This paper is concerned with small parameter asymptotics of magnetic quantum systems. In addition to a semiclassical parameter \eps, the case of small coupling $λ$ to the magnetic vector potential naturally occurs in this context. Magnetic Weyl calculus is adapted to incorporate both parameters, at least one of which needs to be small. Of particular interest is the expansion of the Weyl product which can be used to expand the product of operators in a small parameter, a technique which is prominent to obtain perturbation expansions. Three asymptotic expansions for the magnetic Weyl product of two Hörmander class symbols are proven: (i) \eps \ll 1 and λ\ll 1, (ii) \eps \ll 1 and λ= 1 as well as (iii) \eps = 1 and λ\ll 1. Expansions (i) and (iii) are impossible to obtain with ordinary Weyl calculus. Furthermore, I relate results derived by ordinary Weyl calculus with those obtained with magnetic Weyl calculus by one- and two-parameter expansions. To show the power and versatility of magnetic Weyl calculus, I derive the semirelativistic Pauli equation as a scaling limit from the Dirac equation up to errors of 4th order in 1/c.
We prove criteria for a {\it 'magnetic' Weyl operator} to be in a Schatten-von Neuman class by extending a method developed by H. Cordes, T. Kato and G. Arsu.
We propose a new version of Wigner-Weyl calculus for tight-binding lattice models. It allows to express various physical quantities through Weyl symbols of operators and Green's functions. In particular, Hall conductivity in the presence of varying and arbitrarily strong magnetic field is represented using the proposed formalism as a topological invariant.
Weyl quantization and related semiclassical techniques can be used to study conduction properties of crystalline solids subjected to slowly-varying, external electromagnetic fields. The case where the external magnetic field is constant, is not covered by existing theory as proofs involving usual Weyl calculus break down. This is the regime of the so-called quantum Hall effect where quantization of transverse conductance is observed. To rigorously derive semiclassical equations of motion, one needs to systematically develop a magnetic Weyl calculus which contains a semiclassical parameter. Mat…
We develop a pseudo-differential Weyl calculus on nilpotent Lie groups which allows one to deal with magnetic perturbations of right invariant vector fields. For this purpose we investigate an infinite-dimensional Lie group constructed as the semidirect product of a nilpotent Lie grup and an appropriate function space thereon. We single out an appropriate coadjoint orbit in the semidirect product and construct our pseudo-differential calculus as a Weyl quantization of that orbit.
We show how to relate the full quantum dynamics of a spin-1/2 particle on R^d to a classical Hamiltonian dynamics on the enlarged phase space R^d x S^2 up to errors of second order in the semiclassical parameter. This is done via an Egorov-type theorem for normal Wigner-Weyl calculus for R^d [Lei10,Fol89] combined with the Stratonovich-Weyl calculus for SU(2) [VGB89]. For a specific class of Hamiltonians, including the Rabi- and Jaynes-Cummings model, we prove an Egorov theorem for times much longer than the semiclassical time scale. We illustrate the approach for a simple model of the Stern-G…
We discuss an extension of Toeplitz quantization based on polyanalytic functions. We derive isomorphism theorem for polyanalytic Toeplitz operators between weighted Sobolev-Fock spaces of polyanalytic functions, which are images of modulation spaces under polyanalytic Bargmann transforms. This generalizes well-known results from the analytic setting. Finally, we derive an asymptotic symbol calculus and present an asymptotic expansion of complex Weyl operators in terms of polyanalytic Toeplitz operators.
We show that the Schrödinger equation in phase space proposed by Torres-Vega and Frederick is canonical in the sense that it is a natural consequence of the extended Weyl calculus obtained by letting the Heisenberg group act on functions (or half-densities) defined on phase space. This allows us, in passing, to solve rigorously the TF equation for all quadratic Hamiltonians.
