Symmetries of differential equations. IV

By an application of the geometrical techniques of Lie, Cohen, and Dickson it is shown that a system of differential equations of the form [x^(r_i)]_i = F_i(where r_i > 1 for every i = 1 , ... ,n) cannot admit an infinite number of pointlike symmetry vectors. When r_i = r for every i = 1, ... ,n, upper bounds have been computed for the maximum number of independent symmetry vectors that these systems can possess: The upper bounds are given by 2n_ 2 + nr + 2 (when r> 2), and by 2n_2 + 4n + 2 (when r = 2). The group of symmetries of ͞x^r = ͞0 (r> 1) has also been computed, and the result obtained shows that when n > 1 and r> 2 the number of independent symmetries of these equations does not attain the upper bound 2n _2 + nr + 2, which is a common bound for all systems of differential equations of the form ͞x^r = F[t, ͞x, ... , ͞x^(r - 1 )] when r> 2. On the other hand, when r = 2 the first upper bound obtained has been reduced to the value n^2 + 4n + 3; this number is equal to the number of independent symmetry vectors of the system ͞x= ͞0, and is also a common bound for all systems of the form ͞x = ͞F (t ,͞x, ‾̇x)

The Multidimensional Darboux transformation

A generalization of the classical one-dimensional Darboux transformation to arbitrary n- dimensional oriented Riemannian manifolds is constructed using an intrinsic formulation based on the properties of twisted Hodge Laplacians. The classical two-dimensional Moutard transformation is also generalized to non-compact oriented Riemannian manifolds of dimension n ≥ 2. New examples of quasi-exactly solvable multidimensional matrix Schrödinger operators on curved manifolds are obtained by applying the above results

The open Haldane-Shastry chain: thermodynamics and criticality

We study the thermodynamics and criticality of the su($m|n$) Haldane-Shastry chain of $BC_N$ type with a general chemical potential term. We first derive a complete description of the spectrum of this model in terms of $BC_N$-type motifs, from which we deduce a representation for the partition function as the trace of a product of site-dependent transfer matrices. In the thermodynamic limit, this formula yields a simple expression for the free energy per spin in terms of the Perron-Frobenius eigenvalue of the continuum limit of the transfer matrix. Evaluating this eigenvalue we obtain closed-form expressions for the thermodynamic functions of the chains with $m,n\le2$. Using the motif-based description of the spectrum derived here, we study in detail the ground state of these models and their low energy excitations. In this way we identify the critical intervals in chemical potential space and compute their corresponding Fermi velocities. By contrast with previously studied models of this type, we find in some cases two types of low energy excitations with linear energy-quasimomentum relation. Finally, we determine the central charge of all the critical phases by analyzing the low-temperature behavior of the expression for the free energy per spin.Comment: 46 pages, 6 figures, typeset in LaTe

Thermodynamics and criticality of su(mm) spin chains of Haldane-Shastry type

We study the thermodynamics and critical behavior of su($m$) spin chains of Haldane-Shastry type at zero chemical potential, both in the $A_{N-1}$ and $BC_N$ cases. We evaluate in closed form the free energy per spin for arbitrary values of $m$, from which we derive explicit formulas for the energy, entropy and specific heat per spin. In particular, we find that the specific heat features a single Schottky peak, whose temperature is well approximated for $m\lesssim10$ by the corresponding temperature for an $m$-level system with uniformly spaced levels. We show that at low temperatures the free energy per spin of the models under study behaves as that of a one-dimensional conformal field theory with central charge $c=m-1$ (with the only exception of the Frahm-Inozemtsev chain with zero value of its parameter). However, from a detailed study of the ground state degeneracy and the low-energy excitations, we conclude that these models are only critical in the antiferromagnetic case, with a few exceptions that we fully specify.Comment: 12 pages, 2 figure

Global properties of the spectrum of the Haldane-Shastry spin chain

We derive an exact expression for the partition function of the su(m) Haldane-Shastry spin chain, which we use to study the density of levels and the distribution of the spacing between consecutive levels. Our computations show that when the number of sites N is large enough the level density is Gaussian to a very high degree of approximation. More surprisingly, we also find that the nearest-neighbor spacing distribution is not Poissonian, so that this model departs from the typical behavior for an integrable system. We show that the cumulative spacing distribution of the model can be well approximated by a simple functional law involving only three parameters.Comment: RevTeX 4, 7 pages, 4 figures. To appear in Phys. Rev.

Real Lie Algebras of Differential Operators and Quasi-Exactly Solvable Potentials

We first establish some general results connecting real and complex Lie algebras of first-order differential operators. These are applied to completely classify all finite-dimensional real Lie algebras of first-order differential operators in $R^2$. Furthermore, we find all algebras which are quasi-exactly solvable, along with the associated finite-dimensional modules of analytic functions. The resulting real Lie algebras are used to construct new quasi-exactly solvable Schroedinger operators on $R^2$.Comment: 33 pages, plain TeX. To apper in Phil. Trans. London Math. Soc. Please typeset only the file rf.te

Quasi-Exactly Solvable Spin 1/2 Schr\"odinger Operators

The algebraic structures underlying quasi-exact solvability for spin 1/2 Hamiltonians in one dimension are studied in detail. Necessary and sufficient conditions for a matrix second-order differential operator preserving a space of wave functions with polynomial components to be equivalent to a \sch\ operator are found. Systematic simplifications of these conditions are analyzed, and are then applied to the construction of several new examples of multi-parameter QES spin 1/2 Hamiltonians in one dimension.Comment: 32 pages, LaTeX2e using AMS-LaTeX packag

On form-preserving transformations for the time-dependent Schr\"odinger equation

In this paper we point out a close connection between the Darboux transformation and the group of point transformations which preserve the form of the time-dependent Schr\"odinger equation (TDSE). In our main result, we prove that any pair of time-dependent real potentials related by a Darboux transformation for the TDSE may be transformed by a suitable point transformation into a pair of time-independent potentials related by a usual Darboux transformation for the stationary Schr\"odinger equation. Thus, any (real) potential solvable via a time-dependent Darboux transformation can alternatively be solved by applying an appropriate form-preserving transformation of the TDSE to a time-independent potential. The preeminent role of the latter type of transformations in the solution of the TDSE is illustrated with a family of quasi-exactly solvable time-dependent anharmonic potentials.Comment: LaTeX2e (with amsmath, amssymb, amscd, cite packages), 11 page