Eigenfunctions Concentrated Near a Closed Geodesic

Babich, V. M.; Lazutkin, V. F.

doi:10.1007/978-1-4684-7592-0_2

Cited by 31 publications

(27 citation statements)

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“…If this phenomenon does not take place, i.e., in other words, if the first variation equation of the Hamilton field is stable on 4, then local WKB-asymptotics can be constructed by means of Gaussian packets or excited oscillator modes decreasing along the directions normal to 4. This ansatz was proposed by V. M. Babich with collaborators [6,7] for the case of closed geodesics when dim 4=1 (see also discussions in [32,85]). In the context of general symplectic geometry and Gelfand Lidsky index, the case dim 4=1 was investigated very carefully [16,31,75].…”

Section: =[Q=q(:) P= P(:)]/[h(q P)=const]mentioning

confidence: 93%

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Integral Representations over Isotropic Submanifolds and Equations of Zero Curvature

Karasev

Vorobjev

1998

Advances in Mathematics

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In the phase space over a Riemann manifold we consider a submanifold 4 invariant with respect to a Hamilton flow, isotropic (i.e., the form pdq| 4 is closed), and stable with respect to the first variation equation. For semiclassical wavefunctions of the quantum Hamiltonian we propose a very simple global ansatz with an oscillation front on 4. This ansatz has a form of an integral along 4 from 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 over 4. 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 Poincare Cartan invariant in quantization rule on 4 and calculate spectral series for the quantum Hamiltonian.1998 Academic Press

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Section: =[Q=q(:) P= P(:)]/[h(q P)=const]mentioning

confidence: 93%

“…Such a field will be nonresonant if its frequencies are arithmetically independent in the sense of KAM-theory [6,59].…”

Section: Ii) Resolution Of Quantization Rulementioning

confidence: 99%

Integral Representations over Isotropic Submanifolds and Equations of Zero Curvature

Karasev

Vorobjev

1998

Advances in Mathematics

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“…Theorem 1.2 will be obtained from Theorem 1.3 by a rather direct construction of Gaussian beam quasimodes that concentrate on a given nontangential geodesic. This construction goes back at least to [1], [2], [3], [11], [20] and has been developed further by many authors (often for hyperbolic equations), see for instance [24], [42]. In our case, we need the next result which follows by adapting the methods in the literature in a suitable way.…”

Section: Introductionmentioning

confidence: 97%

The Calderón problem in transversally anisotropic geometries

Ferreira¹,

Kurylev²,

Lassas³

et al. 2016

J. Eur. Math. Soc.

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Abstract. We consider the anisotropic Calderón problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work [13], it was shown that a metric in a fixed conformal class is uniquely determined by boundary measurements under two conditions: (1) the metric is conformally transversally anisotropic (CTA), and (2) the transversal manifold is simple. In this paper we will consider geometries satisfying (1) but not (2). The first main result states that the boundary measurements uniquely determine a mixed Fourier transform / attenuated geodesic ray transform (or integral against a more general semiclassical limit measure) of an unknown coefficient. In particular, one obtains uniqueness results whenever the geodesic ray transform on the transversal manifold is injective. The second result shows that the boundary measurements in an infinite cylinder uniquely determine the transversal metric. The first result is proved by using complex geometrical optics solutions involving Gaussian beam quasimodes, and the second result follows from a connection between the Calderón problem and Gel'fand's inverse problem for the wave equation and the boundary control method.

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“…In the simplest case k = 0 (for stationary points of the Humiltonian system), the semiclassical approximation coincides with the oscillator approximation, which is well known in physics. In the most studied case k = 1 (see [7,[10][11][12][13][14]), the orbital stability of dosed phase trajectories A 1 is a sttfllcient condition for quasimodes to exist and to be constructed (rood O(/ia/2)); a quantization condition of Bohr-Sommerfeld type must also be satisfied (see [8,9]). …”

mentioning

confidence: 99%

“…We shall follow the general scheme of semlclassical quantization for closed trajectories with focal points (in the absence of focal points, formulas for spectral series were obtained in [10,15] for the Beltrami-Laplace operator and in [14] for the SchrSdinger operator). The global formulas for the asy~aptotic eigenfunctions (that also hold in a neighborhood of the turning points) are given by Maslov's canonical operator on the family A2(E, ~) with complex germ r2(A2) (see [8,9]).…”

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confidence: 99%

Quasimodes of the two-dimensional quartic oscillator

Belov

Maximov

1998

Math Notes

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K~y WORDS: quartic potential oscillator, Schr~dinger operator, semiclassical asymptotics of eigenvalues, quasimodes, Gelfand-Lidskii index.For the two parameter Schrfdinger operator with parameters (e, h) and with the family of potentials V'(z)=z~x~2 q-~(zle 4q_m2) ,, m=(mI,x2) ER 2, e_>0,we consider the problem of constructing the semiclassical (/i ~ 0) asymptotics of eigenvalues and eigenhmctions,uniformly with respect to the regular parameter e in a domain ft C R~ + . The quartic potential model (1) is presently one of the most popular models in the problem of semiclassical quantization of nonintegrable Hamiltonian systems with chaotic behavior (e.g., see [1,2]). The nonintegrability of the classical system (corresponding to (1))with the IIamiltonian H~(x, p) --pa/2 + Ve(z) is confirmed by numerical experiments [3][4][5]. In [6] it was proved that there does not exist an additional analytic integral of motion for ~ ~ 1, 1/3. It is well known that the semiclassical approximation to multidimensional spectral problems of quantum mechanics allows one to assign subsequences of asymptotic eigenvalues and asymptotic eigenfunctions (quasimodes of the corresponding Humiltonian) to some invariant objects of the classical system [1, 2]. For nonintegrable systems that possess a family of invariant k-dimensional isotropic tori A t (0 _< k < n, where n is the dimension of the problem), formulas for quasimodes are obtained by the complex WKB method (Maslov's complex germ theory) [7][8][9] under additional stability type conditions (in the linear approximation) for these objects. In the simplest case k = 0 (for stationary points of the Humiltonian system), the semiclassical approximation coincides with the oscillator approximation, which is well known in physics. In the most studied case k = 1 (see [7,[10][11][12][13][14]), the orbital stability of dosed phase trajectories A 1 is a sttfllcient condition for quasimodes to exist and to be constructed (rood O(/ia/2)); a quantization condition of Bohr-Sommerfeld type must also be satisfied (see [8,9]).The nonintegrable system (3) has families of closed trajectories related in a natural way to the group C4~ of discrete symmetries of the potential V~(z). Let us consider the following improper rotations of the plane: reflections Oi: xi ~ -zi, i = 1,2, with respect to the coordinate axes and 04-: xl ~-~ :kzu with respect to the diagonals. Let us also consider the corresponding linear canonical transformations of the phase space, which commute with the Hamiltonian H~(x, p): Gi : xl ~ -xi, Pi ~-* -pi, i = 1,2, and G• ~-* +x2,pl ~ +P2. The fixed points II~ = (xi =pl = 0) and II4-= (xl = +z2, pl = q'p2) of these mappings form two-dimensional symplectic planes. Obviously, these points are invaxiant with

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Eigenfunctions Concentrated Near a Closed Geodesic

Cited by 31 publications

References 2 publications

Integral Representations over Isotropic Submanifolds and Equations of Zero Curvature

Integral Representations over Isotropic Submanifolds and Equations of Zero Curvature

The Calderón problem in transversally anisotropic geometries

Quasimodes of the two-dimensional quartic oscillator

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