strichartz type estimate and we prove local existence and uniqueness of the solution to the original nonlinear problem

ionization for three dimensional time dependent point interactions we study the time evolution of a three dimensional quantum particle under the action of a time dependent point interaction fixed at the origin we assume that the strength of the interaction t is a periodic function with an arbitrary mean under very weak conditions on the fourier coefficients of t we prove that there is complete ionization as t starting from a bound state at time t 0 moreover we prove also that under the same conditions all the states of the system are scattering states

stability and symmetry breaking bifurcation for the ground states of a nls with a δ interaction we determine and study the ground states of a focusing schrodinger equation in dimension one with a power nonlinearity 2 and a strong inhomogeneity represented by a singular point perturbation the so called attractive interaction located at the origin the time dependent problem turns out to be globally well posed in the subcritical regime and locally well posed in the supercritical and critical regime in the appropriate energy space the set of the nonlinear ground states is completely determined for any value of the nonlinearity power it exhibits a symmetry breaking bifurcation structure as a function of the frequency i e the nonlinear eigenvalue more precisely there exists a critical value of the nonlinear eigenvalue such that if 0 then there exist two non symmetric ground states we prove that before bifurcation i e for finally for 2 and omega gg omega all ground states are unstable the branch of odd ground states for obtaining a family of orbitally unstable stationary states existence of ground states is proved by variational techniques and the stability properties of stationary states are investigated by means of the grillakis shatah strauss framework where some non standard techniques have to be used to establish the needed properties of linearization operators

new approach to the semiclassical limit of quantum mechanics i multiple tunnelings in one dimension we propose a new approach for the estimate of the rate of degeneracy of the lowest eigenvalues of the schrodinger operator in the presence of tunneling based on the theory of diffusion processes our method provides lower and upper bounds for the energy splittings and the rates of localization of the wave functions and enables us to discuss cases which as far as we know have never been treated rigorously in the literature in particular we give an analysis of the effect on eigenvalues and eigenfunctions of localized deformations of 1 symmetric double well potentials 2 potentials periodic and symmetric over a finite interval theses situations are characterized by a remarkable dependence on such deformations our probabilistic techniques are inspired by the theory of small random perturbations of dynamical systems

general adiabatic evolution with a gap condition we consider the adiabatic regime of two parameters evolution semigroups generated by linear operators that are analytic in time and satisfy the following gap condition for all times the spectrum of the generator consists in finitely many isolated eigenvalues of finite algebraic multiplicity away from the rest of the spectrum the restriction of the generator to the spectral subspace corresponding to the distinguished eigenvalues is not assumed to be diagonalizable

a minimal energy tracking method for non radially symmetric solutions of coupled nonlinear schrodinger equations we aim at developing methods to track minimal energy solutions of time independent m component coupled discrete nonlinear schrodinger dnls equations we first propose a method to find energy minimizers of the 1 component dnls equation and use it as the initial point of the m component dnls equations in a continuation scheme we then show that the change of local optimality occurs only at the bifurcation points the fact leads to a minimal energy tracking method that guides the choice of bifurcation branch corresponding to the minimal energy solution curve by combining all these techniques with a parameter switching scheme we successfully compute a non radially symmetric energy minimizer that can not be computed by existing numerical schemes straightforwardly

rational equivariant homotopy theory we give an algebraicization of rational s 1 equivariant homotopy theory there is an algebraic category of t systems which is equival nt to the a homotopy category of rational s 1 simply connected s 1 spaces there is is also a theory of minimal models for t systems analogous to sullivan mini nal algebras each s 1 space has an associated minimal t system whic ence des 2 all of its rational homotopy information including its rational equivariant cohomology and postnikov decomposition

a class of nonlinear schrodinger equations with concentrated nonlinearity abstract we consider the nonlinear schrodinger equation in dimension one with a nonlinearity concentrated in a finite number of points detailed results on the local existence of the solution in fractional sobolev spaces h are given we also prove the conservation of the l 2 norm and the energy of the solution and give a global existence result for repulsive and weakly attractive interaction in the space h 1 finally we prove the existence of blow up solutions for strongly attractive interaction

