eigenvalues of the reference operator and semiclassical resonances we prove that the estimate of the number of the eigenvalues in intervals 0 0 and the number of the positive eigenvalues of l h in we apply this result to obtain a breit wigner approximation of the derivative of the spectral shift function near critical energy levels

the spectral norm of a nonnegative matrix abstract given a matrix a a ij define a a ij let 2 denote the spectral norm we show that for any matrix a a 2 min r 1 b c 1 c b c a and show that under mild conditions the minimizers in 1 are essentially unique and are related to the left and right singular vectors of a in a simple way we also show that a 2 a 2 and determine the case of equality

a semi classical trace formula at a non degenerate critical level abstract we study the semi classical trace formula at a critical energy level for a h pseudodifferential operator whose principal symbol has a unique non degenerate critical point for that energy this leads to the study of hamiltonian systems near equilibrium and near the non zero periods of the linearized flow the contributions of these periods to the trace formula are expressed in terms of degenerate oscillatory integrals the new results obtained are formulated in terms of the geometry of the energy surface and the classical dynamics on this surface

solution of toeplitz normal equations by sine transform based preconditioning abstract the normal equations constructed by a toeplitz matrix are studied in order to find a suitable preconditioner related to the discrete sine transform new results are given about the structure of the product of two toeplitz matrices which allow the cgn method to achieve a superlinear rate of convergence this preconditioner outperforms the circulant one for the iterative solution of toeplitz least squares problems such strategy can also be applied to nonsymmetric linear systems a block generalization is discussed

on the spectral norm of the matrix with integer sequences abstract in this study we compute the spectral norm of the matrix a x x i x j i j 1 n where x i are ith components of any integer sequence x n also we give a corollary and examples related to spectral norm

triangular truncation of k fibonacci and k lucas circulant matrices abstract we prove a general theorem that gives tight bounds on the spectral norms of triangularly truncated k fibonacci and k lucas circulant matrices the bounds are good enough to enable the calculation of the limit c c as the dimension n approaches infinity where c denotes the triangular truncation of c and c is any n n circulant matrix built using a sequence s i satisfying s i k s i 1 s i 2 in particular we have that this limit is equal to the golden ratio if c is built using either the ordinary fibonacci or lucas sequence

the spectral norms of g circulant matrices with classical fibonacci and lucas numbers entries abstract in this paper we investigate explicit formulae of spectral norms for g circulant matrices and prove them in details the entries in the first row of the g circulant matrices are a i f i s and a i l i s s 1 2 f i and l i denotes the fibonacci number and lucas number respectively employing these approaches some numerical tests are presented to verify the results

quantum ergodicity and symmetry reduction abstract we study the spectral and ergodic properties of schrodinger operators on a compact connected riemannian manifold m without boundary in case that the examined system possesses certain symmetries more precisely if m carries an isometric and effective action of a compact connected lie group g we prove a generalized equivariant version of the semiclassical weyl law with an estimate for the remainder using a semiclassical functional calculus for h dependent functions and relying on recent results on singular equivariant asymptotics we then deduce an equivariant quantum ergodicity theorem under the assumption that the symmetry reduced hamiltonian flow on the principal stratum of the singular symplectic reduction of m is ergodic in particular we obtain an equivariant version of the shnirelman zelditch colin de verdiere theorem as well as a representation theoretic equidistribution theorem if m g is an orbifold similar results were recently obtained by kordyukov when g is trivial one recovers the classical results

asymptotic approximation of degenerate fiber integrals we study asymptotics of fiber integrals depending on a large parameter when the critical fiber is singular full asymptotic expansions are established in two different cases local extremum and isolated real principal type singularities the main coefficients are computed and invariantly expressed in the most singular cases it is shown that the leading term of the expansion is related to invariant measures on the spherical blow up of the singularity the results can be applied to certain degenerate oscillatory integrals which occur in spectral analysis and quantum mechanics

towards a semiclassical theory of the quantum baker s map abstract we develop semiclassical 0 treatments of the quantum baker s map paying special attention to a the discrete and finite nature of its hilbert space and b the asymptotic analysis of the true quantum objects themselves new techniques are described at the theoretical level exact symbolic decomposition of the propagator and at the numerical level color graphics display of operators in the kirkwood representation both of which are of general interest for other quantized chaotic maps periodic orbit theory is tested numerically and we find that log corrections need to be included before an accurate calculation of the spectrum from periodic orbits can be achieved however the asymptotic analysis of the traces of the propagator reveals unexpected departures from standard semiclassical theory those anomalies which we compute explicitly in a few cases are traced ultimately to the discontinuity of the map and to the compactness of the phase space the analysis is far from complete but the results point towards the necessity of careful assessment of semiclassical folklore when applied to chaotic maps

