Hamilton Flow Research Articles

Propagation of anisotropic Gabor singularities for Schrödinger type equations

We show results on propagation of anisotropic Gabor wave front sets for solutions to a class of evolution equations of Schrödinger type. The Hamiltonian is assumed to have a real-valued principal symbol with the anisotropic homogeneity a(λx,λσξ)=λ1+σa(x,ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a(\\lambda x, \\lambda ^\\sigma \\xi ) = \\lambda ^{1+\\sigma } a(x,\\xi )$$\\end{document} for λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda > 0$$\\end{document} where σ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma > 0$$\\end{document} is a rational anisotropy parameter. We prove that the propagator is continuous on anisotropic Shubin–Sobolev spaces. The main result says that the propagation of the anisotropic Gabor wave front set follows the Hamilton flow of the principal symbol.

Quantitative version of Weyl’s law

We prove a general estimate for the Weyl remainder of an elliptic, semiclassical pseudodifferential operator in terms of volumes of recurrence sets for the Hamilton flow of its principal symbol. This quantifies earlier results of Volovoy (Comm Partial Differential Equations 15:1509–1563, 1990; Ann Global Anal Geom 8:127–136, 1990). Our result particularly improves Weyl remainder exponents for compact Lie groups and surfaces of revolution. And gives a quantitative estimate for Bérard’s Weyl remainder in terms of the maximal expansion rate and topological entropy of the geodesic flow.

Propagation of global analytic singularities for Schrödinger equations with quadratic Hamiltonians

We study the propagation in time of 1/2-Gelfand-Shilov singularities, i.e. global analytic singularities, of tempered distributional solutions of the initial value problem{ut+qw(x,D)u=0u|t=0=u0, on Rn, where u0 is a tempered distribution on Rn, q=q(x,ξ) is a complex-valued quadratic form on R2n=Rxn×Rξn with nonnegative real part Req≥0, and qw(x,D) is the Weyl quantization of q. We prove that the 1/2-Gelfand-Shilov singularities of the initial data that are contained within a distinguished linear subspace of the phase space R2n, called the singular space of q, are transported by the Hamilton flow of Imq, while all other 1/2-Gelfand-Shilov singularities are instantaneously regularized. Our result extends the observation of Hitrik, Pravda-Starov, and Viola '18 that this evolution is instantaneously globally analytically regularizing when the singular space of q is trivial.

Schrödinger trace invariants for homogeneous perturbations of the harmonic oscillator

Let H = H_0 + P denote the harmonic oscillator on \mathbb{R}^d perturbed by an isotropic pseudodifferential operator P of order 1 and let U(t) = \operatorname{exp}(- it H) . We prove a Gutzwiller–Duistermaat–Guillemin type trace formula for \operatorname{Tr} U(t). The singularities occur at times t \in 2 \pi \mathbb{Z} and the coefficients involve the dynamics of the Hamilton flow of the symbol \sigma(P) on the space \mathbb{CP}^{d-1} of harmonic oscillator orbits of energy 1 . This is a novel kind of sub-principal symbol effect on the trace. We generalize the averaging technique of Weinstein and Guillemin to this order of perturbation, and then present two completely different calculations of \operatorname{Tr} U(t) . The first proof directly constructs a parametrix of U(t) in the isotropic calculus, following earlier work of Doll–Gannot–Wunsch. The second proof conjugates the trace to the Bargmann–Fock setting, the order 1 of the perturbation coincides with the 'central limit scaling' studied by Zelditch–Zhou for Toeplitz operators.

Long-Range Scattering for Discrete Schrödinger Operators

In this paper, we define time-independent modifiers to construct a long-range scattering theory for a class of difference operators on $$\mathbb {Z}^d$$ , including the discrete Schrodinger operators on the square lattice. The modifiers are constructed by observing the corresponding Hamilton flow on $$T^*\mathbb {T}^d$$ . We prove the existence and completeness of modified wave operators in terms of the above-mentioned time-independent modifiers.

On semiclassical Fourier integral operators, Schrödinger propagators and coherent states

In this paper, we introduce a class of semiclassical Fourier integral operators using the Fourier-Bargmann transform. We will show that the propagator defined by the solution of the time dependent Schrödinger equation with subquadratic Hamiltonian H(t) is a semiclassical Fourier integral operator of order 0 associated to the Hamilton flow generated by the classical Hamiltonian H(t).

