Volume 55, Issue 1, January 2014

In the present paper, we build a combinatorial invariant, called the “spectral monodromy” from the spectrum of a single (nonselfadjoint) hpseudodifferential operator with two degrees of freedom in the semiclassical limit. Our inspiration comes from the quantum monodromy defined for the joint spectrum of an integrable system of n commuting selfadjoint hpseudodifferential 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 selfadjoint 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 actionangle coordinates,” Commun. Pure Appl. Math.33(6), 687–706 (1980)].
 ARTICLES

 Partial Differential Equations

First Bloch eigenvalue in high contrast media
View Description Hide DescriptionThis paper deals with the asymptotic behavior of the first Bloch eigenvalue in a heterogeneous medium with a high contrast ɛYperiodic conductivity. When the conductivity is bounded in L ^{1} and the constant of the PoincaréWirtinger weighted by the conductivity is very small with respect to ɛ^{−2}, the first Bloch eigenvalue converges as ɛ → 0 to a limit which preserves the secondorder expansion with respect to the Bloch parameter. In dimension two the expansion of the limit can be improved until the fourthorder under the same hypotheses. On the contrary, in dimension three a fibers reinforced medium combined with a L ^{1}unbounded conductivity leads us to a discontinuity of the limit first Bloch eigenvalue as the Bloch parameter tends to zero but remains not orthogonal to the direction of the fibers. Therefore, the high contrast conductivity of the microstructure induces an anomalous effect, since for a given lowcontrast conductivity the first Bloch eigenvalue is known to be analytic with respect to the Bloch parameter around zero.

A short note on the chaoticity of a weight shift on concrete orthonormal basis associated to some FockBargmann space
View Description Hide DescriptionLet be the Hilbert space generated by the orthonormal basis where ν > 0 and α are real numbers, this space is a particular case of (Γ, χ)theta FockBargmann spaces recently constructed by GhanmiIntissar [J. Math. Phys.54, 063514 (2013)]. In the present work, we consider on the weight shift operator defined by where and and we show the chaoticity of the operator where is adjoint of .

Propagation of ultrashort solitons in stochastic Maxwell's equations
View Description Hide DescriptionWe study the propagation of ultrashort short solitons in a cubic nonlinear medium modeled by nonlinear Maxwell's equations with stochastic variations of media. We consider three cases: variations of (a) the dispersion, (b) the phase velocity, (c) the nonlinear coefficient. Using a modified multiscale expansion for stochastic systems, we derive new stochastic generalizations of the short pulse equation that approximate the solutions of stochastic nonlinear Maxwell's equations. Numerical simulations show that soliton solutions of the short pulse equation propagate stably in stochastic nonlinear Maxwell's equations and that the generalized stochastic short pulse equations approximate the solutions to the stochastic Maxwell's equations over the distances under consideration. This holds for both a pathwise comparison of the stochastic equations as well as for a comparison of the resulting probability densities.

A short proof of Weyl's law for fractional differential operators
View Description Hide DescriptionWe study spectral asymptotics for a large class of differential operators on an open subset of with finite volume. This class includes the Dirichlet Laplacian, the fractional Laplacian, and also fractional differential operators with nonhomogeneous symbols. Based on a sharp estimate for the sum of the eigenvalues we establish the first term of the semiclassical asymptotics. This generalizes Weyl's law for the Laplace operator.

Ground states of nonlinear Schrödinger systems with saturable nonlinearity in for two counterpropagating beams
View Description Hide DescriptionCounterpropagating optical beams in nonlinear media give rise to a host of interesting nonlinear phenomena such as the formation of spatial solitons, spatiotemporal instabilities, selffocusing and selftrapping, etc. Here we study the existence of ground state (the energy minimizer under the L ^{2}normalization condition) in twodimensional (2D) nonlinear Schrödinger (NLS) systems with saturable nonlinearity, which describes paraxial counterpropagating beams in isotropic local media. The nonlinear coefficient of saturable nonlinearity exhibits a threshold which is crucial in determining whether the ground state exists. The threshold can be estimated by the GagliardoNirenberg inequality and the ground state existence can be proved by the energy method, but not the concentrationcompactness method. Our results also show the essential difference between 2D NLS equations with cubic and saturable nonlinearities.
 Representation Theory and Algebraic Methods

The tensor hierarchy algebra
View Description Hide DescriptionWe introduce an infinitedimensional Lie superalgebra which is an extension of the Uduality Lie algebra of maximal supergravity in D dimensions, for 3 ⩽ D ⩽ 7. The level decomposition with respect to the Uduality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated BorcherdsKacMoody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of our Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds)KacMoody (super)algebra. Instead the Hodge duality relations between level p and D − 2 − p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.

