In previous papers, a generalization of the Weyl calculus was introduced in connection with the quantization of a particle moving in $\mathbb R^n$ under the influence of a variable magnetic field $B$. It incorporates phase factors defined by $B$ and reproduces the usual Weyl calculus for B=0. In the present article we develop the classical pseudodifferential theory of this formalism for the standard symbol classes $S^m_{ρ,δ}$. Among others, we obtain properties and asymptotic developments for the magnetic symbol multiplication, existence of parametrices, boundedness and positivity results, properties of the magnetic Sobolev spaces. In the case when the vector potential $A$ has all the derivatives of order $\ge 1$ bounded, we show that the resolvent and the fractional powers of an elliptic magnetic pseudodifferential operator are also pseudodifferential. As an application, we get a limiting absorption principle and detailed spectral results for self-adjoint operators of the form $H=h(Q,Π^A)$, where $h$ is an elliptic symbol, $Π^A=D-A$ and $A$ is the vector potential corresponding to a short-range magnetic field.
We characterize the Schwartz kernels of pseudodifferential operators of Shubin type by means of an FBI transform. Based on this we introduce as a generalization a new class of tempered distributions called Shubin conormal distributions. We study their transformation behavior, normal forms and microlocal properties.
Let $H$ be a self-adjoint isotropic elliptic pseudodifferential operator of order $2$. Denote by $u(t)$ the solution of the Schrödinger equation $(i\partial_t - H)u = 0$ with initial data $u(0) = u_0$. If $u_0$ is compactly supported the solution $u(t)$ is smooth for small $t > 0$, but not for all $t$. We determine the wavefront set of $u(t)$ in terms of the wavefront set of $u_0$ and the principal and subprincipal symbol of $H$.
We prove rapid decay (even exponential decay under some stronger assumptions) of the eigenfunctions associated to discrete eigenvalues, for a class of self-adjoint operators in $L^2(\mathbb{R}^d)$ defined by ``magnetic'' pseudodifferential operators (studied in \cite{IMP1}). This class contains the relativistic Schrödinger operator with magnetic field.
The gauge covariant magnetic Weyl calculus has been introduced and studied in previous works. We prove criteria in terms of commutators for operators to be magnetic pseudo-differential operators of suitable symbol classes. The approach is completely intrinsic; neither the statements nor the proofs depend on a choice of a vector potential. We apply this criteria to inversion problems, functional calculus, affiliation results and to the study of the evolution group generated by a magnetic pseudo-differential operator.
Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities on a closed manifold $M$, whose principal symbol is assumed to have simple eigenvalues. We show existence and uniqueness of $m$ orthonormal pseudodifferential projections commuting with the operator $A$ and provide an algorithm for the computation of their full symbols, as well as explicit closed formulae for their subprincipal symbols. Pseudodifferential projections yield a decomposition of $L^2(M)$ into invariant subspaces under the action of $A$ modulo $C^\infty(M)$. Furthermore, they a…
We prove equivariant spectral asymptotics for $ h$-pseudodifferential operators for compact orthogonal group actions generalizing results of El-Houakmi and Helffer (1991) and Cassanas (2006). Using recent results for certain oscillatory integrals with singular critical sets (Ramacher 2010) 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.
Following the work of Duistermaat-Singer \cite{DS} on isomorphisms of algebras of global pseudodifferential operators, we classify isomorphisms of algebras of microlocally defined semiclassical pseudodifferential operators. Specifically, we show that any such isomorphism is given by conjugation by $A = BF$, where $B$ is a microlocally elliptic semiclassical pseudodifferential operator, and $F$ is a microlocal $h$-FIO associated to the graph of a local symplectic transformation.
A new symbol theory for pseudodifferential operators in the complex analytic category is given. This theory provides a cohomological foundation of symbolic calculus.
Mixed-norm Lebesgue spaces found their place in the study of some questions in the theory of partial differential equations, as can be seen from recent interest in the continuity of certain classes of pseudodifferential operators on these spaces. In this paper, we use some recent advances in the pseudodifferential calculus for nonsmooth symbols to prove the boundedness of pseudodifferential operators with such symbols on mixed-norm Lebesgue spaces.