In the presence of a variable magnetic field, the Weyl pseudodifferential calculus must be modified. The usual modification, based on ``the minimal coupling principle'' at the level of the classical symbols, does not lead to gauge invariant formulae if the magnetic field is not constant. We present a gauge covariant quantization, relying on the magnetic canonical commutation relations. The underlying symbolic calculus is a deformation, defined in terms of the magnetic flux through triangles, of the classical Moyal product.
We study the twisted Weyl symbol of metaplectic operators; this requires the definition of an index for symplectic paths related to the Conley-Zehnder index. We thereafter define a metaplectically covariant algebra of pseudo-differential operators acting on functions on symplectic space.
In this work, we consider fixed $1/2$ spin particles interacting with the quantized radiation field in the context of quantum electrodynamics (QED). We investigate the time evolution operator in studying the reduced propagator (interaction picture). We first prove that this propagator belongs to the class of infinite dimensional Weyl pseudodifferential operators recently introduced in \cite {A-J-N} on Wiener spaces. We give a semiclassical expansion of the symbol of the reduced propagator up to any order with estimates on the remainder terms. Next, taking into account analyticity properties fo…
The concern of this article is a semiclassical Weyl calculus on an infinite dimensional Hilbert space $H$. If $(i, H, B)$ is a Wiener triplet associated to $H$, the quantum state space will be the space of $L^2$ functions on $B$ with respect to a Gaussian measure with $h/2$ variance, where $h$ is the semiclassical parameter. We prove the boundedness of our pseudodifferential operators (PDO) in the spirit of Calderón-Vaillancourt with an explicit bound, a Beals type characterization, and metaplectic covariance. An application to a model of quantum electrodynamics (QED) is added in the last sect…
The theme of this work is that the theory of charged particles in a uniform magnetic field can be generalized to a large class of operators if one uses an extended a class of Weyl operators which we call "Landau--Weyl pseudodifferential operators". The link between standard Weyl calculus and Landau--Weyl calculus is made explicit by the use of an infinite family of intertwining "windowed wavepacket transforms"; this makes possible the use of the theory of modulation spaces to study various regularity properties. Our techniques allow us not only to recover easily the eigenvalues and eigenfuncti…
We discuss topics related to finite-dimensional calculus in the context of finite-dimensional quantum mechanics. The truncated Heisenberg-Weyl algebra is called a TAA algebra after Tekin, Aydin, and Arik who formulated it in terms of orthofermions. It is shown how to use a matrix approach to implement analytic representations of the Heisenberg-Weyl algebra in univariate and multivariate settings. We provide examples for the univariate case. Krawtchouk polynomials are presented in detail, including a review of Krawtchouk polynomials that illustrates some curious properties of the Heisenberg-Wey…
One can argue that on flat space $\mathbb{R}^d$ the Weyl quantization is the most natural choice and that it has the best properties (e.g. symplectic covariance, real symbols correspond to Hermitian operators). On a generic manifold, there is no distinguished quantization, and a quantization is typically defined chart-wise. Here we introduce a quantization that, we believe, has the best properties for studying natural operators on pseudo-Riemannian manifolds. It is a generalization of the Weyl quantization - we call it the balanced geodesic Weyl quantization. Among other things, we prove that …
Classical pseudo-differential calculus on $\mathbb{R}^{d}$ can be viewed as a (non-commutative) functional calculus for the standard position and momentum operators $(Q_{1}, \dots , Q_{d})$ and $(P_{1}, \dots , P_{d})$. We generalise this calculus to the setting of two $d$-tuples of operators $A=(A_{1}, \dots , A_{d})$ and $B=(B_{1}, \dots , B_{d})$ acting on a Banach space $X$ such that $iA_{1}, \dots , iA_{d}$ and $iB_{1}, \dots , iB_{d}$ generate bounded $C_0$-groups satisfying the Weyl canonical commutation relations $e^{isA_j}e^{itA_k} = e^{itA_k}e^{isA_j}$, $e^{isB_j}e^{itB_k} = e^{itB_k…
