The topological asymptotic with respect to a singular boundary perturbation

The aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a design functional with respect to the insertion of a small hole in the domain. The question that we address here is what happens if the hole is located at the boundary of the domain and what happens if the boundary is not regular. The adjoint method and the domain truncation technique are proposed to solve this problem. As a model example, we consider the Laplace equation in a domain with a corner. To cite this article: B. Samet, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Eigenvalue asymptotics of a modified Jaynes–Cummings model with periodic modulations

We analyze the influence of additive and multiplicative periodic modulations on the asymptotic behavior of eigenvalues of some Hermitian Jacobi Matrices related to the Jaynes–Cummings model using the so-called “successive diagonalization” method. This approach allows us to find the asymptotics of the nth eigenvalue λ_n as n→∞ stepwise with successively increasing precision. We bring to light the interplay of additive and multiplicative periodic modulations and their influence on the asymptotic behavior of eigenvalues. To cite this article: A. Boutet de Monvel et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Asymptotic expansion of the Faber–Krahn profile of a compact Riemannian manifold

The aim of this Note is to give a proof of a well-known fact: an asymptotic expansion of the isoperimetric profile of a Riemannian manifold for small volumes gives an asymptotic expansion of the Faber–Krahn profile for this same Riemannian manifold. To cite this article: O. Druet, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

A trace formula and high-energy spectral asymptotics for the perturbed Landau Hamiltonian

A two-dimensional Schrödinger operator with a constant magnetic field perturbed by a smooth compactly supported potential is considered. The spectrum of this operator consists of eigenvalues which accumulate to the Landau levels. We call the set of eigenvalues near the nth Landau level an nth eigenvalue cluster, and study the distribution of eigenvalues in the nth cluster as n→∞. A complete asymptotic expansion for the eigenvalue moments in the nth cluster is obtained and some coefficients of this expansion are computed. A trace formula involving the eigenvalue moments is obtained.     

Asymptotics for the Expected Lifetime of Brownian Motion on Thin Domains in ℝ n

We derive a three-term asymptotic expansion for the expected lifetime of Brownian motion and for the torsional rigidity on thin domains in ? ⁿ , and a two-term expansion for the maximum (and corresponding maximizer) of the expected lifetime. The approach is similar to that which we used previously to study the eigenvalues of the Dirichlet Laplacian and consists of scaling the domain in one direction and deriving the corresponding asymptotic expansions as the scaling parameter goes to zero. Apart from being dominated by the one-dimensional Brownian motion along the direction of the scaling, we also see that the symmetry of the perturbation plays a role in the expansion. As in the case of eigenvalues, these expansions may also be used to approximate the exit time for domains where the scaling parameter is not necessarily close to zero.     

Homogenization of eigenvalue problems in perforated domains

In this paper, we treat some eigenvalue problems in periodically perforated domains and study the asymptotic behaviour of the eigenvalues and the eigenvectors when the number of holes in the domain increases to infinity Using the method of asymptotic expansion, we give explicit formula for the homogenized coefficients and expansion for eigenvalues and eigenvectors. If we denote by ε the size of each hole in the domain, then we obtain the following aysmptotic expansion for the eigenvalues: Dirichlet: λ_ε = ε⁻² λ + λ₀ +O (ε), Stekloff: λ_ε = ελ₁ +O (ε²), Neumann: λ_ε = λ₀ + ελ₁ +O (ε²). Using the method of energy, we prove a theorem of convergence in each case considered here. We briefly study correctors in the case of Neumann eigenvalue problem.     

Développement asymptotique q-Gevrey et fonction thêta de Jacobiq-Gevrey asymptotic expansion and Jacobi theta function

Some notions of q-Gevrey asymptotic expansion have been studied in [6,7]. Recently we became interested in a new notion of asymptotic expansion [8]: it is related to a Jacobi theta function and allows one to establish the natural link between the asymptotics of q-difference equations and the theory of elliptic functions. The purpose of this Note is to give some new results related to this notion of asymptotic expansion. To cite this article: J.-P. Ramis, C. Zhang, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 899–902.     