contributions of non extremum critical points to the semi classical trace formula we study the semi classical trace formula at a critical energy level for a h pseudo differential operator on rn whose principal symbol has a totally degenerate critical point for that energy we compute the contribution to the trace formula of isolated non extremum critical points under a condition of real principal type the new contribution to the trace formula is valid for all time in a compact subset of r but the result is modest since we have restrictions on the dimension

ergodicite et limite semi classique consider a self adjoint quantic hamiltonian p h p x hdx whereh 0 is the planck s constant andp some smooth classical observable on the phase space r2n when the classical flow on a compact energy shell p is ergodic we prove that in the limith 0 almost all the eigenfunctions ofp h whose energy is near of are distributed according to the liouville measure on p

l 2 restriction bounds for eigenfunctions along curves in the quantum completely integrable case we show that for a quantum completely integrable system in two dimensions the l2 normalized joint eigenfunctions of the commuting semiclassical pseudodifferential operators satisfy restriction bounds of the form int gamma varphi j hbar 2 ds mathcal o log hbar for generic curves on the surface we also prove that the maximal restriction bounds of burq gerard tzvetkov bgt are generically attained for certain exceptional subsequences of eigenfunctions

the fibonacci identities of orthogonality abstract in even dimensions the orthogonal projection onto the two dimensional space of second order recurrence sequences is particularly nice it is a scaled hankel matrix whose entries consist of the classical fibonacci sequence a new proof is given of this result and new fibonacci identities are derived from it examples are given showing that familiar fibonacci identities can be viewed as special cases we show that the projection in odd dimensions can be written as a rank one lucas perturbation of a scaled lucas hankel matrix from which more fibonacci identities are derived

convolutions of semi classical spectral distributions and periodic orbit theory abstract we study the convolution of semi classical spectral distributions associated to h pseudodifferential operators on r n under standard assumptions the micro support of this object can be characterized via families of periodic orbits correlated simultaneously by energy and periods when all the orbits are non degenerate the convolution admits as h tends to 0 an explicit asymptotic expansion in term of the respective dynamical systems in this setting this result validates the theory of orbits pairs used by physicists in quantum chaos some new contributions related to the crossing of the period functions are also analyzed

circular symmetry and the trace formula let m be a compact n dimensional manifold without boundary and let p be a positive first order elliptic self adjoint classical pseudodifferential operator on lz m the hilbert space of half densities on m the spectrum of p is discrete and accumulates only at we denote the eigenvalues with multiplicities by 2o 21 2 recall that the singularities of the inverse fourier transform of the spectral measure

on the spectral norms of circulant matrices with classical fibonacci and lucas numbers entries abstract this paper is an improving of the work s solak on the norms of circulant matrices with the fibonacci and lucas numbers appl math comp 160 2005 125 132 in which the lower and upper bounds for the spectral norms of the matrices a f mod j i n i j 1 n and b l mod j i n i j 1 n are established in this new paper we compute the spectral norms of these matrices

determinants and inverses of circulant matrices with jacobsthal and jacobsthal lucas numbers abstract let n 3 and let j n circ j 1 j 2 j n and j n circ j 0 j 1 j n 1 be the n n circulant matrices associated with the jacobsthal numbers j 1 j n and the jacobsthal lucas numbers j 0 j n 1 respectively the determinants and the inverses of j n and j n are obtained in terms of j 1 j n and j 1 j n 1 respectively

on the norms of circulant matrices with the fibonacci and lucas numbers let us define a a i j and b b i j as nxn matrices where a i j f m o d j i n and b i j l m o d j i n we find out some bounds for the a and b matrices concerned with the spectral and euclidean norms

some properties of circulant matrices with ducci sequences abstract a ducci sequence is the sequence x d x d 2 x generated by n tuples x x 1 x 2 x n z n where d x d x 1 x 2 x n x 2 x 1 x 3 x 2 x n x 1 equivalently the ducci sequence of n vector x may be defined as d k x where k 0 1 2 and d k x denotes k times iterated vector x in this study we examine properties of the matrix d c n which results from applying the ducci map to the rows of the circulant matrix c n with first row c 0 c 1 c n 1