on the adiabatic theorem for nonself adjoint hamiltonians the authors consider the evolution of a two level system driven by a nonself adjoint hamiltonian h in t and treat the adiabatic limit in to 0 while adiabatic theorem like results do not hold true in general for this case they prove that they are still valid for the subspace corresponding to the eigenvalue having the largest imaginary part least dissipative eigenvalue the theory gives the full asymptotic expansion of the evolution restricted to this subspace the first correction beyond berry s phase is to their best knowledge given explicitly for the first time

on time dependent schrodinger equations global well posedness and growth of sobolev norms in this paper we consider time dependent schrodinger linear pdes of the form i t l t where l t is a continuous family of self adjoint operators we give conditions for well posedness and polynomial growth for the evolution in abstract sobolev spaces if l t h v t where v t is a perturbation smooth in time and h is a self adjoint positive operator whose spectrum can be enclosed in spectral clusters whose distance is increasing we prove that the sobolev norms of the solution grow at most as t when t for any 0 if v t is analytic in time we improve the bound to log t for some 0 the proof follows the strategy due to howland joye and nenciu of the adiabatic approximation of the flow we recover most of known results and obtain new estimates for several models including 1 degree of freedom schrodinger operators on r and schrodinger operators on zoll manifolds

exponentially small adiabatic invariant for the schrodinger equation we study an adiabatic invariant for the time dependent schrfdinger equation which gives the transition probability across a gap from time t to time t when the hamiltonian depends analytically on time and t oo t oo we give sufficient conditions so that this adiabatic invariant tends to zero exponentially fast in the adiabatic limit

reflection coefficient beyond all orders for singular problems ii closed spaced critical points on the nearest critical level line a crucial connection formula with closed spaced critical points on the nearest critical level line is established in this paper in order to use the generalized pokrovskii khalatnikov method to calculate the nontrivial behavior of the reflection coefficient for singular problems

quantum phase corrections from adiabatic iteration the phase change gamma acquired by a quantum state psi t rangle driven by a hamiltonian h 0 t which is taken slowly and smoothly round a cycle is given by a sequence of approximants gamma k obtained by a sequence of unitary transformations the phase sequence is not a perturbation series in the adiabatic parameter epsilon because each gamma k except gamma 0 contains epsilon to infinite order for spin frac 1 2 systems the iteration can be described in terms of the geometry of parallel transport round loops c k on the hamiltonian sphere non adiabatic effects transitions must cause the sequence of gamma k to diverge for spin systems with analytic h 0 t this happens in a universal way the loops c k are sinusoidal spirals which shrink as epsilon k until k sim epsilon 1 and then grow as k the smallest loop has a size exp 1 epsilon comparable with the non adiabaticity

computing the steady states for an asymptotic model of quantum transport in resonant heterostructures in this article we propose a rapid method to compute the steady states including bifurcation diagrams of resonant tunneling heterostructures in the far from equilibrium regime those calculations are made on a simplified model which takes into account the characteristic quantities which arise from an accurate asymptotic analysis of the nonlinear schrodinger poisson system after a summary of the existing theoretical results the asymptotic model is explicitly adapted to physically realistic situations and numerical results are shown in various cases

precise coupling terms in adiabatic quantum evolution the generic case for multi level time dependent quantum systems one can construct superadiabatic representations in which the coupling between separated levels is exponentially small in the adiabatic limit based on results from bete1 for special hamiltonians we explicitly determine the asymptotic behavior of the exponentially small coupling term for generic two state systems with real symmetric hamiltonian the superadiabatic coupling term takes a universal form and depends only on the location and the strength of the complex singularities of the adiabatic coupling function our proof is based on a new norm which allows to rigorously implement darboux principle a heuristic guideline widely used in asymptotic analysis

bounds on the tight binding approximation for the gross pitaevskii equation with a periodic potential abstract we justify the validity of the discrete nonlinear schrodinger equation for the tight binding approximation in the context of the gross pitaevskii equation with a periodic potential both piecewise constant and smooth potentials are considered in the semi classical limit while justification of stationary equations is developed in our previous work pelinovsky et al 2008 11 this work deals with time dependent space decaying solutions on large but finite time intervals

