Results 1  10
of
78
The Riemann Zeros and Eigenvalue Asymptotics
 SIAM Rev
, 1999
"... Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many feat ..."
Abstract

Cited by 61 (10 self)
 Add to MetaCart
(Show Context)
Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t n have a similar structure to those of the semiclassical En ; in particular, they display randommatrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n can be computed accurately from formulae with quantum analogues. The RiemannSiegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.
A Proof of the Gutzwiller Semiclassical Trace Formula using Coherent Sates Decomposition
 Commun. in Math. Phys
, 1999
"... The Gutzwiller trace formula links the eigenvalues of the Schrödinger operator H as Planck’s constant goes to zero (the semiclassical régime) with the closed orbits of the corresponding classical mechanical system. Gutzwiller gave a heuristic proof of this trace formula, using the Feynman integral ..."
Abstract

Cited by 48 (7 self)
 Add to MetaCart
(Show Context)
The Gutzwiller trace formula links the eigenvalues of the Schrödinger operator H as Planck’s constant goes to zero (the semiclassical régime) with the closed orbits of the corresponding classical mechanical system. Gutzwiller gave a heuristic proof of this trace formula, using the Feynman integral representation for the propagator of H. Later, using the theory of Fourier integral operators, mathematicians gave rigorous proofs of the formula in various settings. Here we show how the use of coherent states allows us to give a simple and direct proof. 1
Between classical and quantum
, 2008
"... The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, inclu ..."
Abstract

Cited by 37 (5 self)
 Add to MetaCart
The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, including over 500 references. For example, we sketch how certain intuitive ideas of the founders of quantum theory have fared in the light of current mathematical knowledge. One such idea that has certainly stood the test of time is Heisenberg’s ‘quantumtheoretical Umdeutung (reinterpretation) of classical observables’, which lies at the basis of quantization theory. Similarly, Bohr’s correspondence principle (in somewhat revised form) and Schrödinger’s wave packets (or coherent states) continue to be of great importance in understanding classical behaviour from quantum mechanics. On the other hand, no consensus has been reached on the Copenhagen Interpretation, but in view of the parodies of it one typically finds in the literature we describe it in detail. On the assumption that quantum mechanics is universal and complete, we discuss three ways in which classical physics has so far been believed to emerge from quantum physics, namely
The Semiclassical Trace Formula and Propagation of Wave Packets
 J. Funct. Anal
, 1994
"... We study spectral and propagation properties of operators of the form S ¯ h = P N j=0 ¯h j P j where 8j P j is a differential operator of order j on a manifold M , asymptotically as ¯h ! 0. The estimates are in terms of the flow fOE t g of the classical Hamiltonian H(x; p) = P N j=0 oe P j (x; ..."
Abstract

Cited by 35 (9 self)
 Add to MetaCart
(Show Context)
We study spectral and propagation properties of operators of the form S ¯ h = P N j=0 ¯h j P j where 8j P j is a differential operator of order j on a manifold M , asymptotically as ¯h ! 0. The estimates are in terms of the flow fOE t g of the classical Hamiltonian H(x; p) = P N j=0 oe P j (x; p) on T M , where oe P j is the principal symbol of P j . We present two sets of results. (I) The "semiclassical trace formula", on the asymptotic behavior of eigenvalues and eigenfunctions of S ¯ h in terms of periodic trajectories of H . (II) Associated to certain isotropic submanifolds ae T M we define families of functions f/ ¯ h g and prove that 8t fexp(\Gammait¯hS h )(/ ¯ h )g is a family of the same kind associated to OE t (). Introduction and description of results. In this paper we present some results concerning spectral and propagation properties of a class of differential operators "with small parameter", ¯h (Planck's constant). We have in mind operators of the form a(x...
Semiclassical Form Factor for Chaotic Systems With Spin 1/2
, 1999
"... . We study the properties of the twopoint spectral form factor for classically chaotic systems with spin 1/2 in the semiclassical limit, with a suitable semiclassical trace formula as our principal tool. To this end we introduce a regularized form factor and discuss the limit in which the socalled ..."
Abstract