The Cauchy problem for differential operators with double characteristics

In this monograph we provide a general picture of the Cauchy problem for differential operators with double characteristics from the viewpoint that the Hamilton map and the geometry of orbits of the Hamilton flow completely characterizes the well/ill-posedness of the Cauchy problem.

Solvability and limit bicharacteristics

We shall study the solvability of pseudodifferential operators which are not of principal type. The operator will have real principal symbol and we shall consider the limits of bicharacteristics at the set where the principal symbol vanishes of at least second order. The convergence shall be as smooth curves, then the limit bicharacteristic is a smooth curve. We shall also need uniform bounds on the curvature of the characteristics at the bicharacteristics, but only along the tangents of a given Lagrangean manifold. This gives uniform bounds on the linearization of the normalized Hamilton flow on the tangent space of this manifold at the bicharacteristics. If the quotient of the imaginary part of the subprincipal symbol with the norm of the Hamilton vector field switches sign from $-$ to $+$ on the bicharacteristics and becomes unbounded as they converge to the limit, then the operator is not solvable at the limit bicharacteristic.

Equidistribution of phase shifts in semiclassical potential scattering

Consider a semiclassical Hamiltonian $H := h^{2} \Delta + V - E$ where $\Delta$ is the positive Laplacian on $\mathbb{R}^{d}$, $V \in C^{\infty}_{0}(\mathbb{R}^{d})$ and $E > 0$ is an energy level. We prove that under an appropriate dynamical hypothesis on the Hamilton flow corresponding to $H$, the eigenvalues of the scattering matrix $S_{h}(V)$ define a measure on $\mathbb{S}^{1}$ that converges to Lebesgue measure away from $1 \in \mathbb{S}^{1}$ as $h \to 0$.

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.

Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials

The present paper is concerned with Schr\"odinger equations with variable coefficients and unbounded electromagnetic potentials, where the kinetic energy part is a long-range perturbation of the flat Laplacian and the electric (resp. magnetic) potential can grow subquadratically (resp. sublinearly) at spatial infinity. We prove sharp (local-in-time) Strichartz estimates, outside a large compact ball centered at origin, for any admissible pair including the endpoint. Under the nontrapping condition on the Hamilton flow generated by the kinetic energy, global-in-space estimates are also studied. Finally, under the nontrapping condition, we prove Strichartz estimates with an arbitrarily small derivative loss without asymptotic flatness on the coefficients.

Hamilton flow generated by field lines near a toroidal magnetic surface

A method is described for obtaining the Hamiltonian of a vacuum magnetic field in a given 3D toroidal magnetic surface (superconducting shell). This method is used to derive the expression for the integrable surface Hamiltonian in the form of the expansion of a rotational transform of field lines on embedded near-boundary magnetic surfaces into a Taylor series in the distance from the boundary. This expansion contains the value of the rotational transform and its shear at the boundary surface. It is shown that these quantities are related to the components of the first and second quadratic forms of the boundary surface.

Strichartz estimates for Schrödinger equations with variable coefficients and potentials at most linear at spatial infinity

In the present paper we consider Schrodinger equations with variable coefficients and potentials, where the principal part is a long-range perturbation of the flat Laplacian and potentials have at most linear growth at spatial infinity. We then prove local-in-time Strichartz estimates, outside a large compact set centered at origin, without loss of derivatives. Moreover we also prove global-in-space Strichartz estimates under the non-trapping condition on the Hamilton flow generated by the kinetic energy.

Non-Concentration of Quasimodes for Integrable Systems

We consider the possible concentration in phase space of a sequence of eigenfunctions (or, more generally, a quasimode) of an operator whose principal symbol has completely integrable Hamilton flow. The semiclassical wavefront set WF h of such a sequence is invariant under the Hamilton flow. In principle this may allow concentration of WF h along a single closed orbit if all frequencies of the flow are rationally related. We show that, subject to non-degeneracy hypotheses, this concentration may not in fact occur. Indeed, in the two-dimensional case, we show that WF h must fill out an entire Lagrangian torus. The main tools are the spreading of Lagrangian regularity previously shown by the author, and an analysis of higher order transport equations satisfied by the principal symbol of a Lagrangian quasimode. These yield a unique continuation theorem for the principal symbol of Lagrangian quasimode, which is the principal new result of the paper.