The main purpose in the present paper is to build a Hamiltonian theory for fields which is consistent with the principles of relativity. For this we consider detailed geometric pictures of Lepage theories in the spirit of Dedecker and try to stress out the interplay between the Lepage-Dedecker (LP) description and the (more usual) de Donder-Weyl (dDW) one. One of the main points is the fact that the Legendre transform in the dDW approach is replaced by a Legendre correspondence in the LP theory (This correspondence behaves differently: ignoring the singularities whenever the Lagrangian is dege…
The building blocks of Hudson-Parthasarathy quantum stochastic calculus start with Weyl operators on a symmetric Fock space. To realize a relativistically covariant version of the calculus we construct representations of Poincare group in terms of Weyl operators on suitably constructed, Bosonic or Fermionic based on the mass and spin of the fundamental particle, Fock spaces. We proceed by describing the orbits of homogeneous Lorentz group on R4 and build fiber bundle representations of Poincaré group induced from the stabilizer subgroups (little groups) and build the Boson Fock space of the Hi…
We develop an abstract framework for the investigation of quantization and dequantization procedures based on orthogonality relations that do not necessarily involve group representations. To illustrate the usefulness of our abstract method we show that it behaves well with respect to the infinite tensor products. This construction subsumes examples coming from the study of magnetic Weyl calculus, the magnetic pseudo-differential Weyl calculus, the metaplectic representation on locally compact abelian groups, irreducible representations associated with finite-dimensional coadjoint orbits of so…
This work develops a magnetic pseudodifferential calculus for super operators OpA(F); these map operators onto operators (as opposed to Lp functions onto Lq functions). Here, F could be a tempered distribution or a Hörmander symbol. An important example is Liouville super operators defined in terms of a magnetic pseudodifferential operator. Our work combines ideas from magnetic Weyl calculus developed in [MP04, IMP07, Lei11] and the pseudodifferential calculus on the non-commutative torus from [HLP18a, HLP18b]. Thus, our calculus is inherently gauge-covariant, which means all essential propert…
In a first part we propose an introduction to multisymplectic formalisms, which are generalisations of Hamilton's formulation of Mechanics to the calculus of variations with several variables: we give some physical motivations, related to the quantum field theory, and expound the simplest example, based on a theory due to T. de Donder and H. Weyl. In a second part we explain quickly a work in collaboration with J. Kouneiher (math-ph/0211046) on generalizations of the de Donder--Weyl theory (known as Lepage theories). Lastly we show that in this framework a perturbative classical field theory (…
The usual Weyl calculus is intimately associated with the choice of the standard symplectic structure on $\mathbb{R}^{n}\oplus\mathbb{R}^{n}$. In this paper we will show that the replacement of this structure by an arbitrary symplectic structure leads to a pseudo-differential calculus of operators acting on functions or distributions defined, not on $\mathbb{R}^{n}$ but rather on $\mathbb{R}^{n}\oplus\mathbb{R}^{n}$. These operators are intertwined with the standard Weyl pseudo-differential operators using an infinite family of partial isometries of $L^{2}(\mathbb{R}^{n})\longrightarrow L^{2}(…
Pseudodifferential Weyl calculus on vector bundles
Royal Institute of Technology（皇家理工学院）；University of Vienna, Faculty of Physics（维也纳大学物理学院）；Faculty for Mathematics, University of Regensburg（雷根斯堡大学数学系）；Faculty for Mathematics, TU Chemnitz（开姆尼茨工业大学数学系）；Department of Mathematics, University of Genoa（热那亚大学数学系）；INFN-Section Genoa and INdAM-GNFM（意大利国家核物理研究所热那亚分部及印度数学与应用研究网络）
Contrary to the finite dimensional case, Weyl and Wick quantizations are no more asymptotically equivalent in the infinite dimensional bosonic second quantization. Moreover neither the Weyl calculus defined for cylindrical symbols nor the Wick calculus defined for polynomials are preserved by the action of a nonlinear flow. Nevertheless taking advantage carefully of the information brought by these two calculuses in the mean field asymptotics, the propagation of Wigner measures for general states can be proved, extending to the infinite dimensional case a standard result of semiclassical analy…
We define the Anderson Hamiltonian H on a two-dimensional manifold using high order paracontrolled calculus. It is a self-adjoint operator with pure point spectrum. We get lower and upper bounds on its eigenvalues which imply an almost sure Weyl-type law for H.
