357 research outputs found
Exercises in exact quantization
The formalism of exact 1D quantization is reviewed in detail and applied to
the spectral study of three concrete Schr\"odinger Hamiltonians [-\d^2/\d q^2
+ V(q)]^\pm on the half-line , with a Dirichlet (-) or Neumann (+)
condition at q=0. Emphasis is put on the analytical investigation of the
spectral determinants and spectral zeta functions with respect to singular
perturbation parameters. We first discuss the homogeneous potential
as vs its (solvable) limit (an infinite square well):
useful distinctions are established between regular and singular behaviours of
spectral quantities; various identities among the square-well spectral
functions are unraveled as limits of finite-N properties. The second model is
the quartic anharmonic oscillator: its zero-energy spectral determinants
\det(-\d^2/\d q^2 + q^4 + v q^2)^\pm are explicitly analyzed in detail,
revealing many special values, algebraic identities between Taylor
coefficients, and functional equations of a quartic type coupled to asymptotic
properties of Airy type. The third study addresses the
potentials of even degree: their zero-energy spectral
determinants prove computable in closed form, and the generalized eigenvalue
problems with v as spectral variable admit exact quantization formulae which
are perfect extensions of the harmonic oscillator case (corresponding to N=2);
these results probably reflect the presence of supersymmetric potentials in the
family above.Comment: latex txt.tex, 2 files, 34 pages [SPhT-T00/078]; v2: corrections and
updates as indicated by footnote
Artificial trapping of a stable high-density dipolar exciton fluid
We present compelling experimental evidence for a successful electrostatic
trapping of two-dimensional dipolar excitons that results in stable formation
of a well confined, high-density and spatially uniform dipolar exciton fluid.
We show that, for at least half a microsecond, the exciton fluid sustains a
density higher than the critical density for degeneracy if the exciton fluid
temperature reaches the lattice temperature within that time. This method
should allow for the study of strongly interacting bosons in two dimensions at
low temperatures, and possibly lead towards the observation of quantum phase
transitions of 2D interacting excitons, such as superfluidity and
crystallization.Comment: 11 pages 4 figure
The WKB Approximation without Divergences
In this paper, the WKB approximation to the scattering problem is developed
without the divergences which usually appear at the classical turning points. A
detailed procedure of complexification is shown to generate results identical
to the usual WKB prescription but without the cumbersome connection formulas.Comment: 13 pages, TeX file, to appear in Int. J. Theor. Phy
Trace formula for noise corrections to trace formulas
We consider an evolution operator for a discrete Langevin equation with a
strongly hyperbolic classical dynamics and Gaussian noise. Using an integral
representation of the evolution operator we investigate the high order
corrections to the trace of arbitary power of the operator.
The asymptotic behaviour is found to be controlled by sub-dominant saddle
points previously neglected in the perturbative expansion. We show that a trace
formula can be derived to describe the high order noise corrections.Comment: 4 pages, 2 figure
The Local Time Distribution of a Particle Diffusing on a Graph
We study the local time distribution of a Brownian particle diffusing along
the links on a graph. In particular, we derive an analytic expression of its
Laplace transform in terms of the Green's function on the graph. We show that
the asymptotic behavior of this distribution has non-Gaussian tails
characterized by a nontrivial large deviation function.Comment: 8 pages, two figures (included
Semiclassical short strings in AdS_5 x S^5
We present results for the one-loop correction to the energy of a class of
string solutions in AdS_5 x S^5 in the short string limit. The computation is
based on the observation that, as for rigid spinning string elliptic solutions,
the fluctuation operators can be put into the single-gap Lame' form. Our
computation reveals a remarkable universality of the form of the energy of
short semiclassical strings. This may help to understand better the structure
of the strong coupling expansion of the anomalous dimensions of dual gauge
theory operators.Comment: 12 pages, one pdf figure. Invited Talk at 'Nonlinear Physics. Theory
and Experiment VI', Gallipoli (Italy) - June 23 - July 3, 201
ABJM theory as a Fermi gas
The partition function on the three-sphere of many supersymmetric
Chern-Simons-matter theories reduces, by localization, to a matrix model. We
develop a new method to study these models in the M-theory limit, but at all
orders in the 1/N expansion. The method is based on reformulating the matrix
model as the partition function of an ideal Fermi gas with a non-trivial,
one-particle quantum Hamiltonian. This new approach leads to a completely
elementary derivation of the N^{3/2} behavior for ABJM theory and N=3 quiver
Chern-Simons-matter theories. In addition, the full series of 1/N corrections
to the original matrix integral can be simply determined by a next-to-leading
calculation in the WKB or semiclassical expansion of the quantum gas, and we
show that, for several quiver Chern-Simons-matter theories, it is given by an
Airy function. This generalizes a recent result of Fuji, Hirano and Moriyama
for ABJM theory. It turns out that the semiclassical expansion of the Fermi gas
corresponds to a strong coupling expansion in type IIA theory, and it is dual
to the genus expansion. This allows us to calculate explicitly non-perturbative
effects due to D2-brane instantons in the AdS background.Comment: 52 pages, 11 figures. v3: references, corrections and clarifications
added, plus a footnote on the relation to the recent work by Hanada et a
Some properties of WKB series
We investigate some properties of the WKB series for arbitrary analytic
potentials and then specifically for potentials ( even), where more
explicit formulae for the WKB terms are derived. Our main new results are: (i)
We find the explicit functional form for the general WKB terms ,
where one has only to solve a general recursion relation for the rational
coefficients. (ii) We give a systematic algorithm for a dramatic simplification
of the integrated WKB terms that enter the energy
eigenvalue equation. (iii) We derive almost explicit formulae for the WKB terms
for the energy eigenvalues of the homogeneous power law potentials , where is even. In particular, we obtain effective algorithms to
compute and reduce the terms of these series.Comment: 18 pages, submitted to Journal of Physics A: Mathematical and Genera
Chaos and Quantum Thermalization
We show that a bounded, isolated quantum system of many particles in a
specific initial state will approach thermal equilibrium if the energy
eigenfunctions which are superposed to form that state obey {\it Berry's
conjecture}. Berry's conjecture is expected to hold only if the corresponding
classical system is chaotic, and essentially states that the energy
eigenfunctions behave as if they were gaussian random variables. We review the
existing evidence, and show that previously neglected effects substantially
strengthen the case for Berry's conjecture. We study a rarefied hard-sphere gas
as an explicit example of a many-body system which is known to be classically
chaotic, and show that an energy eigenstate which obeys Berry's conjecture
predicts a Maxwell--Boltzmann, Bose--Einstein, or Fermi--Dirac distribution for
the momentum of each constituent particle, depending on whether the wave
functions are taken to be nonsymmetric, completely symmetric, or completely
antisymmetric functions of the positions of the particles. We call this
phenomenon {\it eigenstate thermalization}. We show that a generic initial
state will approach thermal equilibrium at least as fast as
, where is the uncertainty in the total energy
of the gas. This result holds for an individual initial state; in contrast to
the classical theory, no averaging over an ensemble of initial states is
needed. We argue that these results constitute a new foundation for quantum
statistical mechanics.Comment: 28 pages in Plain TeX plus 2 uuencoded PS figures (included); minor
corrections only, this version will be published in Phys. Rev. E;
UCSB-TH-94-1
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