In this paper we discuss some spectral invariance results for non-smooth pseudodifferential operators with coefficients in Hölder spaces. In analogy to the proof in the smooth case of Beals and Ueberberg, we use the characterization of non-smooth pseudodifferential operators to get such a result. The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols and the fact, that in general the composition of two non-smooth pseudodifferential operators is not a pseudodifferential operator. In order to improve these spectral invariance results …
For a class of zero order pseudodifferential operators we find the asymptotics of eigenvalues converging to a non-isolated tip of the essential spectrum.
In this paper we establish explicit lower bounds for pseudodifferential operators with a radial symbol. The proofs use classical Weyl calculus techniques and some useful, if not celebrated, properties of the Laguerre polynomials.
We investigate properties of pseudodifferential operators on $L^2$ space on manifold with ends including asymptotically conical or hyperbolic ends. Our pseudodifferential operators are a generalization of the canonical quantization which naturally appears in the quantum mechanics on curved spaces. We prove a Calderón-Vaillancourt type theorem for our pseudodifferential operators and discuss a construction of parametrix of elliptic differential operators on manifolds with ends.
We are consider domains in cotangent bundles with the property that the null foliation of their boundary is fibrating and the leaves satisfy a Bohr-Sommerfeld condition (for example, the unit disk bundle of a Zoll metric). Given such a domain, we construct an algebra of associated semiclassical pseudodifferential operators with singular symbols. The Schwartz kernels of the operators have frequency set contained in the union of the diagonal and the flow-out of the null foliation of the boundary of the domain. We develop a symbolic calculus, prove the existence of projectors (under a mild additi…
We report some recent results on analytic pseudodifferential operators, also known as Wick operators. An important tool in our study is the Bargmann transform which provides a coupling between the classical (real) and analytic pseudodifferential calculus. Since the Bargmann transform of Hermite functions gives rise to formal power series in the complex domain, the results are formulated in terms of the Bargmann images of Pilipović spaces.
The aim of this article is to prove a Beals type characterization theorem for pseudodifferential operators in Wiener spaces. The definition of pseudodifferential operators in Wiener spaces and a Calderón-Vaillancourt type result appear in [1]. The set of symbols considered here is the one of [1]. The Weyl calculus in infinite dimension considered here emphasizes the role of the Wick bi-symbols.
We construct parametrices for a class of pseudodifferential operators of infinite order acting on spaces of tempered ultradistributions of Beurling and Roumieu type. As a consequence we obtain a result of hypoellipticity in these spaces.
In this paper we investigate $L^p$ and Sobolev boundedness of a certain class of pseudodifferential operators with non-regular symbols. We employ regularisation methods, namely convolution with a net of mollifiers $(ρ_\eps)_\eps$, and we study the corresponding net of pseudodifferential operators providing $L^p$ and Sobolev estimates which relate the parameter $\eps$ with the non-regularity of the symbol.
Specific global symbol classes and corresponding pseudodifferential operators of infinite order that act continuously on the space of tempered ultradistributions of Beurling and Roumieu type are constructed. For these classes, symbolic calculus is developed.
We study the change of quantization for a class of global pseudodifferential operators of infinite order in the setting of ultradifferentiable functions of Beurling type. The composition of different quantizations as well as the transpose of a quantization are also analysed, with applications to the Weyl calculus. We also compare global $ω$-hypoellipticity and global $ω$-regularity of these classes of pseudodifferential operators.
Motivated by the recent paper of Boggiatto-Garello in J. Pseudo-Differ. Oper. Appl. \textbf{11} (2020), 93-117, where a Gabor operator is regarded as pseudodifferential operator with symbol $p(x,ω)$ periodic on both the variables, we study the continuity and invertibility, on general time frequency invariant spaces, of pseudodifferential operators with completely periodic symbol and general $τ$ quantization.
We investigate global microlocal properties of localization operators and Shubin pseudodifferential operators. The microlocal regularity is measured in terms of a scale of Shubin-type Sobolev spaces. In particular, we prove microlocality and microellipticity of these operators.
A generalization of differential operators are pseudodifferential operators which are used for reasoning about partial differential equations with variable coefficients. A lot of useful properties about classical pseudodifferential operators are already discovered by the works of H. Kumano-Go and L. Hörmander. In this work, we like to transfer their ideas on those operators which symbols are merely Hölder-continous. We focus on the composition of operators, symbol smoothing and operators on manifolds.