There is a connection between the Weyl pseudodifferential calculus and crossed product C*-algebras associated with certain dynamical systems. And in fact both topics are involved in the quantization of a non-relativistic particle moving in R^n. Our paper studies the situation in which a variable magnetic field is also present. The Weyl calculus has to be modified, giving a functional calculus for a family of operators (positions and magnetic momenta) with highly non-trivial commutation relations. On the algebraic side, the dynamical system is twisted by a cocyle defined by the flux of the magn…
Inspired by a recent result of Davies and Pushnitski, we study resonance asymptotics of quantum graphs with general coupling conditions at the vertices. We derive a criterion for the asymptotics to be of a non-Weyl character. We show that for balanced vertices with permutation-invariant couplings the asymptotics is non-Weyl only in case of Kirchhoff or anti-Kirchhoff conditions, while for graphs without permutation numerous examples of non-Weyl behaviour can be constructed. Furthermore, we present an insight helping to understand what makes the Kirchhoff/anti-Kirchhoff coupling particular from…
We study asymptotical behaviour of resonances for a quantum graph consisting of a finite internal part and external leads placed into a magnetic field, in particular, the question whether their number follows the Weyl law. We prove that the presence of a magnetic field cannot change a non-Weyl asymptotics into a Weyl one and vice versa. On the other hand, we present examples demonstrating that for some non-Weyl graphs the ``effective size'' of the graph, and therefore the resonance asymptotics, can be affected by the magnetic field.
We derive an asymptotic expansion for the Weyl function of a one-dimensional Schrödinger operator which generalizes the classical formula by Atkinson. Moreover, we show that the asymptotic formula can also be interpreted in the sense of distributions.
The asymptotic behaviour of the eigenvalue counting function of Laplacians on Hanoi attractors is determined. To this end, Dirichlet and resistance forms are constructed. Due to the non self-similarity of these sets, the classical construction of the Laplacian for p.c.f. self-similar fractals has to be modified by combining discrete and quantum graphs methods.
We prove a Tauberian theorem for singular values of noncommuting operators which allows us to prove exact asymptotic formulas in noncommutative geometry at a high degree of generality. We explain how, via the Birman--Schwinger principle, these asymptotics imply that a semiclassical Weyl law holds for many interesting noncommutative examples. In Connes' notation for quantized calculus, we prove that for a wide class of $p$-summable spectral triples $(\mathcal{A},H,D)$ and self-adjoint $V \in \mathcal{A}$, there holds \[\lim_{h\downarrow 0} h^p\mathrm{Tr}(χ_{(-\infty,0)}(h^2D^2+V)) = \int V_-^{\…
This paper aims to study the q-analogue of the Sturm Liouville problem and to give an asymptotic behaviour at infinity for its solution '. Additionally, we establish an asymptotic expansion of the q-Bessel function $j_α$ for $α>-{1/2}. We are not in situation to claim that our results are new but they have the advantage to show that the method used by Agranovich and Marchenko remain true.
We show how to find the coefficient by the leading term of the resonance asymptotics using the method of pseudo orbit expansion for quantum graphs which do not obey the Weyl asymptotics. For a non-Weyl graph we develop a method how to reduce the number of edges of a corresponding directed graph. Through this method we prove bounds on the above coefficient depending on the structure of the graph for graphs with the same lengths of internal edges. We explicitly find the positions of the resolvent resonances.
