Band asymptotics in two dimensions

Guillemin, Victor

doi:10.1016/0001-8708(81)90042-6

Cited by 47 publications

(31 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The study of the spectral properties of the operator − 1 2 ∆ g + V in this geometric context is a problem which has a long history in microlocal analysis starting with the works of Duistermaat-Guillemin [14,23,24], Weinstein [51] and Colin de Verdière [9]. Many other important results on the fine structure of the spectrum of Zoll manifolds were obtained both in the microlocal framework [25,47,48,53,54], and in the semiclassical setting [8,28,26] -see also [30,31] in the nonselfadjoint setting.…”

Section: Fm Takes Part Into the Visiting Faculty Program Of Icmat Andmentioning

confidence: 99%

See 1 more Smart Citation

Concentration and Non-Concentration for the Schrödinger Evolution on Zoll Manifolds

Macià

Rivière

2015

Commun. Math. Phys.

View full text Add to dashboard Cite

Abstract. We study the long time dynamics of the Schrödinger equation on Zoll manifolds. We establish criteria under which the solutions of the Schrödinger equation can or cannot concentrate on a given closed geodesic. As an application, we derive some results on the set of semiclassical measures for eigenfunctions of Schrödinger operators: we prove that adding a potential to the Laplacian on the sphere results on the the existence of geodesics γ such that δ γ cannot be obtain as a semiclassical measure for some sequence of eigenfunctions. We also show that the same phenomenon occurs for the free Laplacian on certain Zoll surfaces.

show abstract

Section: Fm Takes Part Into the Visiting Faculty Program Of Icmat Andmentioning

confidence: 99%

“…Integrating again (25) with respect to t, letting → 0 + and using the composition formula for pseudodifferential operators gives that, for every τ ∈ R the following holds…”

Section: 2mentioning

confidence: 99%

Concentration and Non-Concentration for the Schrödinger Evolution on Zoll Manifolds

Macià

Rivière

2015

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…The following results have been obtained by Weinstein and Guillemin (see [29,15,17,16] and also [6,7]): Proofs can be found in [15] and [17]. In such a way, we get a quantum integrable system H 1 = h 2 ∆, H 2 .…”

Section: U (−T)v U(t) Dtmentioning

confidence: 69%

Singular Bohr–Sommerfeld rules for 2D integrable systems

Verdìère

Ngọc

2003

Annales Scientifiques de lʼÉcole Normale Supérieure

View full text Add to dashboard Cite

ABSTRACT. -This article gives Bohr-Sommerfeld rules for semiclassical completely integrable systems with two degrees of freedom with non-degenerate singularities (Morse-Bott singularities) under the assumption that the energy level of the first Hamiltonian is non-singular. The more singular case of focusfocus singularities was treated in previous works by San Vũ Ngo . c. The case of one degree of freedom has been studied by Colin de Verdière and Parisse.The results are applied to some famous examples: the geodesics of the ellipsoid, the 1 : 2-resonance, and Schrödinger operators on the sphere S 2 . A numerical test shows that the semiclassical Bohr-Sommerfeld rules match very accurately the "purely quantum" computations. Les résultats sont appliqués à quelques exemples célèbres : les géodésiques de l'ellipsoïde, la résonance 1 : 2 et les opérateurs de Schrödinger sur la sphère S 2 . Un test numérique montre que les règles de BohrSommerfeld semi-classiques sont en parfait accord avec les calculs "purement quantiques". 2003 Éditions scientifiques et médicales Elsevier SAS

show abstract

“…Therefore, see [11, proposition 1], [12], or Theorems 1.1, 1.2 in [13], there is a change of coordinates X, Y → X ′ , Y ′ which, if H ′ = (X ′2 +Y ′2 ) (2Ar) 2 and H = (X 2 +Y 2 ) (2Ar) 2 , keeps H equal to H ′ and makes ω proportional to dX ′ ∧ dY ′ :…”

Section: Normal Form Without Determination Of Actionangle Variablesmentioning

confidence: 99%

Rigid motions: Action-angles, relative cohomology and polynomials with roots on the unit circle

Françoise

Garrido

Gallavotti

2013

Journal of Mathematical Physics

View full text Add to dashboard Cite

Revisiting canonical integration of the classical solid near a uniform rotation, canonical action angle coordinates, hyperbolic and elliptic, are constructed in terms of various power series with coefficients which are polynomials in a variable r 2 depending on the inertia moments. Normal forms are derived via the analysis of a relative cohomology problem and shown to be obtainable without the use of ellitptic integrals (unlike the derivation of the action-angles). Results and conjectures also emerge about the properties of the above polynomials and the location of their roots. In particular a class of polynomials with all roots on the unit circle arises. OverviewIntegration of a rigid body motions is well known. In this paper we recall in Sec.2 the kinematic description of the coordinates in which the system is obviously integrable by reduction to quadratures (Deprit's coordinates, described in Eq.(2.5)). We try to treat simultaneously the motions near all proper rotations, at the expense of having to append labels a = ±1 to most quantities thus making the notation a little heavier. This is done to keep always clear that the stable and unstable motions are in some sense described by the same analytic functions.Our first aim is to obtain canonical coordinates p, q, in the vicinity of the proper rotations, such that the Hamiltonian, at fixed total angular momentum A, becomes a function of the product pq (near unstable proper rotations) or of p 2 + q 2 (near the stable ones). Although existence of such coordinates is well known and their determination is in principle controlled by appropriate (elliptic) integrals, [1, Sec.50], it remains, to our knowledge, only implicit in the literature, [2, Sec.69].In Sec.3 explicit expressions for the elliptic integrals are derived, Eq.(3.7), which give the motions in terms of variables p ′ , q ′ evolving exponentially in time on the hyperbolae p ′ q ′ = const or rotating uniformly on the circles p ′2 + q ′2 = const and "reducing" the problem to determine the nontrivial scaling function C (depending on p ′ q ′ or p ′2 + q ′2 ) which leads to the above mentioned canonical coordinates via p = p ′ C, q = q ′ C. In Appendix A given details on the computation of appropriate elliptic integrals: this should also clarify the advantages of the cohomological analysis of the following Sec.4 and 6.In Sec.4 the scaling function C is related to a Jacobian determinant between two differential forms, and it is computed: this is done, avoiding evaluation of other elliptic integrals, by computing the cohomology of the forms dB ∧ dβ (where B, β are two Deprit's coordinates, see Sec.2) and dp ′ ∧ dq ′ relative to the functions p ′ q ′ or p ′2 + q ′2 . The canonical coordinates are therefore completely determined, Eq.(4.10),(4.11) (together with Eq.(3.7)).The latter expressions are still quite implicit and in Sec.5 we bring them to a form suitable for the evaluation of the integrating coordinates via computable power series; and as an example a few terms of the scaling functions C and of t...

show abstract

Band asymptotics in two dimensions

Cited by 47 publications

References 9 publications

Concentration and Non-Concentration for the Schrödinger Evolution on Zoll Manifolds

Concentration and Non-Concentration for the Schrödinger Evolution on Zoll Manifolds

Singular Bohr–Sommerfeld rules for 2D integrable systems

Rigid motions: Action-angles, relative cohomology and polynomials with roots on the unit circle

Contact Info

Product

Resources

About