262 research outputs found
Solutions for the General, Confluent and Biconfluent Heun equations and their connection with Abel equations
In a recent paper, the canonical forms of a new multi-parameter class of Abel
differential equations, so-called AIR, all of whose members can be mapped into
Riccati equations, were shown to be related to the differential equations for
the hypergeometric 2F1, 1F1 and 0F1 functions. In this paper, a connection
between the AIR canonical forms and the Heun General (GHE), Confluent (CHE) and
Biconfluent (BHE) equations is presented. This connection fixes the value of
one of the Heun parameters, expresses another one in terms of those remaining,
and provides closed form solutions in terms of pFq functions for the resulting
GHE, CHE and BHE, respectively depending on four, three and two irreducible
parameters. This connection also turns evident what is the relation between the
Heun parameters such that the solutions admit Liouvillian form, and suggests a
mechanism for relating linear equations with N and N-1 singularities through
the canonical forms of a non-linear equation of one order less.Comment: Original version submitted to Journal of Physics A: 16 pages, related
to math.GM/0002059 and math-ph/0402040. Revised version according to
referee's comments: 23 pages. Sign corrected (June/17) in formula (79).
Second revised version (July/25): 25 pages. See also
http://lie.uwaterloo.ca/odetools.ht
Dynamical spin-spin correlation functions in the Kondo model out of equilibrium
We calculate the dynamical spin-spin correlation functions of a Kondo dot
coupled to two noninteracting leads held at different chemical potentials. To
this end we generalize a recently developed real-time renormalization group
method in frequency space (RTRG-FS) to allow the calculation of dynamical
correlation functions of arbitrary dot operators in systems describing spin
and/or orbital fluctuations. The resulting two-loop RG equations are
analytically solved in the weak-coupling regime. This implies that the method
can be applied provided either the voltage through the dot or the external
magnetic field are sufficiently large, , where the
Kondo temperature is the scale where the system enters the
strong-coupling regime. Explicitly, we calculate the longitudinal and
transverse spin-spin correlation and response functions as well as the
resulting fluctuation-dissipation ratios. The correlation functions in
real-frequency space can be calculated in Matsubara space without the need of
any analytical continuation. We obtain analytic results for the line-shape, the
small- and large-frequency limits and several other features like the height
and width of the peak in the transverse susceptibility at
, where denotes the reduced magnetic field.
Furthermore, we discuss how the developed method can be generalized to
calculate dynamical correlation functions of other operators involving
reservoir degrees of freedom as well.Comment: 30 page
Extensions of differential representations of SL(2) and tori
Linear differential algebraic groups (LDAGs) measure differential algebraic
dependencies among solutions of linear differential and difference equations
with parameters, for which LDAGs are Galois groups. The differential
representation theory is a key to developing algorithms computing these groups.
In the rational representation theory of algebraic groups, one starts with
SL(2) and tori to develop the rest of the theory. In this paper, we give an
explicit description of differential representations of tori and differential
extensions of irreducible representation of SL(2). In these extensions, the two
irreducible representations can be non-isomorphic. This is in contrast to
differential representations of tori, which turn out to be direct sums of
isotypic representations.Comment: 21 pages; few misprints corrected; Lemma 4.6 adde
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