The aim of these notes is to describe some recent results concerning dispersive estimates for principally normal pseudodifferential operators. The main motivation for this comes from unique continuation problems. Such estimates can be used to prove Carleman inequalities, which in turn yield unique continuation results for various partial differential operators with rough potentials.
We characterize the action of isotropic pseudodifferential operators on functions in terms of their action on Hermite functions. We show that an operator $A : S(\mathbb{R}) \to S(\mathbb{R})$ is an isotropic pseudodifferential operator of order r if and only if its "matrix" $(K(A))_{m,n} := < Aϕ_n,ϕ_m>_{L^2(\mathbb{R})}$ is rapidly decreasing away from the diagonal $\{m = n\}$, order $\frac {r}{2}$ in $m + n$, and where applying the discrete difference operator along the diagonal decreases the order by one. Additionally, we use this result to prove an analogue of Beal's theorem for isotropic p…
We establish the existence of a bounded $H_\infty$-calculus for a large class of hypoelliptic pseudodifferential operators on R^n and closed manifolds.
We develop a theory of pseudodifferential operators of infinite order for the global classes $\mathcal{S}_ω$ of ultradifferentiable functions in the sense of Björck, following the previous ideas given by Prangoski for ultradifferentiable classes in the sense of Komatsu. We study the composition and the transpose of such operators with symbolic calculus and provide several examples.
We study the boundedness of rough Fourier integral and pseudodifferential operators, defined by general rough Hörmander class amplitudes, on Banach and quasi-Banach $L^p$ spaces. Thereafter we apply the aforementioned boundedness in order to improve on some of the existing boundedness results for Hörmander class bilinear pseudodifferential operators and certain classes of bilinear (as well as multilinear) Fourier integral operators. For these classes of amplitudes, the boundedness of the aforementioned Fourier integral operators turn out to be sharp. Furthermore we also obtain results for roug…
Spectral asymmetry via pseudodifferential projections: the massless Dirac operator
Università degli Studi di Milano（米兰大学）；Maxwell Institute for Mathematical Sciences, Edinburgh（爱丁堡马克斯韦尔数学科学研究所）；Heriot-Watt University, Edinburgh（爱丁堡赫瑞-瓦特大学）；Università degli Studi di Pavia（帕维亚大学）；INFN, Sezione di Pavia（意大利国家核物理研究所帕维亚分部）；INdAM, Sezione di Pavia（意大利国家高等数学研究所帕维亚分部）
We give a procedure for reconstructing a magnetic field and electric potential from boundary measurements related to the magnetic Schroedinger operator in three and higher dimensions.
Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities on a closed manifold $M$, whose principal symbol is assumed to have simple eigenvalues. Relying on a basis of pseudodifferential projections commuting with $A$, we construct an almost-unitary pseudodifferential operator that diagonalizes $A$ modulo an infinitely smoothing operator. We provide an invariant algorithm for the computation of its full symbol, as well as an explicit closed formula for its subprincipal symbol. Finally, we give a quantitative description of the relation between th…
We construct a parametrix of a resolvent of elliptic differential operators acting on half-densities on manifolds with ends. The construction is carried out by introducing suitable pseudodifferential operators compatible with the end structure. Our class of pseudodifferential operators and symbols is independent of the choice of Riemannian metric on the manifold and applicable to both of asymptotically conical and hyperbolic manifolds. As an application, we prove the essential self-adjointness of elliptic symmetric differential operators on manifolds.
By using a time slicing procedure, we represent the solution operator of a second-order parabolic pseudodifferential equation on $\R^n$ as an infinite product of zero-order pseudodifferential operators. A similar representation formula is proven for parabolic differential equations on a compact Riemannian manifold. Each operator in the multi-product is given by a simple explicit Ansatz. The proof is based on an effective use of the Weyl calculus and the Fefferman-Phong inequality.