on the inverse of circulant matrix via generalized k horadam numbers in this paper by considering the circulant matrix c n h circ h k 1 h k 2 h k n whose entries are the generalized k horadam numbers we present a new generalization to compute determinants and inverses of c n h in fact the results in here are the most general statements to obtain the inverses and determinants in such matrices having the elements of all second order sequences

preconditioners for image restoration by reblurring techniques it is well known that iterative algorithms for image deblurring that involve the normal equations show usually a slow convergence a variant of the normal equations which replaces the conjugate transpose a h of the system matrix a with a new matrix is proposed this approach which is linked with regularization preconditioning theory and reblurring processes can be applied to a wide set of iterative methods here we examine landweber steepest descent richardson lucy and image space reconstruction algorithm several computational tests show that this strategy leads to a significant improvement of the convergence speed of the methods moreover it can be naturally combined with other widely used acceleration techniques

on the bounds for the norms of r circulant matrices with the fibonacci and lucas numbers abstract in this paper we have found upper and lower bounds for the spectral norms of r circulant matrices in the forms a c r f 0 f 1 f n 1 b c r l 0 l 1 l n 1 and we have obtained some bounds for the spectral norms of kronecker and hadamard products of a and b matrices

a note on spectral norms of even order r circulant matrices abstract in this paper we preset some improved techniques for proofs within zhou and jiang 2014 meanwhile focusing on n and r are co prime we generalize those results to investigate formulae of even order r circulant matrices with alternately positive and negative entrywise and we deduce explicit formulae for their spectral norms with the entries in the first row

quasi newton approach to nonnegative image restorations abstract image restoration or deblurring is the process of attempting to correct for degradation in a recorded image typically the blurring system is assumed to be linear and spatially invariant and fast fourier transform fft based schemes result in efficient computational image restoration methods however real images have properties that cannot always be handled by linear methods in particular an image consists of positive light intensities and thus a nonnegativity constraint should be enforced this constraint and other ways of incorporating a priori information have been suggested in various applications and can lead to substantial improvements in the reconstructions nevertheless such constraints are rarely implemented because they lead to nonlinear problems which require demanding computations we suggest efficient implementations for three nonnegatively constrained restorations schemes constrained least squares maximum likelihood and maximum entropy we show that with a certain parameterization and using a quasi newton scheme these methods are very similar in addition our formulation reveals a connection between our approach for maximum likelihood and the expectation maximization em method used extensively by astronomers numerical experiments illustrate that our approach is superior to em both in terms of accuracy and efficiency

a quantitative version of the observation that the hadamard product is a principal submatrix of the kronecker product abstract the hadamard and kronecker products of two n m matrices a b are related by a b p t 1 a b p 2 where p 1 p 2 are partial permutation matrices after establishing several properties of the p matrices this relationship is employed to demonstrate how a simplified theory of the hadamard product can be developed during this process the well known result a b a b aa bb is extended to a b a b 1 2 aa bb ab ba aa bb showing an inherent link between the hadamard product and conventional product of two matrices this leads to a sharper bound on the spectral norm of a b a b 1 2 a 2 b 2 ab 2 1 2 a b and an improvement on the weak majorization of a b 2 a b w 1 2 2 a 2 b 2 ab w 2 a 2 b for a real non singular matrix x and invertible diagonal matrices d e the spectral condition number is shown to be if scaled bounded below as follows dxe 2 x x t 2 x x t 2 1 2 x x t 1 for a 0 we have i a 2 1 2 i a 2 a a i a 2 and a 1 2 a 1 2 2 1 2 i a a 1 a a 1 when a 0 the latter inequality is compared to styan s inequality a a 1 1 2 i a a 1 when a is a correlation matrix and is shown to possess stronger properties of ordering finally the relationship a b p t 1 a b p 2 is applied to determine conditions of singularity of certain orderings of the hadamard products of matrices

spectral fluctuations of schrodinger operators generated by critical points of the potential starting from the spectrum of schrodinger operators on mathbb r n we propose a method to detect critical points of the potential we argue semi classically on the basis of a mathematically rigorous version of gutzwiller s trace formula which expresses spectral statistics in term of classical orbits a critical point of the potential with zero momentum is an equilibrium of the flow and generates certain singularities in the spectrum via sharp spectral estimates this fluctuation indicates the presence of a critical point and allows to reconstruct partially the local shape of the potential some generalizations of this approach are also proposed