Asymptotics of eigenvalues for regular Sturm-Liouville problems with eigenvalue parameter in the boundary condition

In this paper we obtain asymptotic estimates of eigenvalues for regular Sturm-Liouville problems having the eigenparameter in the boundary condition. The method is based on an iterative procedure solving the associated Riccati equation and producing an asymptotic expansion of the solution in the higher powers of 1/λ1/2 as λ→∞.     

On a test of equality of two-parameter exponential distributions

In this paper, an asymptotic expansion of the distribution of the statistic for testing the equality of p two-parameter exponential distributions is obtained upto the order n^?4 with the second term of the order n^?3 where n is the size of the sample drawn from the i th exponential population. The asymptotic expansion can therefore be used to obtain accurate approximations to the critical values of the test statistic even for comparatively small values of n. Also we have shown that F-tables can be used to test the hypothesis when the sample size is moderately large.     

Exact formulae and matrix-less eigensolvers for block banded symmetric Toeplitz matrices

Precise asymptotic expansions for the eigenvalues of a Toeplitz matrix \(T_n(f)\), as the matrix size n tends to infinity, have recently been obtained, under suitable assumptions on the associated generating function f. A restriction is that f has to be polynomial, monotone, and scalar-valued. In this paper we focus on the case where \(\mathbf {f}\) is an \(s\times s\) matrix-valued trigonometric polynomial with \(s\ge 1\), and \(T_n(\mathbf {f})\) is the block Toeplitz matrix generated by \(\mathbf {f}\), whose size is \(N(n,s)=sn\). The case \(s=1\) corresponds to that already treated in the literature. We numerically derive conditions which ensure the existence of an asymptotic expansion for the eigenvalues. Such conditions generalize those known for the scalar-valued setting. Furthermore, following a proposal in the scalar-valued case by the first author, Garoni, and the third author, we devise an extrapolation algorithm for computing the eigenvalues of banded symmetric block Toeplitz matrices with a high level of accuracy and a low computational cost. The resulting algorithm is an eigensolver that does not need to store the original matrix, does not need to perform matrix-vector products, and for this reason is called matrix-less. We use the asymptotic expansion for the efficient computation of the spectrum of special block Toeplitz structures and we provide exact formulae for the eigenvalues of the matrices coming from the \(\mathbb {Q}_p\) Lagrangian Finite Element approximation of a second order elliptic differential problem. Numerical results are presented and critically discussed.     

Reproducing kernels of Sobolev spaces via a green kernel approach with differential operators and boundary operators

We introduce a vector differential operator P and a vector boundary operator B to derive a reproducing kernel along with its associated Hilbert space which is shown to be embedded in a classical Sobolev space. This reproducing kernel is a Green kernel of differential operator L:?=?P ^???T P with homogeneous or nonhomogeneous boundary conditions given by B, where we ensure that the distributional adjoint operator P ^??? of P is well-defined in the distributional sense. We represent the inner product of the reproducing-kernel Hilbert space in terms of the operators P and B. In addition, we find relationships for the eigenfunctions and eigenvalues of the reproducing kernel and the operators with homogeneous or nonhomogeneous boundary conditions. These eigenfunctions and eigenvalues are used to compute a series expansion of the reproducing kernel and an orthonormal basis of the reproducing-kernel Hilbert space. Our theoretical results provide perhaps a more intuitive way of understanding what kind of functions are well approximated by the reproducing kernel-based interpolant to a given multivariate data sample.     

Estimation semi-classique du courant quantique en présence d'un grand champ magnétique variable

We calculate an asymptotic expression of the quantum current in the presence of a strong non-constant magnetic field. Thanks to a commutator identity for the current operator, we are led to estimate the sum of negative eigenvalues of a modified Pauli operator. To cite this article: S. Negra, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Non-uniform bounds for short asymptotic expansions in the CLT for balls in a Hilbert space

We consider short asymptotic expansions for the probability of a sum of i.i.d. random elements to hit a ball in a Hilbert space H. The error bound for the expansion is of order O(n^-1). It depends on the first 12 eigenvalues of the covariance operator only. Moreover, the bound is non-uniform, i.e. the accuracy of the approximation becomes better as the distance between a boundary of the ball and the origin in H grows.     

Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle

An asymptotic expansion including error bounds is given for polynomials {P _n, Q_n} that are biorthogonal on the unit circle with respect to the weight function (1?e^iθ)^α+β(1?e^?iθ)^α?β. The asymptotic parameter isn; the expansion is uniform with respect toz in compact subsets ofC{0}. The pointz=1 is an interesting point, where the asymptotic behavior of the polynomials strongly changes. The approximants in the expansions are confluent hyper-geometric functions. The polynomials are special cases of the Gauss hyper-geometric functions. In fact, with the results of the paper it follows how (in a uniform way) the confluent hypergeometric function is obtained as the limit of the hypergeometric function₂ F ₁(a, b; c; z/b), asb→±∞,z≠b, withz=0 as “transition” point in the uniform expansion.     

Matching and multiscale expansions for a model singular perturbation problem

We consider the Laplace–Dirichlet equation in a polygonal domain which is perturbed at the scale ε near one of its vertices. We assume that this perturbation is self-similar, that is, derives from the same pattern for all values of ε. On the base of this model problem, we compare two different approaches: the method of matched asymptotic expansions and the method of multiscale expansion. We enlighten the specificities of both techniques, and show how to switch from one expansion to the other. To cite this article: S. Tordeux et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Discrete spectrum of a model Schr?dinger operator on the half-plane with Neumann conditions

We study the eigenpairs of a model Schr?dinger operator with a quadratic potential and Neumann boundary conditions on a half-plane. The potential is degenerate in the sense that it reaches its minimum all along a line that makes the angle ?? with the boundary of the half-plane. We show that the first eigenfunctions satisfy localization properties related to the distance to the minimum line of the potential. We investigate the densification of the eigenvalues below the essential spectrum in the limit ?? ?? 0, and we prove a full asymptotic expansion for these eigenvalues and their associated eigenvectors. We conclude the paper by numerical experiments obtained by a finite element method. The numerical results confirm and enlighten the theoretical approach.     

Bound states of a two-boson system on a two-dimensional lattice

We consider a Hamiltonian of a two-boson system on a two-dimensional lattice Z². The Schrödinger operator H(k₁, k₂) of the system for k₁ = k₂ = π, where k = (k₁, k₂) is the total quasimomentum, has an infinite number of eigenvalues. In the case of a special potential, one eigenvalue is simple, another one is double, and the other eigenvalues have multiplicity three. We prove that the double eigenvalue of H(π,π) splits into two nondegenerate eigenvalues of H(π, π ? 2β) for small β > 0 and the eigenvalues of multiplicity three similarly split into three different nondegenerate eigenvalues. We obtain asymptotic formulas with the accuracy of β² and also an explicit form of the eigenfunctions of H(π, π ?2β) for these eigenvalues.     

Comportement à très basses énergies de la densité d'états intégrée du modèle d'Anderson non borné inférieurement

We obtain a complete asymptotic expansion of the integrated density of states of the unbounded Anderson model at low energies. We also study the evolution of this asymptotic when the decay of the tail of the distribution of the random potential increases. To cite this article: O. Saad, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On a separation theorem for generalized eigenvalues and a problem in the analysis of sample surveys

We obtain usable bounds for the asymptotic percentage points of chi-squared tests of fit for log-linear models fitted to contingency tables estimated from survey data, by applying some new separation inequalities for the generalized eigenvalues of a matrix X′AX with respect to a matrix X′BX, when both the matrices A and B are nonnegative definite. We also present some historical remarks on the Poincaré separation theorem for eigenvalues from which our new inequalities are shown to follow.     

Asymptotic distribution of the OLS estimator for a purely autoregressive spatial model

We derive the asymptotics of the OLS estimator for a purely autoregressive spatial model. Only low-level conditions are used. As the sample size increases, the spatial matrix is assumed to approach a square-integrable function on the square (0,1)². The asymptotic distribution is a ratio of two infinite linear combinations of χ² variables. The formula involves eigenvalues of an integral operator associated with the function approached by the spatial matrices. Under the conditions imposed identification conditions for the maximum likelihood method and method of moments fail. A corrective two-step procedure using the OLS estimator is proposed.