Smooth pseudodifferential operators on $\mathbb{R}^n$ can be characterized by their mapping properties between $L^p-$Sobolev spaces due to Beals and Ueberberg. In applications such a characterization would also be useful in the non-smooth case, for example to show the regularity of solutions of a partial differential equation. Therefore, we will show that every linear operator $P$, which satisfies some specific continuity assumptions, is a non-smooth pseudodifferential operator of the symbol-class $C^τ S^m_{1,0}(\mathbb{R}^n \times \mathbb{R}^n)$. The main new difficulties are the limited mapp…
We study the dynamics of resonances of analytic perturbations of 0th order pseudodifferential operators $P(s)$. In particular, we prove a Fermi golden rule for resonances of $P(s)$ at embedded eigenvalues of $P=P(0)$. We also study the dynamics of eigenvalues of $P(t)=P+itΔ$ as the eigenvalues converge to simple eigenvalues of $P$. The 0th order pseudodifferential operators we consider satisfy natural dynamical assumptions and are used as microlocal models of internal waves.
We consider toroidal pseudodifferential operators with operator-valued symbols, their mapping properties and the generation of analytic semigroups on vector-valued Besov and Sobolev spaces. We show that a parabolic toroiodal pseudodifferential operator generates an analytic semigroup both on Banach space-valued Besov spaces and on Banach space-valued Sobolev spaces. Here, no condition on the Banach space is imposed. For the proof of the Sobolev space result, we establish a uniform estimate on the kernel which is given as an infinite parameter-dependent sum. An application to abstract non-auton…
In the first part of the paper the authors study the minimal and maximal extension of a class of weighted pseudodifferential operators in the Fréchet space $L^p_{\rm loc}(Ω)$. In the second one non homogeneous microlocal properties are introduced and propagation of Sobolev singularities for solutions to (pseudo)differential equations is given. For both the arguments actual examples are provided.
The L2-norm closure of the algebra of all zero-order classical (or polyhomogeneous) pseudodifferential operators on a closed manifold $X$ is proven not to contain Hörmander's class $Ψ^0(X)$. A set of generators (for that closed algebra) smaller than all zero-order classical pseudos is also described.
In this paper, we consider pseudodifferential operators on the torus with operator-valued symbols and prove continuity properties on vector-valued toroidal Besov spaces, without assumptions on the underlying Banach spaces. The symbols are of limited smoothness with respect to $x$ and satisfy a finite number of estimates on the discrete derivatives. The proof of the main result is based on a description of the operator as a convolution operator with a kernel representation which is related to the dyadic decomposition appearing in the definition of the Besov space.
We characterize the groupoids for which an operator is Fredholm if, and only if, its principal symbol and all its boundary restrictions are invertible. A groupoid with this property is called {\em Fredholm}. Using results on the Effros-Hahn conjecture, we show that an almost amenable, Hausdorff, second countable groupoid is Fredholm. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. In particular, one can use these results to characterize the Fredholm operators on manifolds with cylindrical and poly-cylindrical ends, on manifolds that a…
We study the asymptotic behavior of the counting function of tensor products of operators, in the cases where the factors are either pseudodifferential operators on closed manifolds, or pseudodifferential operators of Shubin type on $\mathbb{R}^n$, respectively. We obtain, in particular, the sharpness of the remainder term in the corresponding Weyl formulae, which we prove by means of the analysis of some explicit examples.
Starting out from a new description of a class of parameter-dependent pseudodifferential operators with finite regularity number due to G. Grubb, we introduce a calculus of parameter-dependent, poly-homogeneous symbols whose homogeneous components have a particular type of point-singularity in the covariable-parameter space. Such symbols admit intrinsically a second kind of expansion which is closely related to the expansion in the Grubb-Seeley calculus and permits to recover the resolvent-trace expansion for elliptic pseudodifferential oerators originally proved by Grubb-Seeley. Another appli…
The paper concerns algebras of almost periodic pseudodifferential operators on $\mathbb R^d$ with symbols in Hörmander classes. We study three representations of such algebras, one of which was introduced by Coburn, Moyer and Singer and the other two inspired by results in probability theory by Gladyshev. Two of the representations are shown to be unitarily equivalent for nonpositive order. We apply the results to spectral theory for almost periodic pseudodifferential operators acting on $L^2$ and on the Besicovitch Hilbert space of almost periodic functions.
An approximation Ansatz for the operator solution, $U(z',z)$, of a hyperbolic first-order pseudodifferential equation, $\d_z + a(z,x,D_x)$ with $\Re (a) \geq 0$, is constructed as the composition of global Fourier integral operators with complex phases. An estimate of the operator norm in $L(H^{(s)},H^{(s)})$ of these operators is provided which allows to prove a convergence result for the Ansatz to $U(z',z)$ in some Sobolev space as the number of operators in the composition goes to $\infty$.