sharp semiclassical estimates for the number of eigenvalues below a totally degenerate critical level we consider the semiclassical asymptotic behaviour of the number of eigenvalues smaller than e for elliptic operators in l2 rd we describe a method of obtaining remainder estimates related to the volume of the region of the phase space in which the principal symbol takes values belonging to the intervals e e h where e is close to e if the volume of this region is o h then we obtain remainder estimates o h1 d with no assumptions on the hessian of the principal symbol at the critical level moreover we do not assume that the coefficients are smooth all results hold if second order derivatives of coefficients are holder continuous

semi classical spectral estimates for schrodinger operators at a critical level case of a degenerate maximum of the potential we study the semi classical trace formula at a critical energy level for a schrodinger operator on r n we assume here that the potential has a totally degenerate crit ical point associated to a local maximum the main result which establishes the contribution of the associated equilibrium in the trace formula is valid for all time in a compact subset of r and includes the singularity in t 0 for these new contributions the asymptotic expansion involves the logarithm of the parameter h depending on an explicit arithmetic condition on the dimension and the order of the critical point this logarithmic contribution can appear in the leading term

a semi classical trace formula for schrodinger operators in the case of a critical energy level let 2 2 v x be a schrodinger operator on rn with smooth potentialv x as x the spectrum of is discrete and one can study the asymptotic of the smoothed spectral density e k ek e as 0 here ek k nis the spectrum of and c 0 r we shall investigate the case whereeis a critical value of the symbolhof and extending the work of brummelhuis paul and uribe in 3 we will prove the existence of a full asymptotic expansion for in terms of and ln and compute the leading coefficient we will consider new weyl type estimates for the counting function nec k n ek e

semiclassical spectral estimates for schrodinger operators at a critical energy level case of a degenerate minimum of the potential abstract we study the semi classical trace formula at a critical energy level for a schrodinger operator on r n we assume here that the potential has a totally degenerate critical point associated to a local minimum the main result which computes the contribution of this equilibrium is valid for all time in a compact and establishes the existence of a total asymptotic expansion whose top order coefficient depends only on the germ of the potential at the critical point

a unifying approach to some old and new theorems on distribution and clustering abstract a unifying approach is proposed to studying the distributions of eigenvalues and singular values of toeplitz matrices associated with a fourier series and multilevel toeplitz matrices associated with a multidimensional fourier series obtained are the extensions of the szego and avram parter theorems where the generating function is now required to belong to l 2 and not necessarily to l analogous extensions are given for multilevel toeplitz matrices in particular it is proved that if f x 1 x p l 2 then the p level complex toeplitz matrices allied with f have their singular values distributed as f x 1 x p the distribution results for the cesaro optimal circulants hold even if f l 1 also suggested are new theorems on clustering that have to do with the preconditioning of multilevel toeplitz matrices by multilevel circulants

closed natural splines an effective solution an algorithm for construction closed curvature continuous splines named as closed natural splines or cn splines based on cubic bezier representation along with supporting fuz zui theorem which provides an effective solution for the inversion of fuz zu a special case of almost three diagonal circulant matrix in explicit form is presented characteristics of fuz zui matrices examined exponentially relaxing oscillations for single point disturbance of cn spline demonstrated

on the determinants and inverses of circulant matrices with fibonacci and lucas numbers abstract let a n circ f 1 f 2 f n and b n circ l 1 l 2 l n be circulant matrices where f n is the fibonacci number and l n is the lucas number we prove that a n is invertible for n 2 and b n is invertible for any positive integer n afterwards the values of the determinants of matrices a n and b n can be expressed by utilizing only the fibonacci and lucas numbers in addition the inverses of matrices a n and b n are derived