the adiabatic limit of schrodinger operators on fibre bundles we consider schrodinger operators h delta g varepsilon v on a fibre bundle m mathop rightarrow limits pi b with compact fibres and a metric g varepsilon that blows up directions perpendicular to the fibres by a factor varepsilon 1 gg 1 we show that for an eigenvalue lambda of the fibre wise part of h satisfying a local gap condition and every n in mathbb n there exists a subspace of l 2 m that is invariant under h up to errors of order varepsilon n 1 the dynamical and spectral features of h on this subspace can be described by an effective operator on the fibre wise lambda eigenspace bundle mathcal e rightarrow b giving detailed asymptotics for h

fisher information kinetic energy and uncertainty relation inequalities by interpolating between fisher information and mechanical kinetic energy we introduce a general notion of kinetic energy with respect to a parameter of schrodinger wavefunctions from a statistical inference perspective kinetic energy is the sum of fisher information and an integral of a parametrized analogue of quantum mechanical current density related to phase a family of integral inequalities concerning kinetic energy and moments are established among which the cramer rao inequality and the weyl heisenberg inequality are special cases in particular the integral inequalities involving the negative order moments are relevant to the study of electron systems moreover by specifying the parameter to a scale we obtain a family of inequalities of uncertainty relation type which incorporate the position and momentum observables symmetrically in a single quantity

proof of the landau zener formula in an adiabatic limit with small eigenvalue gaps we consider a smooth operator valued functionh t that has two isolated non degenerate eigenvaluese a t ande b t for 0 we assume these eigenvalues are bounded away from the rest of the spectrum ofh t but have an avoided crossing with one another with a closest approach that iso as tends to zero under these circumstances we study the small e limit for the adiabatic schrodinger equation i varepsilon frac partial psi partial t h t varepsilon 1 2 psi we prove that the landau zener formula correctly describes the coupling between the adiabatic states associated with the eigenvaluese a t ande b t as the system propagates through the avoided crossing

the stationary current voltage characteristics of the quantum drift diffusion model this paper is concerned with numerical algorithms for the bipolar quantum drift diffusion model for the thermal equilibrium case a quasi gradient method minimizing the energy functional is introduced and strong convergence is proved the computation of current voltage characteristics is performed by means of an extended gummel iteration it is shown that the involved fixed point mapping is a contraction for small applied voltages in this case the model equations are uniquely solvable and convergence of the proposed iteration scheme follows numerical simulations of a one dimensional resonant tunneling diode are presented the computed current voltage characteristics are in good qualitative agreement with experimental measurements the appearance of negative differential resistances is verified for the first time in a quantum drift diffusion model

time development of exponentially small non adiabatic transitions optimal truncations of asymptotic expansions are known to yield approximations to adiabatic quantum evolutions that are accurate up to exponentially small errors in this paper we rigorously determine the leading order non adiabatic corrections to these approximations for a particular family of two level analytic hamiltonian functions our results capture the time development of the exponentially small transition that takes place between optimal states by means of a particular switching function our results confirm the physics predictions of sir michael berry in the sense that the switching function for this family of hamiltonians has the form that he argues is universal

non trivial prefactors in adiabatic transition probabilities induced by high order complex degeneracies the adiabatic limit of the transition probability for two level systems driven by real symmetric time dependent hamiltonians is considered when the hamiltonian depends analytically on time the transition probability is governed in simple cases by a complex eigenvalue crossing point which is generically a square root branching point for the eigenvalues in such a case the transition probability is given by the dykhne formula the recently discovered geometric prefactor being equal to one in this paper we deal with the general situation where the relevant eigenvalue crossing point is a branching point of order n 2 n or 1 for the eigenvalues and a zero of order m or 0 for the hamiltonian itself the analysis shows that the dykhne formula must be completed by a novel prefactor which depends on both n and m in particular this prefactor can take the value zero in contrast to the geometrical prefactor we also consider the case where the transition probability is governed by n complex eigenvalue crossing points of different orders nj and mj j 1 n the end result displays an interference phenomenon between the individual prefactors similar to the case of generic eigenvalue crossing points nj 1 and mj 0 considered earlier

purely nonlinear instability of standing waves with minimal energy we consider hamiltonian systems with u 1 symmetry we prove that in the generic situation the standing wave that has the minimal energy among all other standing waves is unstable in spite of the absence of linear instability essentially the instability is caused by higher algebraic degeneracy of the zero eigenvalue in the spectrum of the linearized system we apply our theory to the non