Gsystems and deformation of Gactions on
View Description Hide DescriptionGiven a (smooth) action φ of a Lie group G on we construct a differential graded algebra whose Maurer–Cartan elements are in onetoone correspondence with some class of deformations of the (induced) Gaction on . In the final part of this note we discuss the cohomological obstructions to the existence and to the uniqueness (in a sense to be clarified) of such deformations.

Matrix elements for type 1 unitary irreducible representations of the Lie superalgebra gl(mn)
View Description Hide DescriptionUsing our recent results on eigenvalues of invariants associated to the Lie superalgebra gl(mn), we use characteristic identities to derive explicit matrix element formulae for all gl(mn) generators, particularly nonelementary generators, on finite dimensional type 1 unitary irreducible representations. We compare our results with existing works that deal with only subsets of the class of type 1 unitary representations, all of which only present explicit matrix elements for elementary generators. Our work therefore provides an important extension to existing methods, and thus highlights the strength of our techniques which exploit the characteristic identities.

Algebraic vs physical N = 6 3algebras
View Description Hide DescriptionIn our previous paper, we classified linearly compact algebraic simple N = 6 3algebras. In the present paper, we classify their “physical” counterparts, which actually appear in the N = 6 supersymmetric 3dimensional ChernSimons theories.

Graded vertex algebras and conformal flow
View Description Hide DescriptionWe consider graded vertex algebras, which are vertex algebras V with a grading such that V is an admissible Vmodule generated by “lowest weight vectors.” We show that such vertex algebras have a “good” representation theory in the sense that there is a Zhu algebra A(V) and a bijection between simple admissible Vmodules and simple A(V)modules. We also consider pseudo vertex operator algebras (PVOAs), which are graded vertex algebras with a conformal vector such that the homogeneous subspaces of V are generalized eigenspaces for L(0); essentially, these are VOAs that lack any semisimplicity or integrality assumptions on L(0). As a motivating example, we show that deformation of the conformal structure (conformal flow) of a strongly regular VOA (e.g., a lattice theory, or WessZuminoWitten model) is a path in a space whose points are PVOAs.

Loop Virasoro Lie conformal algebra
View Description Hide DescriptionThe Lie conformal algebra of loop Virasoro algebra, denoted by , is introduced in this paper. Explicitly, is a Lie conformal algebra with basis and λbrackets [L i λ L j ] = (−∂−2λ)L i+j . Then conformal derivations of are determined. Finally, rank one conformal modules and graded free intermediate series modules over are classified.
 Quantum Mechanics

Generalization of Lieb's variational principle to Bogoliubov–Hartree–Fock theory
View Description Hide DescriptionIn its original formulation, Lieb's variational principle holds for fermion systems with purely repulsive pair interactions. As a generalization we prove for both fermion and boson systems with semibounded Hamiltonian that the infimum of the energy over quasifree states coincides with the infimum over pure quasifree states. In particular, the Hamiltonian is not assumed to preserve the number of particles. To shed light on the relation between our result and the usual formulation of Lieb's variational principle in terms of oneparticle density matrices, we also include a characterization of pure quasifree states by means of their generalized oneparticle density matrices.

Optimum phase space probabilities from quantum tomography
View Description Hide DescriptionWe determine a positive normalised phase space probability distribution P with minimum mean square fractional deviation from the Wigner distribution W. The minimum deviation, an invariant under phase space rotations, is a quantitative measure of the quantumness of the state. The positive distribution closest to W will be useful in quantum mechanics and in time frequency analysis. The positionmomentum correlations given by the distribution can be tested experimentally in quantum optics.

Quantum driven dissipative parametric oscillator in a blackbody radiation field
View Description Hide DescriptionWe consider the general open system problem of a charged quantum oscillator confined in a harmonic trap, whose frequency can be arbitrarily modulated in time, that interacts with both an incoherent quantized (blackbody) radiation field and with an arbitrary coherent laser field. We assume that the oscillator is initially in thermodynamic equilibrium with its environment, a nonfactorized initial density matrix of the system and the environment, and that at t = 0 the modulation of the frequency, the coupling to the incoherent and the coherent radiation are switched on. The subsequent dynamics, induced by the presence of the blackbody radiation, the laser field, and the frequency modulation, is studied in the framework of the influence functional approach. This approach allows incorporating, in analytic closed formulae, the nonMarkovian character of the oscillatorenvironment interaction at any temperature as well the nonMarkovian character of the blackbody radiation and its zeropoint fluctuations. Expressions for the time evolution of the covariance matrix elements of the quantum fluctuations and the reduced densityoperator are obtained.