fine granularity and spatially adaptive regularization for projection based image deblurring this paper studies two classes of regularization strategies to achieve an improved tradeoff between image recovery and noise suppression in projection based image deblurring the first is based on a simple fact that r times landweber iteration leads to a fixed level of regularization which allows us to achieve fine granularity control of projection based iterative deblurring by varying the value r the regularization behavior is explained by using the theory of lagrangian multiplier for variational schemes the second class of regularization strategy is based on the observation that various regularized filters can be viewed as nonexpansive mappings in the metric space a deeper understanding about different regularization filters can be gained by probing into their asymptotic behavior the fixed point of nonexpansive mappings by making an analogy to the states of matter in statistical physics we can observe that different image structures smooth regions regular edges and textures correspond to different fixed points of nonexpansive mappings when the temperature regularization parameter varies such an analogy motivates us to propose a deterministic annealing based approach toward spatial adaptation in projection based image deblurring significant performance improvements over the current state of the art schemes have been observed in our experiments which substantiates the effectiveness of the proposed regularization strategies

bohr sommerfeld conditions for integrable systems with critical manifolds of focus focus type we present a detailed study in the semi classical regime h of microlocal properties of systems of two commuting h pseudo dierential operators ph ph such that the joint principal symbol p p p has a special kind of singularity called a focusfocus singularity typical examples include the quantum spherical pendulum or the quantum champagne bottle in the spirit of colin de verdiere and parisse we show that such systems have a universal behavior described by singular quantization conditions of bohr sommerfeld type these conditions are used to give a precise description of the joint spectrum of such systems including the phenomenon of quantum monodromy and dierent formulations of the counting function for the joint eigenvalues close to the singularity in which a logarithm of the semi classical constant h appears thanks to numerical computations done by ms child for the case of the champagne bottle we are able to accurately illustrate our statements

improvement of space invariant image deblurring by preconditioned landweber iterations the landweber method is a simple and flexible iterative regularization algorithm whose projected variant provides nonnegative image reconstructions since the method is usually very slow we apply circulant preconditioners exploiting the shift invariance of many deblurring problems in order to accelerate the convergence this way reasonable reconstructions can be obtained within a few iterations the method becomes competitive and more robust than other approaches that although faster sometimes lead to lower accuracy some theoretical analysis of convergence is given together with numerical validations

on the spectral norms of the matrices connected to integer number sequences in this paper we compute the spectral norms of the matrices related with integer sequences and we give two examples related with fibonacci lucas pell and perrin numbers

semiconvergence and relaxation parameters for projected sirt algorithms we give a detailed study of the semiconvergence behavior of projected nonstationary simultaneous iterative reconstruction technique sirt algorithms including the projected landweber algorithm we also consider the use of a relaxation parameter strategy proposed recently for the standard algorithms for controlling the semiconvergence of the projected algorithms we demonstrate the semiconvergence and the performance of our strategies by examples taken from tomographic imaging

inverse spectral problems for schrodinger operators in this article we find some explicit formulas for the semi classical wave invariants at the bottom of the well of a schrodinger operator as an application of these new formulas for the wave invariants we improve the inverse spectral results proved by guillemin and uribe in gu they proved that under some symmetry assumptions on the potential v x the taylor expansion of v x near a non degenerate global minimum can be recovered from the knowledge of the low lying eigenvalues of the associated schrodinger operator in mathbb r n we prove similar inverse spectral results using fewer symmetry assumptions we also show that in dimension 1 no symmetry assumption is needed to recover the taylor coefficients of v x

a semiclassical egorov theorem and quantum ergodicity for matrix valued operators we study the semiclassical time evolution of observables given by matrix valued pseudodifferential operators and construct a decomposition of the hilbert space l2 open image in new windowd open image in new windown into a finite number of almost invariant subspaces for a certain class of observables that is preserved by the time evolution we prove an egorov theorem we then associate with each almost invariant subspace of l2 open image in new windowd open image in new windown a classical system on a product phase space t open image in new windowd open image in new window where open image in new window is a compact symplectic manifold on which the classical counterpart of the matrix degrees of freedom is represented for the projections of eigenvectors of the quantum hamiltonian to the almost invariant subspaces we finally prove quantum ergodicity to hold if the associated classical systems are ergodic

on the small scale mass concentration of modes let be commuting jointly elliptic pseudodifferential operators on a compact manifold x of dimension n d suppose is the limit set of the bicharacteristic flow of the classical hamiltonian p 1 restricted to the variety we discuss the corresponding concentration of mass as for a subsequence of joint eigenfunctions of the p j s with eigenvalues sufficiently close to

uniform distribution of eigenfunctions on compact hyperbolic surfaces on demontre que les fonctions propres k du laplacien sur une surface hyperbolique compacte x devient uniformement distribuee sur x quand k

