430 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
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
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
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.
Fermi Edge Singularities in the Mesoscopic Regime: II. Photo-absorption Spectra
We study Fermi edge singularities in photo-absorption spectra of generic
mesoscopic systems such as quantum dots or nanoparticles. We predict deviations
from macroscopic-metallic behavior and propose experimental setups for the
observation of these effects. The theory is based on the model of a localized,
or rank one, perturbation caused by the (core) hole left behind after the
photo-excitation of an electron into the conduction band. The photo-absorption
spectra result from the competition between two many-body responses, Anderson's
orthogonality catastrophe and the Mahan-Nozieres-DeDominicis contribution. Both
mechanisms depend on the system size through the number of particles and, more
importantly, fluctuations produced by the coherence characteristic of
mesoscopic samples. The latter lead to a modification of the dipole matrix
element and trigger one of our key results: a rounded K-edge typically found in
metals will turn into a (slightly) peaked edge on average in the mesoscopic
regime. We consider in detail the effect of the "bound state" produced by the
core hole.Comment: 16 page
On the Convergence of the WKB Series for the Angular Momentum Operator
In this paper we prove a recent conjecture [Robnik M and Salasnich L 1997 J.
Phys. A: Math. Gen. 30 1719] about the convergence of the WKB series for the
angular momentum operator. We demonstrate that the WKB algorithm for the
angular momentum gives the exact quantization formula if all orders are summed.Comment: latex, 9 pages, no figures, to be published in Journal of Physics A:
Math. and Ge
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
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
Exact computation of one-loop correction to energy of pulsating strings in AdS_5 x S^5
In the present paper, which is a sequel to arXiv:1001:4018, we compute the
one-loop correction to the energy of pulsating string solutions in AdS_5 x S^5.
We show that, as for rigid spinning string elliptic solutions, the fluctuation
operators for pulsating solutions can be also put into the single-gap Lame'
form. A novel aspect of pulsating solutions is that the one-loop correction to
their energy is expressed in terms of the stability angles of the quadratic
fluctuation operators. We explicitly study the "short string" limit of the
corresponding one-loop energies, demonstrating a certain universality of the
form of the energy of "small" semiclassical strings. Our results may help to
shed light on the structure of strong-coupling expansion of anomalous
dimensions of dual gauge theory operators.Comment: 49 pages; v2: appendix F and note about antiperiodic fermions added,
typos corrected, references adde
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