共查询到20条相似文献,搜索用时 15 毫秒
1.
John A. Toth 《Communications in Mathematical Physics》2009,288(1):379-401
We show that for a quantum completely integrable system in two dimensions, the L
2-normalized joint eigenfunctions of the commuting semiclassical pseudodifferential operators satisfy restriction bounds of
the form for generic curves γ on the surface. We also prove that the maximal restriction bounds of Burq-Gerard-Tzvetkov [BGT] are generically attained
for certain exceptional subsequences of eigenfunctions.
The author was supported by a William Dawson Fellowship and NSERC Grant OGP0170280. 相似文献
2.
Quantum systems whose classical counterpart have ergodic dynamics are quantum ergodic in the sense that almost all eigenstates are uniformly distributed in phase space. In contrast, when the classical dynamics is integrable, there is concentration of eigenfunctions on invariant structures in phase space. In this paper we study eigenfunction statistics for the Laplacian perturbed by a delta-potential (also known as a point scatterer) on a flat torus, a popular model used to study the transition between integrability and chaos in quantum mechanics. The eigenfunctions of this operator consist of eigenfunctions of the Laplacian which vanish at the scatterer, and new, or perturbed, eigenfunctions. We show that almost all of the perturbed eigenfunctions are uniformly distributed in configuration space. 相似文献
3.
Jiao-Kai Chen 《Few-Body Systems》2013,54(11):2081-2095
In this paper, we show that the simple Nyström method can yield very accurate eigenvalues and eigenfunctions not only for large principal quantum number but also for large angular momentum quantum number. We demonstrate that the furcation phenomenon emerging in the calculated eigenfunctions can be regarded as an indicator for the bad behavior of the integral equation and the unreliability of the obtained results. 相似文献
4.
We calculate statistical properties of the eigenfunctions of two quantum systems that exhibit intermediate spectral statistics: star graphs and Seba billiards. First, we show that these eigenfunctions are not quantum ergodic, and calculate the corresponding limit distribution. Second, we find that they can be strongly scarred, in the case of star graphs by short (unstable) periodic orbits and, in the case of Seba billiards, by certain families of orbits. We construct sequences of states which have such a limit. Our results are illustrated by numerical computations. 相似文献
5.
Biswas D 《Physical review letters》2004,93(20):204102
A comparison of classical and quantum evolution usually involves a quasiprobability distribution as a quantum analogue of the classical phase space distribution. In an alternate approach that we adopt here, the classical density is projected on to the configuration space. We show that for billiards, the eigenfunctions of the coarse-grained projected classical evolution operator are identical to a first approximation to the quantum Neumann eigenfunctions. However, even though there exists a correspondence between the respective eigenvalues, their time evolutions differ. This is demonstrated numerically for the stadium and lemon-shaped billiards. 相似文献
6.
Siegfried Böcherer Peter Sarnak Rainer Schulze-Pillot 《Communications in Mathematical Physics》2003,242(1-2):67-80
Motivated by problems of mathematical physics (quantum chaos) questions of equidistribution of eigenfunctions of the Laplace operator on a Riemannian manifold have been studied by several authors. We consider here, in analogy with arithmetic hyperbolic surfaces, orthonormal bases of eigenfunctions of the Laplace operator on the two dimensional unit sphere which are also eigenfunctions of an algebra of Hecke operators which act on these spherical harmonics. We formulate an analogue of the equidistribution of mass conjecture for these eigenfunctions as well as of the conjecture that their moments tend to moments of the Gaussian as the eigenvalue increases. For such orthonormal bases we show that these conjectures are related to the analytic properties of degree eight arithmetic L-functions associated to triples of eigenfunctions. Moreover we establish the conjecture for the third moments and give a conditional (on standard analytic conjectures about these arithmetic L-functions) proof of the equidistribution of mass conjecture.
Electronic Supplementary Material: Supplementary material is available in the online version of this article at http://dx.doi.org/10.1007/s00220-003-0922-5Part of this work was done at the Institute for Advanced Study, Princeton, NJ. 相似文献
7.
Daniel An 《Letters in Mathematical Physics》2009,87(3):209-223
The integral representation of the eigenfunctions of quantum periodic Toda chain constructed by Kharchev and Lebedev is revisited.
We prove that Pasquier and Gaudin’s solutions of the Baxter equation provides a complete set of eigenfunctions under this
integral representation. This will, in addition, show that the joint spectrum of commuting Hamiltonians of the quantum periodic
Toda chain is simple.