Semiclassical wavepackets emerging from interaction with an environment
View Description Hide DescriptionWe study the quantum evolution in dimension three of a system composed by a test particle interacting with an environment made of N harmonic oscillators. At time zero the test particle is described by a spherical wave, i.e., a highly correlated continuous superposition of states with well localized position and momentum, and the oscillators are in the ground state. Furthermore, we assume that the positions of the oscillators are not collinear with the center of the spherical wave. Under suitable assumptions on the physical parameters characterizing the model, we give an asymptotic expression of the solution of the Schrödinger equation of the system with an explicit control of the error. The result shows that the approximate expression of the wave function is the sum of two terms, orthogonal in and describing rather different situations. In the first one, all the oscillators remain in their ground state and the test particle is described by the free evolution of a slightly deformed spherical wave. The second one consists of a sum of N terms where in each term there is only one excited oscillator and the test particle is correspondingly described by the free evolution of a wave packet, well concentrated in position and momentum. Moreover, the wave packet emerges from the excited oscillator with an average momentum parallel to the line joining the oscillator with the center of the initial spherical wave. Such wave packet represents a semiclassical state for the test particle, propagating along the corresponding classical trajectory. The main result of our analysis is to show how such a semiclassical state can be produced, starting from the original spherical wave, as a result of the interaction with the environment.

Bound states in a hyperbolic asymmetric doublewell
View Description Hide DescriptionWe report a new class of hyperbolic asymmetric doublewell whose bound state wavefunctions can be expressed in terms of confluent Heun functions. An analytic procedure is used to obtain the energy eigenvalues and the criterion for the potential to support bound states is discussed.

Bound states for multiple Diracδ wells in spacefractional quantum mechanics
View Description Hide DescriptionUsing the momentumspace approach, we obtain bound states for multiple Diracδ wells in the framework of spacefractional quantum mechanics. Introducing first an attractive Diraccomb potential, i.e., Dirac comb with strength −g (g > 0), in the spacefractional Schrödinger equation we show that the problem of obtaining eigenenergies of a system with N Diracδ wells can be reduced to a problem of obtaining the eigenvalues of an N × N matrix. As an illustration we use the present matrix formulation to derive expressions satisfied by the boundstate energies of N = 1, 2, 3 delta wells. We also obtain the corresponding wave functions and express them in terms of Fox's Hfunction.

Squeezed states and Hermite polynomials in a complex variable
View Description Hide DescriptionFollowing the lines of the recent paper of J.P. Gazeau and F. H. Szafraniec [J. Phys. A: Math. Theor.44, 495201 (2011)], we construct here three types of coherent states, related to the Hermite polynomials in a complex variable which are orthogonal with respect to a nonrotationally invariant measure. We investigate relations between these coherent states and obtain the relationship between them and the squeezed states of quantum optics. We also obtain a second realization of the canonical coherent states in the Bargmann space of analytic functions, in terms of a squeezed basis. All this is done in the flavor of the classical approach of V. Bargmann [Commun. Pure Appl. Math.14, 187 (1961)].

Completeness for sparse potential scattering
View Description Hide DescriptionThe present paper is devoted to the scattering theory of a class of continuum Schrödinger operators with deterministic sparse potentials. We first establish the limiting absorption principle for both modified free resolvents and modified perturbed resolvents. This actually is a weak form of the classical limiting absorption principle. We then prove the existence and completeness of local wave operators, which, in particular, imply the existence of wave operators. Under additional assumptions on the sparse potential, we prove the completeness of wave operators. In the context of continuum Schrödinger operators with sparse potentials, this paper gives the first proof of the completeness of wave operators.

From the attempt of certain classical reformulations of quantum mechanics to quasiprobability representations
View Description Hide DescriptionThe concept of an injective affine embedding of the quantum states into a set of classical states, i.e., into the set of the probability measures on some measurable space, as well as its relation to statistically complete observables is revisited, and its limitation in view of a classical reformulation of the statistical scheme of quantum mechanics is discussed. In particular, on the basis of a theorem concerning a nondenseness property of a set of coexistent effects, it is shown that an injective classical embedding of the quantum states cannot be supplemented by an at least approximate classical description of the quantum mechanical effects. As an alternative approach, the concept of quasiprobability representations of quantum mechanics is considered.