H-pseudodifferential Operators Research Articles

Weyl composition of symbols in large dimension

This paper is concerned with the Weyl composition of symbols in large dimension. We specify a class of symbols in order to estimate the Weyl symbol of the product of two Weyl h-pseudodifferential operators, with constants independent of the dimension. The proof includes regularized and hybrid compositions, together with a decomposition formula. We also analyze, in this context, the remainder term of the semiclassical expansion of the Weyl composition. The class of symbols contains symbols of Schrodinger semigroups in large dimension, typically for nearest neighbors or mean field interaction potentials. The Weyl composition is applied with Kac operators.

On generalized Bohr–Sommerfeld quantization rules for operators with PT symmetry

We give Bohr–Sommerfeld rules corresponding to quasi-eigenvalues in the pseudospectrum for a one-dimensional h-pseudodifferential operator verifying PT symmetry.

Equivariant spectral asymptotics for h-pseudodifferential operators

We prove equivariant spectral asymptotics for h-pseudodifferential operators for compact orthogonal group actions generalizing results of El Houakmi and Helffer [“Comportement semi-classique en présence de symétries: Action d'un groupe de Lie compact,” Asymp. Anal. 5(2), 91–113 (1991)] and Cassanas [“Reduced Gutzwiller formula with symmetry: Case of a Lie group,” J. Math. Pures Appl. 85(6), 719–742 (2006)]. Using recent results for certain oscillatory integrals with singular critical sets [P. Ramacher, “Singular equivariant asymptotics and Weyl's law: On the distribution of eigenvalues of an invariant elliptic operator,” J. Reine Angew. Math. (Crelles J.) (to be published)], we can deduce a weak equivariant Weyl law. Furthermore, we can prove a complete asymptotic expansion for the Gutzwiller trace formula without any additional condition on the group action by a suitable generalization of the dynamical assumptions on the Hamilton flow.

Spectral monodromy of non-self-adjoint operators

In the present paper, we build a combinatorial invariant, called the “spectral monodromy” from the spectrum of a single (non-self-adjoint) h-pseudodifferential operator with two degrees of freedom in the semi-classical limit. Our inspiration comes from the quantum monodromy defined for the joint spectrum of an integrable system of n commuting self-adjoint h-pseudodifferential operators, given by S. Vu Ngoc [“Quantum monodromy in integrable systems,” Commun. Math. Phys. 203(2), 465–479 (1999)]. The first simple case that we treat in this work is a normal operator. In this case, the discrete spectrum can be identified with the joint spectrum of an integrable quantum system. The second more complex case we propose is a small perturbation of a self-adjoint operator with a classical integrability property. We show that the discrete spectrum (in a small band around the real axis) also has a combinatorial monodromy. The main difficulty in this case is that we do not know the description of the spectrum everywhere, but only in a Cantor type set. In addition, we also show that the corresponding monodromy can be identified with the classical monodromy, defined by J. Duistermaat [“On global action-angle coordinates,” Commun. Pure Appl. Math. 33(6), 687–706 (1980)].

Local exponential estimates for h-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators

We consider the class of matrix h-pseudodifferential operators Op h (a) with symbols a = (a ij ) i,j=1 N , where the coefficients a ij ∈ C ∞(ℝ x n × ℝ ξ n ⊗ C(0, 1] satisfy the estimates |ϖ x β g6 ξ α α ij (x, ξ, h)| ⩽ C αβ 〈ξ〉 m and 〈ξ〉 = (1 + |ξ|2)1/2 for every multi-indices α, β. We also assume that a ij (x, ξ) is analytically continued with respect to ξ to a tube domain ℝ n + i \( \mathcal{B} \), where \( \mathcal{B} \) is a bounded domain in ℝ n containing the origin. The main results of the paper are the local estimates for solutions of h-pseudodifferential equations.

Non-Selfadjoint Perturbations of Selfadjoint Operators in Two Dimensions IIIa. One Branching Point

AbstractThis is the third in a series of works devoted to spectral asymptotics for non-selfadjoint perturbations of selfadjointh-pseudodifferential operators in dimension 2, having a periodic classical flow. Assuming that the strength ॉ of the perturbation is in the rangeh2≪ ॉ ≪ h1/2(and may sometimes reach even smaller values), we get an asymptotic description of the eigenvalues in rectangles [−1/C, 1/C] +iॉ[F0− 1/C,F0 + 1/C],C≫ 1, when ॉF0is a saddle point value of the flow average of the leading perturbation.

Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators

We consider quite general h-pseudodifferential operators on R n with small random perturbations and show that in the limit h → 0 the eigenvalues are distributed according to a Weyl law with a probabality that tends to 1. The first author has previously obtained a similar result in dimension 1. Our class of perturbations is different.

Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point

We study the microlocal kernel of h-pseudodifferential operators Op h ( p ) − z , where z belongs to some neighborhood of size O ( h ) of a critical value of its principal symbol p 0 ( x , ξ ) . We suppose that this critical value corresponds to a hyperbolic fixed point of the Hamiltonian flow H p 0 . First we describe propagation of singularities at such a hyperbolic fixed point, both in the analytic and in the C ∞ category. In both cases, we show that the null solution is the only element of this microlocal kernel which vanishes on the stable incoming manifold, but for energies z in some discrete set. For energies z out of this set, we build the element of the microlocal kernel with given data on the incoming manifold. We describe completely the operator which associate the value of this null solution on the outgoing manifold to the initial data on the incoming one. In particular it appears to be a semiclassical Fourier integral operator associated to some natural canonical relation.

Diophantine tori and spectral asymptotics for nonselfadjoint operators

We study spectral asymptotics for small nonselfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part possesses several invariant Lagrangian tori enjoying a Diophantine property. We get complete asymptotic expansions for all eigenvalues in certain rectangles in the complex plane in two different cases: in the first case, we assume that the strength ε of the perturbation is O(hδ) for some δ > 0 and is bounded from below by a fixed positive power of h. In the second case, ε is assumed to be sufficiently small but independent of h, and we describe the eigenvalues completely in a fixed h-independent domain in the complex spectral plane.

Convolutions of semi-classical spectral distributions and periodic-orbit theory

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.

Thermodynamic Limits for Hamiltonians Defined as Pseudodifferential Operators

If P(h) is a h-pseudodifferential operator in R n associated to an holomorphic semi-bounded symbol in some neighborhood of the real phase space, with bounded derivatives, we describe the symbol of e −tP(h), by inequalities where the constants depend on the bounds for the derivatives of the symbol of P(h), but not on the dimension n. Some applications to thermodynamic limits (free energy) are given.

A semi-classical trace formula at a non-degenerate critical level

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.

Calcul pseudodifférentiel en grande dimension et limites thermodynamiques

If P n ( h) is a h-pseudodifferential operator in R n associated to an holomorphic semi-bounded symbol in some neighborhood of the real phase space, with bounded derivatives, we describe the symbol of e − tP n ( h) , by inegalities where the constants depend on the bounds for the derivatives of the symbol of P n ( h), but not on the dimension n. Some applications to thermodynamic limits are given. To cite this article: C. Royer, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

QUASI-INVARIANT TORI AND SEMI-EXCITED STATES FOR SCHRO¨DINGER OPERATORS. I. ASYMPTOTICS

ABSTRACT Let be a h-Pseudodifferential operator with Weyl symbol such that p 0(x,ξ) has a non degenerate minimum at (0, 0) with non resonant frequencies and is uniformly elliptic outside this point. We are interested in the case of semi-excited eigenstates localized near quasi-invariant tori, with energies . We construct quasi-modes giving all energies of P in such an interval, and give their asymptotics in the classically forbidden region near an umbilic. We consider next the structure of the caustic near the umbilic, that we identify with a Whitney crease, except when P enjoys the property, that p(x, ξ, h) is invariant under the symmetry in which case the caustic is the product of simple folds.

Quantum monodromy and semi-classical trace formulæ

We consider an h-pseudodifferential operator whose symbol has a closed Hamiltonian trajectory. There exists a Fourier integral operator which quantizes in a natural way the Poincaré map. With the help of this monodromy operator, we give a trace formula which leads to a new proof of the trace formula of Duistermaat–Guillemin and Gutzwiller.

A semi-classical trace formula for several commuting operators

We consider several selfadjoint h-pseudodifferential operators which commute to each other. Let be a regular value of the principal symbol of A(h) and K be a compact neighborhood of E. Under suitable assumptions the joint spectrum of A(h) in K consists of finitely many joint eigenvalues . Let ζ be a smooth function in Rk having a Fourier transform with compact support. The aim of this article is to investigate the semiclassical asymptotics of the trace in terms of the periodic orbits of the joint Hamiltoian flow of a0 lying on the energy surface .

ON MASLOV'S METHOD FOR CONSTRUCTING COMBINED ASYMPTOTICS FORh-PSEUDODIFFERENTIAL EQUATIONS

The authors discuss a scheme proposed by V.?P.?Maslov for constructing combined (with respect to smoothness and a small parameter h) asymptotics of the solution of the Cauchy problem for h-pseudodifferential equations. The exposition is carried out by means of examples of equations for the oscillations of a crystal lattice and for water waves. The main attention is given to the isolation of the leading term of the asymptotics. A number of estimates are proved for the remainders in formulas for the action of an h-pseudodifferential operator on the exponential function, with respect to smoothness and the parameter. Bibliography: 9 titles.