This is a review of some coordinate-free calculi of pseudodifferential operators developed in the last years. As an application, we use a coordinate-free calculus to obtain new results on the behaviour of the spectral projections of a self-adjoint elliptic second order differential operator under perturbation of coefficients.
In this paper a bisingular pseudodifferential calculus, along the lines of the one introduced by L. Rodino in [12], is developed in the global setting of a product of compact Lie groups. The approach follows that introduced by M. Ruzhansky and V. Turunen [13] (see also V. Fischer [5]), in that it exploits the harmonic analysis of the groups involved.
We obtain new semiclassical estimates for pseudodifferential operators with low regular symbols. Such symbols appear naturally in a Cauchy Problem related to recent weak solutions to the unstable Muskat problem constructed via convex integration in [CCF16]. In particular, our new estimates reveal the tight relation between the speed of opening of the mixing zone and the regularity of the interphase.
We give a complete characterization of the continuity of pseudodifferential operators with symbols in modulation spaces $M^{p,q}$, acting on a given Lebesgue space $L^r$. Namely, we find the full range of triples $(p,q,r)$, for which such a boundedness occurs. More generally, we completely characterize the same problem for operators acting on Wiener amalgam space $W(L^r,L^s)$ and even on modulation spaces $M^{r,s}$. Finally the action of pseudodifferential operators with symbols in $W(\Fur L^1,L^\infty)$ is also investigated.
In this paper, we study the boundedness of pseudodifferential operators with symbols in the Hörmander class $S^0_{ρ,ρ}$ on $α$-modulation spaces $M_{p,q}^{s,α}$, and consider the relation between $α$ and $ρ$. In particular, we show that pseudodifferential operators with symbols in $S^0_{α,α}$ are bounded on all $α$-modulation spaces $M^{s,α}_{p,q}$, for arbitrary $s\in\mathbb{R}$ and for the whole range of exponents $0 < p,q \leq \infty$.
Motivated by the work of Colin de Verdière and Saint-Raymond on spectral theory 0th order pseudodifferential operators on tori we consider viscosity limits in which 0th order operators $ P $ are replaced by $ P + i νΔ$, $ ν> 0 $. By adapting the Helffer--Sjöstrand theory of scattering resonances we show that in a complex neighbourhood of the continuous spectrum eigenvalues of $ P + i νΔ$ have limits as viscosity $ ν$ goes to 0. In the simplified setting of tori this justifies claims made in the physics literature.
Classically, Gohberg-type Lemmas provide lower bounds for the distance of suitable pseudodifferential operators acting in a Hilbert space to the ideal of compact operators, in terms of "the behavior of the symbol at infinity". In this article the pseudodifferential operators are associated to a compact Abelian group $X$ and an important role is played by its Pontryagin dual $\widehat X$. Hörmander-type classes of symbols are not always available; they will be replaced by crossed product $C^*$-algebras involving a vanishing oscillation condition, which anyway is more general even in the particu…
We study the phenomena that arise when we combine the standard pseudodifferential operators with those operators that appear in the study of some sub-elliptic estimates, and on strongly pseudoconvex domains. The algebra of operators we introduce is geometrically invariant, and is adapted to a smooth distribution of tangent subspaces of constant rank. We isolate certain ideals in the algebra whose analysis is of particular interest.
We provide evidence for the conjecture that the Wodzicki-Chern classes vanish for all bundles with the group Z of invertible zeroth order pseudodifferential operators as structure group. In particular, we prove this vanishing if the structure group reduces to pseudodifferential operators with leading order symbol the identity. The leading order Chern classes are nonzero in general, and they detect elements of the de Rham cohomology groups of the classifying space BZ.
The purpose of this article is to establish upper and lower estimates for the integral kernel of the semigroup exp(-tP) associated to a classical, strongly elliptic pseudodifferential operator P of positive order on a closed manifold. The Poissonian bounds generalize those obtained for perturbations of fractional powers of the Laplacian. In the selfadjoint case, extensions to t in C_+ are studied. In particular, our results apply to the Dirichlet-to-Neumann semigroup.