finitely correlated states on quantum spin chains we study a construction that yields a class of translation invariant states on quantum spin chains characterized by the property that the correlations across any bond can be modeled on a finite dimensional vector space these states can be considered as generalized valence bond states and they are dense in the set of all translation invariant states we develop a complete theory of the ergodic decomposition of such states including the decomposition into periodic neel ordered states the ergodic components have exponential decay of correlations all states considered can be obtained as local functions of states of a special kind so called purely generated states which are shown to be ground states for suitably chosen finite range vbs interactions we show that all these generalized vbs models have a spectral gap our theory does not require symmetry of the state with respect to a local gauge group in particular we illustrate our results with a one parameter family of examples which are not isotropic except for one special case this isotropic model coincides with the one dimensional antiferromagnet recently studied by affleck kennedy lieb and tasaki

exponential times in the one dimensional gross pitaevskii equation with multiple well potential we consider the gross pitaevskii equation in 1 space dimension with a n well trapping potential we prove in the semiclassical limit that the finite dimensional eigenspace associated to the lowest n eigenvalues of the linear operator is slightly deformed by the nonlinear term into an almost invariant manifold mathcal m precisely one has that solutions starting on mathcal m or close to it will remain close to mathcal m for times exponentially long with the inverse of the size of the nonlinearity as heuristically expected the effective equation on mathcal m is a perturbation of a discrete nonlinear schrodinger equation we deduce that when the size of the nonlinearity is large enough then tunneling among the wells essentially disappears that is for almost all solutions starting close to mathcal m their restriction to each of the wells has norm approximatively constant over the considered time scale in the particular case of a double well potential we give a more precise result showing persistence or destruction of the beating motions over exponentially long times the proof is based on canonical perturbation theory surprisingly enough due to the gauge invariance of the system no non resonance condition is required

some integral identities and inequalities for entire functions and their application to the coherent state transform let be an entire function on cn and for any h 0 and r 0 define fr z r e 2 z 2h let d h denote h n times lebesgue measure on cn frs2 2 d h n sh frsd h from this and a logarithmic sobolev inequality we easily deduce qnq fr q pnp fr p for all 0 p q t8 where the lp norms are taken with respect to the measure d h above we apply these results to the study of the spaces ap consisting of all entire functions satisfying z pe 2 z 2h d h obtaining sharp bounds for some associated operators and proving denseness of analytic polynomials in ap for 1 p we then apply our results to the coherent state transform extending and simplifying some previously known results

histories of adiabatic quantum transitions the way in which the transition amplitude to an initially unoccupied state increases to its exponentially small final value is studied in detail in the adiabatic approximation for a 2 state quantum system by transforming to a series of superadiabatic bases clinging ever closer to the exact evolving state it is shown that transition histories renormalize onto a universal one in which the amplitude grows to its final value as an error function rather than via large oscillations as in the ordinary adiabatic basis the time for the universal transition is of order surd hslash delta where delta is the small adiabatic slowness parameter in perturbation theory the pre exponential factor of the final amplitude renormalizes superadiabatically from the incorrect value textstyle frac 1 3 pi for the ordinary adiabatic basis to the correct value unity the various histories could be observed in spin experiments

local smoothing property and strichartz inequality for schrodinger equations with potentials superquadratic at infinity abstract we study smoothing properties for time dependent schrodinger equations i u t 1 2 u v x u u 0 l 2 r n with potentials which satisfy v x o x m at infinity m 2 we show that the solution u t x is 1 m times differentiable with respect to x at almost all t r and explain that this is the result of the fact that the sojourn time of classical particles with energy in arbitrary compact set is less than ct 1 m during 0 t when is very large we also show strichartz s inequality with derivative loss for such potentials and give its application to nonlinear schrodinger equations

horseshoes for coupled discrete nonlinear schrodinger equations in this paper we study the spatial disorder of coupled discrete nonlinear schrodinger cdnls equations with piecewise monotone nonlinearities by the construction of horseshoes we show that the cdnls equation possesses a hyperbolic invariant cantor set on which it is topological conjugate to the full shift on n symbols the cdnls equation exhibits spatial disorder resulting from the strong amplitudes and stiffness of the nonlinearities in the system the complexity of the disorder is determined by the oscillations of the nonlinearities we then apply our results to cdnls equations with kerr like nonlinearity we shall also show some patterns of the localized solutions of the cdnls equation