相似文献
8.
G. Berkolaiko J. P. Keating U. Smilansky 《Communications in Mathematical Physics》2007,273(1):137-159
We prove quantum ergodicity for a family of graphs that are obtained from ergodic one-dimensional maps of an interval using
a procedure introduced by Pakónski et al (J. Phys. A, 34, 9303-9317 (2001)). As observables we take the L
2 functions on the interval. The proof is based on the periodic orbit expansion of a majorant of the quantum variance. Specifically,
given a one-dimensional, Lebesgue-measure-preserving map of an interval, we consider an increasingly refined sequence of partitions
of the interval. To this sequence we associate a sequence of graphs, whose directed edges correspond to elements of the partitions
and on which the classical dynamics approximates the Perron-Frobenius operator corresponding to the map. We show that, except
possibly for subsequences of density 0, the eigenstates of the quantum graphs equidistribute in the limit of large graphs.
For a smaller class of observables we also show that the Egorov property, a correspondence between classical and quantum evolution
in the semiclassical limit, holds for the quantum graphs in question. 相似文献
9.
We present a Chaplygin gas Friedmann-Robertson-Walker quantum cosmological model in the presence of the cosmological constant.
We apply the Schutz’s variational formalism to recover the notion of time, and this gives rise to Wheeler-DeWitt equation
for the scale factor. We study the early and late time universes and show that the presence of the Chaplygin gas leads to
an effective positive cosmological constant for the late times. This suggests the possibility of changing the sign of the
effective cosmological constant during the transition from the early times to the late times. For the case of an effective
negative cosmological constant for both epoches, we solve the resulting Wheeler-DeWitt equation using the Spectral Method
and find the eigenvalues and eigenfunctions for positive, zero, and negative constant spatial curvatures. Then, we use the
eigenfunctions in order to construct wave packets for each case and obtain the time-dependent expectation value of the scale
factors, which are found to oscillate between finite maximum and minimum values. Since the expectation value of the scale
factors never tend to the singular point, we have an initial indication that this model may not have singularities at the
quantum level. 相似文献
10.
We investigate the spatial statistics of the energy eigenfunctions on large quantum graphs. It has previously been conjectured that these should be described by a Gaussian Random Wave Model, by analogy with quantum chaotic systems, for which such a model was proposed by Berry in 1977. The autocorrelation functions we calculate for an individual quantum graph exhibit a universal component, which completely determines a Gaussian Random Wave Model, and a system-dependent deviation. This deviation depends on the graph only through its underlying classical dynamics. Classical criteria for quantum universality to be met asymptotically in the large graph limit (i.e. for the non-universal deviation to vanish) are then extracted. We use an exact field theoretic expression in terms of a variant of a supersymmetric σ model. A saddle-point analysis of this expression leads to the estimates. In particular, intensity correlations are used to discuss the possible equidistribution of the energy eigenfunctions in the large graph limit. When equidistribution is asymptotically realized, our theory predicts a rate of convergence that is a significant refinement of previous estimates. The universal and system-dependent components of intensity correlation functions are recovered by means of an exact trace formula which we analyse in the diagonal approximation, drawing in this way a parallel between the field theory and semiclassics. Our results provide the first instance where an asymptotic Gaussian Random Wave Model has been established microscopically for eigenfunctions in a system with no disorder. 相似文献
11.
Debabrata Biswas 《Pramana》2005,64(4):563-575
The classical phase space density projected on to the configuration space offers a means of comparing classical and quantum
evolution. In this alternate approach that we adopt here, we show that for billiards, the eigenfunctions of the coarse-grained
projected classical evolution operator are identical to a first approximation to the quantum Neumann eigenfunctions. Moreover,
there exists a correspondence between the respective eigenvalues although their time evolutions differ. 相似文献
12.
O. Brodier T. Neicu A. Kudrolli 《The European Physical Journal B - Condensed Matter and Complex Systems》2001,23(3):365-372
We report a numerical study of the flexural modes of a plate using semi-classical analysis developed in the context of quantum
systems. We first introduce the Clover billiard as a paradigm for a system inside which rays exhibit stable and chaotic trajectories.
The resulting phase space explored by the ray trajectories is illustrated using the Poincare surface of section, and shows
that it has both integrable and chaotic regions. Examples of the stable and the unstable periodic orbits in the geometry are
presented. We numerically solve the biharmonic equation for the flexural vibrations of the Clover shaped plate with clamped
boundary conditions. The first few hundred eigenvalues and the eigenfunctions are obtained using a boundary elements method.
