19,540 research outputs found
Signatures of S-wave bound-state formation in finite volume
We discuss formation of an S-wave bound-state in finite volume on the basis
of L\"uscher's phase-shift formula.It is found that although a bound-state pole
condition is fulfilled only in the infinite volume limit, its modification by
the finite size corrections is exponentially suppressed by the spatial extent
in a finite box . We also confirm that the appearance of the S-wave
bound state is accompanied by an abrupt sign change of the S-wave scattering
length even in finite volume through numerical simulations. This distinctive
behavior may help us to discriminate the loosely bound state from the lowest
energy level of the scattering state in finite volume simulations.Comment: 25 pages, 30 figures; v2: typos corrected and two references added,
v3: final version to appear in PR
Non-polynomial extensions of solvable potentials a la Abraham-Moses
Abraham-Moses transformations, besides Darboux transformations, are
well-known procedures to generate extensions of solvable potentials in
one-dimensional quantum mechanics. Here we present the explicit forms of
infinitely many seed solutions for adding eigenstates at arbitrary real energy
through the Abraham-Moses transformations for typical solvable potentials, e.g.
the radial oscillator, the Darboux-P\"oschl-Teller and some others. These seed
solutions are simple generalisations of the virtual state wavefunctions, which
are obtained from the eigenfunctions by discrete symmetries of the potentials.
The virtual state wavefunctions have been an essential ingredient for
constructing multi-indexed Laguerre and Jacobi polynomials through multiple
Darboux-Crum transformations. In contrast to the Darboux transformations, the
virtual state wavefunctions generate non-polynomial extensions of solvable
potentials through the Abraham-Moses transformations.Comment: 29 page
Unified Theory of Annihilation-Creation Operators for Solvable (`Discrete') Quantum Mechanics
The annihilation-creation operators are defined as the
positive/negative frequency parts of the exact Heisenberg operator solution for
the `sinusoidal coordinate'. Thus are hermitian conjugate to each
other and the relative weights of various terms in them are solely determined
by the energy spectrum. This unified method applies to most of the solvable
quantum mechanics of single degree of freedom including those belonging to the
`discrete' quantum mechanics.Comment: 43 pages, no figures, LaTeX2e, with amsmath, amssym
A new family of shape invariantly deformed Darboux-P\"oschl-Teller potentials with continuous \ell
We present a new family of shape invariant potentials which could be called a
``continuous \ell version" of the potentials corresponding to the exceptional
(X_{\ell}) J1 Jacobi polynomials constructed recently by the present authors.
In a certain limit, it reduces to a continuous \ell family of shape invariant
potentials related to the exceptional (X_{\ell}) L1 Laguerre polynomials. The
latter was known as one example of the `conditionally exactly solvable
potentials' on a half line.Comment: 19 pages. Sec.5(Summary and Comments): one sentence added in the
first paragraph, several sentences modified in the last paragraph.
References: one reference ([25]) adde
The phase structure of a chiral model with dilatons in hot and dense matter
We explore the phase structure of a chiral model of constituent quarks and
gluons implementing scale symmetry breaking at finite temperature and chemical
potential. In this model the chiral dynamics is intimately linked to the trace
anomaly saturated by a dilaton field. The thermodynamics is governed by two
condensates, thermal expectation values of sigma and dilaton fields, which are
the order parameters responsible for the phase transitions associated with the
chiral and scale symmetries. Within the mean field approximation, we find that
increasing temperature a system experiences a chiral phase transition and then
a first-order phase transition of partial scale symmetry restoration
characterized by a melting gluon-condensate takes place at a higher
temperature. There exists a region at finite chemical potential where the scale
symmetry remains dynamically broken while the chiral symmetry is restored. We
also give a brief discussion on the sigma-meson mass constrained from Lattice
QCD.Comment: 6 pages, 5 figures; v2) new figures and references adde
Orthogonal Polynomials from Hermitian Matrices
A unified theory of orthogonal polynomials of a discrete variable is
presented through the eigenvalue problem of hermitian matrices of finite or
infinite dimensions. It can be considered as a matrix version of exactly
solvable Schr\"odinger equations. The hermitian matrices (factorisable
Hamiltonians) are real symmetric tri-diagonal (Jacobi) matrices corresponding
to second order difference equations. By solving the eigenvalue problem in two
different ways, the duality relation of the eigenpolynomials and their dual
polynomials is explicitly established. Through the techniques of exact
Heisenberg operator solution and shape invariance, various quantities, the two
types of eigenvalues (the eigenvalues and the sinusoidal coordinates), the
coefficients of the three term recurrence, the normalisation measures and the
normalisation constants etc. are determined explicitly.Comment: 53 pages, no figures. Several sentences and a reference are added. To
be published in J. Math. Phy
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