66,314 research outputs found
Quasi-Exactly Solvable Potentials on the Line and Orthogonal Polynomials
In this paper we show that a quasi-exactly solvable (normalizable or
periodic) one-dimensional Hamiltonian satisfying very mild conditions defines a
family of weakly orthogonal polynomials which obey a three-term recursion
relation. In particular, we prove that (normalizable) exactly-solvable
one-dimensional systems are characterized by the fact that their associated
polynomials satisfy a two-term recursion relation. We study the properties of
the family of weakly orthogonal polynomials defined by an arbitrary
one-dimensional quasi-exactly solvable Hamiltonian, showing in particular that
its associated Stieltjes measure is supported on a finite set. From this we
deduce that the corresponding moment problem is determined, and that the -th
moment grows like the -th power of a constant as tends to infinity. We
also show that the moments satisfy a constant coefficient linear difference
equation, and that this property actually characterizes weakly orthogonal
polynomial systems.Comment: 22 pages, plain TeX. Please typeset only the file orth.te
Unbounded Recursion and Non-size-increasing Functions
We investigate the computing power of function algebras defined by means of unbounded recursion on notation. We introduce two function algebras which contain respectively the regressive logspace computable functions and the non-size-increasing logspace computable functions. However, such algebras are unlikely to be contained in the set of logspace computable functions because this is equivalent to L=P . Finally, we introduce a function algebra based on simultaneous recursion on notation for the non-size-increasing functions computable in polynomial time and linear space
Novel Properties of Frustrated Low Dimensional Magnets with Pentagonal Symmetry
In the context of magnetism, frustration arises when a group of spins cannot
find a configuration that minimizes all of their pairwise interactions
simultaneously. We consider the effects of the geometric frustration that
arises in a structure having pentagonal loops. Such five-fold loops can be
expected to occur naturally in quasicrystals, as seen for example in a number
of experimental studies of surfaces of icosahedral alloys. Our model considers
classical vector spins placed on vertices of a subtiling of the two dimensional
Penrose tiling, and interacting with nearest neighbors via antiferromagnetic
bonds. We give a set of recursion relations for this system, which consists of
an infinite set of embedded clusters with sizes that increase as a power of the
golden mean. The magnetic ground states of this fractal system are studied
analytically, and by Monte Carlo simulation.Comment: 7 pages, 7 figures, contribution to ICQ11 (Sapporo, Japan 2010)
conference proceeding
Exact Dynamics of the SU(K) Haldane-Shastry Model
The dynamical structure factor of the SU(K) (K=2,3,4)
Haldane-Shastry model is derived exactly at zero temperature for arbitrary size
of the system. The result is interpreted in terms of free quasi-particles which
are generalization of spinons in the SU(2) case; the excited states relevant to
consist of K quasi-particles each of which is characterized by a
set of K-1 quantum numbers. Near the boundaries of the region where
is nonzero, shows the power-law singularity. It is
found that the divergent singularity occurs only in the lowest edges starting
from toward positive and negative q. The analytic result
is checked numerically for finite systems via exact diagonalization and
recursion methods.Comment: 35 pages, 3 figures, youngtab.sty (version 1.1
Recursions for rational q,t-Catalan numbers
We give a simple recursion labeled by binary sequences which computes rational q,t-Catalan power series, both in relatively prime and non relatively prime cases. It is inspired by, but not identical to recursions due to B. Elias, M. Hogancamp, and A. Mellit, obtained in their study of link homology. We also compare our recursion with the Hogancamp-Mellit's recursion and verify a connection between the Khovanov-Rozansky homology of N,M-torus links and the rational q,t-Catalan power series for general positive N,M
MATAD: a program package for the computation of MAssive TADpoles
In the recent years there has been an enormous development in the evaluation
of higher order quantum corrections. An essential ingredient in the practical
calculations is provided by vacuum diagrams, i.e. integrals without external
momenta. In this paper a program package is described which can deal with one-,
two- and three-loop vacuum integrals with one non-zero mass parameter. The
principle structure is introduced and the main parts of the package are
described in detail. Explicit examples demonstrate the fields of application.Comment: 37 pages, to be published in Comp. Phys. Commu
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