The Fourier transform of the eigenvalues show strong peaks which correspond to ray periodic orbits. However, the peaks corresponding
to the shortest stable periodic orbits are not stronger than the peaks associated with unstable periodic orbits. We also perform
statistics on the obtained eigenvalues and the eigenfunctions. The eigenvalue spacing distribution P(s) shows a strong peak and therefore deviates from both the Poisson and the Wigner distribution of random matrix theory at
small spacings because of the C4v symmetry of the Clover geometry. The density distribution of the eigenfunctions is observed to agree with the Porter-Thomas
distribution of random matrix theory.
Received 12 February 2001 and Received in final form 17 April 2001 相似文献
13.
P. Pedram M. Mirzaei S. Jalalzadeh S. S. Gousheh 《General Relativity and Gravitation》2008,40(8):1663-1681
We present perfect fluid Friedmann–Robertson–Walker quantum cosmological models in the presence of negative cosmological constant.
In this work the Schutz’s variational formalism is applied for radiation, dust, cosmic string, and domain wall dominated Universes
with positive, negative, and zero constant spatial curvature. In this approach the notion of time can be recovered. These
give rise to Wheeler–DeWitt equations for the scale factor. We find their eigenvalues and eigenfunctions by using Spectral
Method. After that, we use the eigenfunctions in order to construct wave packets for each case and evaluate the time-dependent
expectation value of the scale factors, which are found to oscillate between finite maximum and minimum values. Since the
expectation values of the scale factors never tends to the singular point, we have an initial indication that these models
may not have singularities at the quantum level. 相似文献
14.
We prove a strong version of quantum ergodicity for linear hyperbolic maps of the torus (“cat maps”). We show that there
is a density one sequence of integers so that as N tends to infinity along this sequence, all eigenfunctions of the quantum propagator at inverse Planck constant N are uniformly distributed.
A key step in the argument is to show that for a hyperbolic matrix in the modular group, there is a density one sequence of
integers N for which its order (or period) modulo N is somewhat larger than .
Received: 15 October 1999 / Accepted: 4 June 2001 相似文献
15.
Cheng-Hung Chang Tyll Krüger Roman Schubert Serge Troubetzkoy 《Communications in Mathematical Physics》2008,282(2):395-418
For general quantum systems the semiclassical behaviour of eigenfunctions in relation to the ergodic properties of the underlying
classical system is quite difficult to understand. The Wignerfunctions of eigenstates converge weakly to invariant measures
of the classical system, the so-called quantum limits, and one would like to understand which invariant measures can occur
that way, thereby classifying the semiclassical behaviour of eigenfunctions.
We introduce a class of maps on the torus for whose quantisations we can understand the set of quantum limits in great detail.
In particular we can construct examples of ergodic maps which have singular ergodic measures as quantum limits, and examples
of non-ergodic maps where arbitrary convex combinations of absolutely continuous ergodic measures can occur as quantum limits.
The maps we quantise are obtained by cutting and stacking. 相似文献
16.
We use the Whittaker vectors and the Drinfeld Casimir element to show that eigenfunctions of the difference Toda Hamiltonian can be expressed via fermionic formulas. Motivated by the combinatorics of the fermionic formulas we use the representation theory of the quantum groups to prove a number of identities for the coefficients of the eigenfunctions. 相似文献
17.
18.
19.
We discuss isospectral quantum graphs which are not
isometric. These graphs are the analogues of the isospectral
domains in R2 which were introduced recently in [1–5] all
based on Sunada's construction of isospectral domains
[6]. After discussing some of the properties of these
graphs, we present an example which support the conjecture that by
counting the nodal domains of the corresponding eigenfunctions one
can resolve the isospectral ambiguity. 相似文献
20.
S. Sree Ranjani 《Annals of Physics》2005,320(1):164-174
We analyze the Scarf potential, which exhibits both discrete energy bound states and energy bands, through the quantum Hamilton-Jacobi approach. The singularity structure and the boundary conditions in the above approach, naturally isolate the bound and periodic states, once the problem is mapped to the zero energy sector of another quasi-exactly solvable quantum problem. The energy eigenvalues are obtained without having to solve for the corresponding eigenfunctions explicitly. We also demonstrate how to find the eigenfunctions through this method. 相似文献