353 research outputs found
An exact solution method for 1D polynomial Schr\"odinger equations
Stationary 1D Schr\"odinger equations with polynomial potentials are reduced
to explicit countable closed systems of exact quantization conditions, which
are selfconsistent constraints upon the zeros of zeta-regularized spectral
determinants, complementing the usual asymptotic (Bohr--Sommerfeld)
constraints. (This reduction is currently completed under a certain vanishing
condition.) In particular, the symmetric quartic oscillators are admissible
systems, and the formalism is tested upon them. Enforcing the exact and
asymptotic constraints by suitable iterative schemes, we numerically observe
geometric convergence to the correct eigenvalues/functions in some test cases,
suggesting that the output of the reduction should define a contractive
fixed-point problem (at least in some vicinity of the pure case).Comment: flatex text.tex, 4 file
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
Anharmonic Oscillators, Spectral Determinant and Short Exact Sequence of affine U_q(sl_2)
We prove one of conjectures, raised by Dorey and Tateo in the connection
among the spectral determinant of anharmonic oscillator and vacuum eigenvalues
of transfer matrices in field theory and statistical mechanics. The exact
sequence of plays a fundamental role in the proof.Comment: 8 pages, Latex 2
Spectral zeta functions of a 1D Schr\"odinger problem
We study the spectral zeta functions associated to the radial Schr\"odinger
problem with potential V(x)=x^{2M}+alpha x^{M-1}+(lambda^2-1/4)/x^2. Using the
quantum Wronskian equation, we provide results such as closed-form evaluations
for some of the second zeta functions i.e. the sum over the inverse eigenvalues
squared. Also we discuss how our results can be used to derive relationships
and identities involving special functions, using a particular 5F_4
hypergeometric series as an example. Our work is then extended to a class of
related PT-symmetric eigenvalue problems. Using the fused quantum Wronskian we
give a simple method for calculating the related spectral zeta functions. This
method has a number of applications including the use of the ODE/IM
correspondence to compute the (vacuum) nonlocal integrals of motion G_n which
appear in an associated integrable quantum field theory.Comment: 15 pages, version
Functional Relations in Stokes Multipliers and Solvable Models related to U_q(A^{(1)}_n)
Recently, Dorey and Tateo have investigated functional relations among Stokes
multipliers for a Schr{\"o}dinger equation (second order differential equation)
with a polynomial potential term in view of solvable models. Here we extend
their studies to a restricted case of n+1-th order linear differential
equations.Comment: 20 pages, some explanations improved, To appear in J. Phys.
A nonextensive entropy approach to solar wind intermittency
The probability distributions (PDFs) of the differences of any physical
variable in the intermittent, turbulent interplanetary medium are scale
dependent. Strong non-Gaussianity of solar wind fluctuations applies for short
time-lag spacecraft observations, corresponding to small-scale spatial
separations, whereas for large scales the differences turn into a Gaussian
normal distribution. These characteristics were hitherto described in the
context of the log-normal, the Castaing distribution or the shell model. On the
other hand, a possible explanation for nonlocality in turbulence is offered
within the context of nonextensive entropy generalization by a recently
introduced bi-kappa distribution, generating through a convolution of a
negative-kappa core and positive-kappa halo pronounced non-Gaussian structures.
The PDFs of solar wind scalar field differences are computed from WIND and ACE
data for different time lags and compared with the characteristics of the
theoretical bi-kappa functional, well representing the overall scale dependence
of the spatial solar wind intermittency. The observed PDF characteristics for
increased spatial scales are manifest in the theoretical distribution
functional by enhancing the only tuning parameter , measuring the
degree of nonextensivity where the large-scale Gaussian is approached for
. The nonextensive approach assures for experimental studies
of solar wind intermittency independence from influence of a priori model
assumptions. It is argued that the intermittency of the turbulent fluctuations
should be related physically to the nonextensive character of the
interplanetary medium counting for nonlocal interactions via the entropy
generalization.Comment: 17 pages, 7 figures, accepted for publication in Astrophys.
Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics
The knowledge of quantum phase flow induced under the Weyl's association rule
by the evolution of Heisenberg operators of canonical coordinates and momenta
allows to find the evolution of symbols of generic Heisenberg operators. The
quantum phase flow curves obey the quantum Hamilton's equations and play the
role of characteristics. At any fixed level of accuracy of semiclassical
expansion, quantum characteristics can be constructed by solving a coupled
system of first-order ordinary differential equations for quantum trajectories
and generalized Jacobi fields. Classical and quantum constraint systems are
discussed. The phase-space analytic geometry based on the star-product
operation can hardly be visualized. The statement "quantum trajectory belongs
to a constraint submanifold" can be changed e.g. to the opposite by a unitary
transformation. Some of relations among quantum objects in phase space are,
however, left invariant by unitary transformations and support partly geometric
relations of belonging and intersection. Quantum phase flow satisfies the
star-composition law and preserves hamiltonian and constraint star-functions.Comment: 27 pages REVTeX, 6 EPS Figures. New references added. Accepted for
publication to JM
Scar Intensity Statistics in the Position Representation
We obtain general predictions for the distribution of wave function
intensities in position space on the periodic orbits of chaotic ballistic
systems. The expressions depend on effective system size N, instability
exponent lambda of the periodic orbit, and proximity to a focal point of the
orbit. Limiting expressions are obtained that include the asymptotic
probability distribution of rare high-intensity events and a perturbative
formula valid in the limit of weak scarring. For finite system sizes, a single
scaling variable lambda N describes deviations from the semiclassical N ->
infinity limit.Comment: To appear in Phys. Rev. E, 10 pages, including 4 figure
Universality in the flooding of regular islands by chaotic states
We investigate the structure of eigenstates in systems with a mixed phase
space in terms of their projection onto individual regular tori. Depending on
dynamical tunneling rates and the Heisenberg time, regular states disappear and
chaotic states flood the regular tori. For a quantitative understanding we
introduce a random matrix model. The resulting statistical properties of
eigenstates as a function of an effective coupling strength are in very good
agreement with numerical results for a kicked system. We discuss the
implications of these results for the applicability of the semiclassical
eigenfunction hypothesis.Comment: 11 pages, 12 figure
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