ON THE CAUCHY PROBLEM FOR NON-EFFECTIVELY HYPERBOLIC OPERATORS: THE GEVREY 3 WELL-POSEDNESS

For hyperbolic differential operators P with double characteristics we study the relations between the maximal Gevrey index for the strong Gevrey well-posedness and the Hamilton map and flow of the associated principal symbol p. If the Hamilton map admits a Jordan block of size 4 on the double characteristic manifold denoted by Σ and by assuming that the Hamilton flow does not approach Σ tangentially, we proved earlier that the Cauchy problem for P is well-posed in the Gevrey class 1 ≤ s < 4 for any lower order term. In the present paper, we remove this restriction on the Hamilton flow and establish that the Cauchy problem for P is well-posed in the Gevrey class 1 ≤ s < 3 for any lower order term and we check that the Gevrey index 3 is optimal. Combining this with results already proved for the other cases, we conclude that the Hamilton map and flow completely characterizes the threshold for the strong Gevrey well-posedness and vice versa.

Manipulation of semiclassical photon states

Calvo and Picón defined a class of operators, for use in quantum communication, that allows arbitrary manipulations of the three lowest two-dimensional Hermite–Gaussian modes HT={|0,0⟩,|1,0⟩,|0,1⟩}. Our paper continues the study of those operators, and our results fall into two categories. For one, we show that the generators of the operators have infinite deficiency indices, and we explicitly describe all self-adjoint realizations. And second, we investigate semiclassical approximations of the propagators. The basic method is to start from a semiclassical Fourier integral operator ansatz and then construct approximate solutions of the corresponding evolution equations. In doing so, we give a complete description of the Hamilton flow, which in most cases is given by elliptic functions. We find that the semiclassical approximation behaves well when acting on sufficiently localized initial conditions, for example, finite sums of semiclassical Hermite–Gaussian modes, since near the origin the Hamilton trajectories trace out the bounded components of elliptic curves.

Semiclassical Resolvent Estimates for Schrödinger Operators with Coulomb Singularities

Consider the Schrödinger operator with semiclassical parameter h, in the limit where h goes to zero. When the involved long-range potential is smooth, it is well known that the boundary values of the operator’s resolvent at a positive energy λ are bounded by O(h −1) if and only if the associated Hamilton flow is non-trapping at energy λ. In the present paper, we extend this result to the case where the potential may possess Coulomb singularities. Since the Hamilton flow then is not complete in general, our analysis requires the use of an appropriate regularization.

Wave packet parametrices for evolutions governed by pdo's with rough symbols

Author(s): Marzuola, Jeremy; Metcalfe, Jason; Tataru, Daniel | Abstract: In this note, we consider wave packet parametrices for Schrodinger-like evolution equations. Under an integrability condition along the flow, we prove that the flow is then globally well-defined and bilipschitz. Under an additional smallness assumption, we prove that the kernel of the phase space operator decays rapidly away from the graph of the Hamilton flow. This can be used to prove that smoothness is preserved for finite times by the flow of the equation.

Generalized Hamilton flow and Poisson relation for the scattering kernel

The generalized Hamilton flow determined by a Hamilton function on a symplectic manifold with boundary is considered. A regularity property of this flow is proved, which for “Sard's theorem type applications” is as good as smoothness. It implies in particular that the generalized Hamilton flow preserves the Hausdorff dimension of Borel subsets of its phase space. As an application, it is shown that in scattering by obstacles, the so-called Poisson relation for the scattering kernel s( t, θ, ω) becomes an equality for almost all pairs of unit vectors ( θ, ω) .

Integral Representations over Isotropic Submanifolds and Equations of Zero Curvature

In the phase space over a Riemann manifold we consider a submanifold 4 invariant with respect to a Hamilton flow, isotropic (i.e., the formpdq|Λis closed), and stable with respect to the first variation equation. For semiclassical wave- functions of the quantum Hamiltonian we propose a very simple global ansatz with an oscillation front onΛ. This ansatz has a form of an integral alongΛfrom Gaussian packets framed by an “amplitude.” The amplitude is a parallel section of a bundle of polynomials. The connection on this bundle is generated by a symplectic connection with zero curvature on the normal symplectic bundle overΛ. The coefficients of such a connection are calculated explicitly in terms of infinitesimal symmetries, and also in adiabatic approximation. We investigate topological and geometrical objects arising as corrections to the Poincaré–Cartan invariant in quantization rule onΛand calculate spectral series for the quantum Hamiltonian.