fisher information matrix of husimi distribution we calculate the fisher information matrix of husimi distribution in the fock bargmann representation it turns out that the fisher information of the position and that of the momentum move in opposite directions and that a weighted trace of the fisher information matrix is a constant independent of the husimi distribution this may be interpreted as a kind of uncertainty relation in the spirit of heisenberg uncertainty principle from the statistical inference point of view

nonlinear schrodinger operators and molecular structure we minimize the energy of a quantum hamiltonian on the tensor product of two hilbert spaces within the class of product states this yields a nonlinear schrodinger equation whose ground state may bifurcate producing symmetry breaking we describe a procedure for computing the ground states numerically and prove that it converges we argue that the nonlinear schrodinger equation has relevance to the issue of molecular structure in quantum chemistry and study an exactly soluble example in detail to support this claim the paper concludes with a brief discussion of other approaches to molecular structure

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

linear adiabatic theory exponential estimates a general adiabatic expansion is written down the basic result is that under appropriate smoothness conditions the adiabatic estimations can be pushed up to exponential order the results imply exponential estimates not only for hamiltonians analytic in a neighbourhood of the real axis but also for hamiltonians which in an appropriate sense beong to gevrey classes

on asymptotic perturbation theory for quantum mechanics almost invariant subspaces and gauge invariant magnetic perturbation theory singular perturbation theory for quantum mechanics is considered in a framework generalizing the spectral concentration theory under very general conditions asymptotic estimations on the rayleigh schrodinger expansions of the perturbed spectral projections are obtained as a consequence almost invariant subspaces of exponential order are constructed the results cover practically all singular perturbations considered in nonrelativistic quantum mechanics in the magnetic field case under the condition that the magnetic field does not increase at infinity a gauge invariant perturbation theory leading to convergent series with field dependent coefficients is developed

semiclassical limit for multistate klein gordon systems almost invariant subspaces and scattering theory by using the method of helffer and sjostrand to construct moyal projections we extend the almost invariant subspace theory to the semiclassical context applications to the semiclassical limit for two component klein gordon hamiltonian are given more precisely under the conditions that the potential is analytic and its eigenvalues never cross we prove that the scattering matrix is block diagonal up to exponentially small errors also we show how the existence of almost invariant subspaces leads to the existence of quasimodes with exponentially long lifetimes

precise exponential estimates in adiabatic theory general adiabatic evolutions associated to hamiltonians which admit a holomorphic extension with respect to the time variable in a complex strip and whose spectrum satisfies a gap condition are studied an explicit rate of exponential decay is given which is related to simple geometric quantities associated to the spectrum of the hamiltonian for the transition probability between the two parts of the spectrum when the evolution is taken from to

destruction of the beating effect for a non linear schrodinger equation we consider a non linear perturbation of a symmetric double well potential as a model for molecular localization in the semiclassical limit we prove the existence of a critical value of the perturbation parameter giving the destruction of the beating effect this value is twice the one corresponding to the first bifurcation of the fundamental state here we make use of a particular projection operator introduced by g nenciu in order to extend to an infinite dimensional space some known results for a two level system

the discrete spectrum of the spinless one dimensional salpeter hamiltonian perturbed by δ interactions we rigorously define the self adjoint one dimensional salpeter hamiltonian perturbed by an attractive of strength centred at the origin by explicitly providing its resolvent our approach is based on a coupling constant renormalization a technique used first heuristically in quantum field theory and implemented in the rigorous mathematical construction of the self adjoint operator representing the negative laplacian perturbed by the in two and three dimensions we show that the spectrum of the self adjoint operator consists of the absolutely continuous spectrum of the free salpeter hamiltonian and an eigenvalue given by a smooth function of the parameter the method is extended to the model with two twin attractive deltas symmetrically situated with respect to the origin in order to show that the discrete spectrum of the related self adjoint hamiltonian consists of two eigenvalues namely the ground state energy and that of the excited antisymmetric state we investigate in detail the dependence of these two eigenvalues on the two parameters of the model that is to say both the aforementioned strength and the separation distance with regard to the latter a remarkable phenomenon is observed differently from the well behaved schrodinger case the 1d salpeter hamiltonian with two identical dirac distributions symmetrically situated with respect to the origin does not converge as the separation distance shrinks to zero to the one with a single centred at the origin having twice the strength however the expected behaviour in the limit in the norm resolvent sense can be achieved by making the coupling of the twin deltas suitably dependent on the separation distance itself