schultz matrix iteration based method for stable solution of discrete ill posed problems abstract in this paper we propose an iterative method for solving discrete ill posed problems based on matrix iterations generated by schultz method and known to converge to the moore penrose pseudoinverse of a matrix practically letting the schultz matrix iteration be x k we construct the vector x k x k b where b is a data vector hence by construction the iterates converge to the minimum 2 norm solution of a least squares problem with coefficient matrix a and data vector b we derive theoretical properties of the sequence x k and show that it is quadratically convergent in the case of corrupted data we analyze the semi convergence behavior of the iterates and conclude that the iteration must be truncated to control the propagation of the noise error as a result we derive an error estimate for the case where the truncation parameter is chosen by the discrepancy principle in addition combining a projected approach with the new method we propose variants of the method that are well suited for large scale problems several numerical results are presented to illustrate the effectiveness of the method on well known test problems

a proof of the gutzwiller semiclassical trace formula using coherent states decomposition the gutzwiller trace formula links the eigenvalues of the schrodinger operator as planck s constant goes to zero the semiclassical regime with the closed orbits of the corresponding classical mechanical system gutzwiller gave a heuristic proof of this trace formula using the feynman integral representation for the propagator of later using the theory of fourier integral operators mathematicians gave rigorous proofs of the formula in various settings here we show how the use of coherent states allows us to give a simple and direct proof

on hessenberg and pentadiagonal determinants related with fibonacci and fibonacci like numbers in this paper we establish several new connections between the generalizations of fibonacci and lucas sequences and hessenberg determinants we also give an interesting conjecture related to the determinant of an infinite pentadiagonal matrix with the classical fibonacci and gaussian fibonacci numbers

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

the spectrum of a family of circulant preconditioned toeplitz systems the solutions of symmetric positive definite toeplitz systems ax b are studied by the preconditioned conjugate gradient method the preconditioner is the circulant matrix c that minimizes the frobenius norm c a f t chan an optimal circulant preconditioner for toeplitz systems ucla department of mathematics cam report 87 06 june 1987 the convergence rate of these iterative methods is known to depend on the distribution of the eigenvalues of c 1 a for toeplitz matrix a with entries which are fourier coefficients of a positive function in the wiener class this paper establishes the invertibility of c finds the asymptotic behaviour of the eigenvalues of the preconditioned matrix c 1 a as the dimension increases and proves that they are clustered around 1

a semi classical trace formula at a totally degenerate critical level we study the semi classical trace formula at a critical energy level for an h pseudo differential operator on open image in new window n whose principal symbol has a totally degenerate critical point for that energy this problem is studied for a large time behavior and under the hypothesis that the principal symbol of the operator has a local extremum at the critical point

on the regularizing properties of the gmres method the gmres method is a popular iterative method for the solution of large linear systems of equations with a nonsymmetric nonsingular matrix however little is known about the behavior of this method when it is applied to the solution of nonsymmetric linear ill posed problems with a right hand side that is contaminated by errors we show that when the associated error free right hand side lies in a finite dimensional krylov subspace the gmres method is a regularization method the iterations are terminated by a stopping rule based on the discrepancy principle

inverse toeplitz preconditioners for ill posed problems abstract it has been shown recently that iterative regularization using conjugate gradient type methods for image restoration problems can be effectively preconditioned with circulant approximations here it is shown that the theoretical properties of this approach are not restricted to circulant matrices specifically a toeplitz approximate inverse preconditioning scheme for discrete ill posed problems is considered it is proved that the preconditioned system approximates the prolate matrix and that this property implies that fast convergence of conjugate gradient type methods can be expected in addition it is shown that these results can be generalized to two dimensional problems an image restoration application is used to demonstrate the properties of the preconditioner

the semi classical trace formula and propagation of wave packets abstract we study spectral and propagation properties of operators of the form s h n j 0 h j p j where j p j is a differential operator of order j on a manifold m asymptotically as h 0 the estimates are in terms of the flow t of the classical hamiltonian h x p n j 0 p j x p on t m where p j is the principal symbol of p j we present two sets of results i the semiclassical trace formula on the asymptotic behavior of eigenvalues and eigenfunctions of s h in terms of periodic trajectors of h ii associated to certain isotropic submanifolds t m we define families of functions h and prove that t exp iths is a family of the same kind associated to t