We revisit the celebrated Peierls-Onsager substitution employing the magnetic pseudo-differential calculus for weak magnetic fields with no spatial decay conditions, when the non-magnetic symbols have a certain spatial periodicity. We show in great generality that the symbol of the magnetic band Hamiltonian admits a convergent expansion. Moreover, if the non-magnetic band Hamiltonian admits a localized composite Wannier basis, we show that the magnetic band Hamiltonian is unitarily equivalent to a Hofstadter-like magnetic matrix. In addition, if the magnetic field perturbation is slowly variab…
We investigate Weyl type asymptotics of functional-difference operators associated to mirror curves of special del Pezzo Calabi-Yau threefolds. These operators are $H(ζ)=U+U^{-1}+V+ζV^{-1}$ and $H_{m,n}=U+V+q^{-mn}U^{-m}V^{-n}$, where $U$ and $V$ are self-adjoint Weyl operators satisfying $UV=q^{2}VU$ with $q=e^{iπb^{2}}$, $b>0$ and $ζ>0$, $m,n\in\mathbb{N}$. We prove that $H(ζ)$ and $H_{m,n}$ are self-adjoint operators with purely discrete spectrum on $L^{2}(\mathbb{R})$. Using the coherent state transform we find the asymptotical behaviour for the Riesz mean $\sum_{j\ge 1}(λ-λ_{j})_{+}$ as $…
This papers is concerned with multisymplectic formalisms which are the frameworks for Hamiltonian theories for fields theory. Our main purpose is to study the observable $(n-1)$-forms which allows one to construct observable functionals on the set of solutions of the Hamilton equations by integration. We develop here two different points of view: generalizing the law $\{p,q\} = 1$ or the law $dF/dt = \{H,F\}$. This leads to two possible definitions; we explore the relationships and the differences between these two concepts. We show that -- in contrast with the de Donder--Weyl theory -- the tw…
By Weyl's asymptotic formula, for any potential $V$ whose negative part $V_-$ is an $L^{1+d/2}$-function, \begin{align*} \operatorname{Tr} [-h^2 Δ+ V]_- &= L_d^{\mathrm{cl}} h^{-d} \int \mathrm{d} x\,[V]_-^{1+\frac d 2} + \mathrm{o} (h^{-d})_{h \to 0} , \end{align*} with the semiclassical constant $L^{\mathrm{cl}}_d = 2^{-d} π^{-d/2} / Γ(2 + \frac d 2)$. In this paper, we show that, even if $[V_1]_-, [V_2]_- \notin L^{1+d/2}$, but the difference $[V_1]_-^{1+d/2}-[V_2]_-^{1+d/2}$ is integrable, then we still have the asymptotic formula \[ \operatorname{Tr} [-h^2 Δ+ V_1 ]_- - \operatorname{Tr} […
We consider a simple model of an open partially expanding map. Its trapped set K in phase space is a fractal set. We first show that there is a well defined discrete spectrum of Ruelle resonances which describes the asymptotics of correlation functions for large time and which is parametrized by the Fourier component νon the neutral direction of the dynamics. We introduce a specific hypothesis on the dynamics that we call "minimal captivity". This hypothesis is stable under perturbations and means that the dynamics is univalued on a neighborhood of K. Under this hypothesis we show the existenc…
We study spectral asymptotics for a large class of differential operators on an open subset of $\R^d$ with finite volume. This class includes the Dirichlet Laplacian, the fractional Laplacian, and also fractional differential operators with non-homogeneous 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.
Asymptotic results are derived for the number of random walks in alcoves of affine Weyl groups (which are certain regions in $n$-dimensional Euclidean space bounded by hyperplanes), thus solving problems posed by Grabiner [J. Combin. Theory Ser. A 97 (2002), 285-306]. These results include asymptotic expressions for the number of vicious walkers on a circle, and as well for the number of vicious walkers in an interval. The proofs depart from the exact results of Grabiner [loc. cit.], and require as diverse means as results from symmetric function theory and the saddle point method, among other…