an informational characterization of schrodinger s uncertainty relations heisenberg s uncertainty relations employ commutators of observables to set fundamental limits on quantum measurement the information concerning incompatibility non commutativity of observables is well included but that concerning correlation is missing schrodinger s uncertainty relations remedy this defect by supplementing the correlation in terms of anti commutators however both heisenberg s uncertainty relations and schrodinger s uncertainty relations are expressed in terms of variances which are not good measures of uncertainty in general situations e g when mixed states are involved by virtue of the wigner yanase skew information we will establish an uncertainty relation along the spirit of schrodinger from a statistical inference perspective and propose a conjecture the result may be interpreted as a quantification of certain aspect of the celebrated wigner araki yanase theorem for quantum measurement which states that observables not commuting with a conserved quantity cannot be measured exactly

normal forms and quantization formulae we consider the schrodinger operator where as is gevrey of order and has a unique non degenerate minimum a quantization formula up to an error of order is obtained for all eigenvalues of q lying in any interval with a 1 and 0 0 for eigenvalues in 0 1 the error is of order the proof is based upon uniform nekhoroshev estimates on the quantum normal form constructed quantizing the lie transformation

existence of dynamics for a 1d nls equation perturbed with a generalized point defect in the present paper we study the well posedness for the one dimensional cubic nls perturbed by a generic point interaction point interactions are described as the 4 parameter family of self adjoint extensions of the symmetric 1d laplacian defined on the regular functions vanishing at a point and in the present context can be interpreted as localized defects interacting with the nls field a previously treated special case is given by an nls equation with a defect which we generalize and extend as far as well posedness is concerned to the whole family of point interactions we prove existence and uniqueness of the local cauchy problem in strong form initial data and evolution in the operator domain of point interactions weak form initial data and evolution in the form domain of point interactions and conservation laws of mass and energy are proved for finite energy weak solutions of the problem which imply global existence of the dynamics a technical difficulty arises due to the fact that a power nonlinearity does not preserve the form domain for a subclass of point interactions to overcome it a technique based on the extension of resolvents of the linear part of the generator to maps between a suitable hilbert space and the energy space is devised and estimates are given which show the needed regularization properties of the nonlinear flow

normal form for the symmetry breaking bifurcation in the nonlinear schrodinger equation abstract we derive and justify a normal form reduction of the nonlinear schrodinger equation for a general pitchfork bifurcation of the symmetric bound state that occurs in a double well symmetric potential we prove persistence of normal form dynamics for both supercritical and subcritical pitchfork bifurcations in the time dependent solutions of the nonlinear schrodinger equation over long but finite time intervals

blow up for the pointwise nls in dimension two absence of critical power abstract we consider the schrodinger equation in dimension two with a fixed pointwise focusing nonlinearity and show the occurrence of a blow up phenomenon with two peculiar features first the energy threshold under which all solutions blow up is strictly negative and coincides with the infimum of the energy of the standing waves second there is no critical power nonlinearity i e for every power there exist blow up solutions this last property is uncommon among the conservative schrodinger equations with local nonlinearity

power law corrections to the kubo formula vanish in quantum hall systems in first order perturbation theory conductivity is given by the kubo formula which in a quantum hall system equals the first chern class of a vector bundle we apply the adiabatic theorem to show that these topological constraints quantize the averaged conductivity to all orders of perturbation theory hence the kubo formula is valid to all orders

splitting instability the unstable double wells in this paper we perform the semiclassical analysis of a pair of resonances in the case of a quasi symmetrical unstable double well we consider two kinds of asymmetric perturbations one supported in the infinite external well the other one of the stark kind we prove that the first perturbation is able to localize each state inside one of the internal wells so that we have linear stark effect and vanishing of the splitting at the crossing point of the two resonances this phenomenon is critical in the ratio between the internal and external barrier lengths and the critical value of the ratio is close to two possible applications to the molecular structure and to the vanishing of the inversion frequency are briefly discussed