Cited by 22 (17 self)
 Add to MetaCart
. We study the properties of the twopoint spectral form factor for classically chaotic systems with spin 1/2 in the semiclassical limit, with a suitable semiclassical trace formula as our principal tool. To this end we introduce a regularized form factor and discuss the limit in which the socalled diagonal approximation can be recovered. The incorporation of the spin contribution to the trace formula requires an appropriate variant of the equidistribution principle of long periodic orbits as well as the notion of a skew product of the classical translational and spin dynamics. Provided this skew product is mixing, we show that generically the diagonal approximation of the form factor coincides with the respective predictions from random matrix theory. PACS numbers: 03.65.Sq, 05.45.Mt k Email address: bol@physik.uniulm.de  Email address: kep@physik.uniulm.de + Address after 1 October 1999: Abteilung Theoretische Physik, Universitat Ulm, AlbertEinsteinAllee 11, D89069 Ulm, G...
A Scaling Theory of Bifurcations in the Symmetric WeakNoise Escape Problem
, 1995
"... We consider the overdamped limit of twodimensional double well systems perturbed by weak noise. In the weak noise limit the most probable fluctuational path leading from either point attractor to the separatrix (the most probable escape path, or MPEP) must terminate on the saddle between the two we ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
(Show Context)
We consider the overdamped limit of twodimensional double well systems perturbed by weak noise. In the weak noise limit the most probable fluctuational path leading from either point attractor to the separatrix (the most probable escape path, or MPEP) must terminate on the saddle between the two wells. However, as the parameters of a symmetric double well system are varied, a unique MPEP may bifurcate into two equally likely MPEP’s. At the bifurcation point in parameter space, the activation kinetics of the system become nonArrhenius. We quantify the nonArrhenius behavior of a system at the bifurcation point, by using the MaslovWKB method to construct an approximation to the quasistationary probability distribution of the system that is valid in a boundary layer near the separatrix. The approximation is a formal asymptotic solution of the Smoluchowski equation. Our analysis relies on the construction of a new scaling theory, which yields ‘critical exponents’ describing weaknoise behavior at the bifurcation point, near the saddle.
Spectral Statistics in the Quantized Cardioid Billiard
, 1994
"... : The spectral statistics in the strongly chaotic cardioid billiard are studied. The analysis is based on the first 11000 quantal energy levels for odd and even symmetry respectively. It is found that the levelspacing distribution is in good agreement with the GOE distribution of randommatrix the ..."
Abstract

Cited by 12 (8 self)
 Add to MetaCart
: The spectral statistics in the strongly chaotic cardioid billiard are studied. The analysis is based on the first 11000 quantal energy levels for odd and even symmetry respectively. It is found that the levelspacing distribution is in good agreement with the GOE distribution of randommatrix theory. In case of the number variance and rigidity we observe agreement with the randommatrix model for shortrange correlations only, whereas for longrange correlations both statistics saturate in agreement with semiclassical expectations. Furthermore the conjecture that for classically chaotic systems the normalized mode fluctuations have a universal Gaussian distribution with unit variance is tested and found to be in very good agreement for both symmetry classes. By means of the Gutzwiller trace formula the trace of the cosinemodulated heat kernel is studied. Since the billiard boundary is focusing there are conjugate points giving rise to zeros at the locations of the periodic orbits in...
F.: Pseudosymmetries of Anosov maps and Spectral statistics. Nonlinearity 13
, 2000
"... The statistics of the quantum eigenvalues of certain families of nonlinear maps on the twotorus are found not to belong to the universality classes one would expect from the symmetries of the (classical) dynamics the maps generate. These anomalies are shown to be caused by arithmetical quantum sym ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
(Show Context)
The statistics of the quantum eigenvalues of certain families of nonlinear maps on the twotorus are found not to belong to the universality classes one would expect from the symmetries of the (classical) dynamics the maps generate. These anomalies are shown to be caused by arithmetical quantum symmetries which do not have a classical limit. They are related to the dynamics generated by associated linear torus maps on particular rational lattices that form the support of the quantum Wigner functions.
Magnetic Edge States
, 2002
"... Magnetic edge states are responsible for various phenomena of magnetotransport. Their importance is due to the fact that, unlike the bulk of the eigenstates in a magnetic system, they carry electric current along the boundary of a confined domain. Edge states can exist both as interior (quantum dot ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
Magnetic edge states are responsible for various phenomena of magnetotransport. Their importance is due to the fact that, unlike the bulk of the eigenstates in a magnetic system, they carry electric current along the boundary of a confined domain. Edge states can exist both as interior (quantum dot) and exterior (antidot) states. In the present report we develop a consistent and practical spectral theory for the edge states encountered in magnetic billiards. It provides an objective definition for the notion of edge states, is applicable for interior and exterior problems, facilitates efficient quantization schemes, and forms a convenient starting point for both the semiclassical description and the statistical analysis. After elaborating these topics we use the semiclassical spectral theory to uncover nontrivial spectral correlations between the interior and the exterior edge states. We show that they are the quantum manifestation of a classical duality between the trajectories in an interior and
Bifurcations of periodic orbits and uniform approximations
 J. Phys. AMath. Gen
, 1997
"... We derive uniform approximations for contributions to Gutzwiller’s periodicorbit sum for the spectral density which are valid close to bifurcations of periodic orbits in systems with mixed phase space. There, orbits lie close together and give collective contributions, while the individual contribu ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
(Show Context)
We derive uniform approximations for contributions to Gutzwiller’s periodicorbit sum for the spectral density which are valid close to bifurcations of periodic orbits in systems with mixed phase space. There, orbits lie close together and give collective contributions, while the individual contributions of Gutzwiller’s type would diverge at the bifurcation. New results for the tangent, the period doubling and the period tripling bifurcation are given. They are obtained by going beyond the local approximation and including higher order terms in the normal form of the action. The uniform approximations obtained are tested on the kicked top and are found to be in excellent agreement with exact quantum results. PACS numbers: 03.20.+i Classical mechanics of discrete systems: general mathematical aspects. 03.65.Sq Semiclassical theories and applications. 05.45.+b Theory and models of chaotic systems. Submitted to Journal of Physics A 1 1