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

geometric currents in piezoelectricity as a simple model for piezoelectricity we consider a gas of infinitely many non interacting electrons subject to a slowly time dependent periodic potential we show that in the adiabatic limit the macroscopic current is determined by the geometry of the bloch bundle as a consequence we obtain the king smith and vanderbilt formula up to errors smaller than any power of the adiabatic parameter

universal lower bounds on eigenvalue splittings for one dimensional schrodinger operators we provide lower bounds on the eigenvalue splitting for d2 dx2 v x depending only on qualitative properties ofv for example ifv is c on a b ande n en 1 are two successive eigenvalues of d2 dx2 v withu a u b 0 boundary conditions and if lambda mathop max limits e in e n 1 e n x in a b e v x 1 2 then e n e n 1 geqq pi lambda 2 exp left lambda b a right the exponential factor in such bounds are saturated precisely in tunneling examples our results arenot restricted tov s of compact support but only require e n mathop lim limits overline x to infty v x

bifurcation and stability for nonlinear schrodinger equations with double well potential in the semiclassical limit we consider the stationary solutions for a class of schrodinger equations with a symmetric double well potential and a nonlinear perturbation here in the semiclassical limit we prove that the reduction to a finite mode approximation give the stationary solutions up to an exponentially small term and that symmetry breaking bifurcation occurs at a given value for the strength of the nonlinear term the kind of bifurcation picture only depends on the nonlinearity power we then discuss the stability instability properties of each branch of the stationary solutions finally we consider an explicit one dimensional toy model where the double well potential is given by means of a couple of attractive dirac s delta pointwise interactions

full asymptotic expansion of transition probabilities in the adiabatic limit the authors consider a two level quantum mechanical system driven by an analytic time dependent hamiltonian of the form h in t in the adiabatic limit in or 0

reflection coefficient beyond all orders for singular problems this paper generalizes the pokrovskii khalatnikov method to calculate the actual behavior of the reflection coefficient for singular problems which vanishes to all orders in the small parameter e two different classes of reflection coefficient problems are considered

exponentially accurate error estimates of quasiclassical eigenvalues we study the behaviour of truncated rayleigh schrodinger series for the low lying eigenvalues of the one dimensional time independent schrodinger equation in the semiclassical limit 0 under certain hypotheses on the potential v x we prove that for any given small 0 there is an optimal truncation of the series for the approximate eigenvalues such that the difference between an approximate and exact eigenvalue is smaller than exp c for some positive constant c we also prove the analogous results concerning the eigenfunctions

a model category structure for equivariant algebraic models in the equivariant category of spaces with an action of a finite group algebraic minimal models exist which describe the rational homotopy for g spaces which are 1 connected and of finite type these models are diagrams of commutative differential graded algebras in this paper we prove that a model category structure exists on this diagram category in such a way that the equivariant minimal models are cofibrant objects we show that with this model structure there is a quillen equivalence between the equivariant category of rational g spaces satisfying the above conditions and the algebraic category of the models

linearized instability for nonlinear schrodinger and klein gordon equations in this paper i am introducing a new technique for proving instability of bound states for hamiltonian systems there are already two disparate types of instability results in the literature the approach developed by strauss shatah 20 gave an instability criterion coming from the variational structure of the problem on the other hand jones approach 11 produced a complementary criterion related to the difference between the number of negative eigenvalues of two selfadjoint operators using quite different techniques it turns out that with the methods developed in this paper these two criteria can be derived within a single framework that also leads to a generalization of the previous results finally in order to demonstrate how this method works i apply the instability criterion in some specific examples

birkhoff normal form for partial differential equations with tame modulus we prove an abstract birkhoff normal form theorem for hamiltonian partial differential equations pdes the theorem applies to semilinear equations with nonlinearity satisfying a property that we call tame modulus such a property is related to the classical tame inequality by moser in the nonresonant case we deduce that any small amplitude solution remains very close to a torus for very long times we also develop a general scheme to apply the abstract theory to pdes in one space dimensions and we use it to study some concrete equations nonlinear wave nlw equation nonlinear schrodinger nls equation with different boundary conditions an application to an nls equation on the d dimensional torus is also given in all cases we deduce bounds on the growth of high sobolev norms in particular we get lower bounds on the existence time of solutions