Algebraic curves for commuting elements in the q-deformed Heisenberg algebra

In this paper we extend the eliminant construction of Burchnall and Chaundy for commuting differential operators in the Heisenberg algebra to the q-deformed Heisenberg algebra and show that it again provides annihilating curves for commuting elements, provided q satisfies a natural condition. As a side result we obtain estimates on the dimensions of the eigenspaces of elements of this algebra in its faithful module of Laurent series.Comment: 18 pages, 2 figures, LaTeX. Final version with some improvements in presentation. To appear in Journal of Algebra

General methods for constructing bispectral operators

We present methods for obtaining new solutions to the bispectral problem. We achieve this by giving its abstract algebraic version suitable for generalizations. All methods are illustrated by new classes of bispectral operators.Comment: 11 pages, LaTeX2e, uses amsfonts.sty and latexsym.sty, no figure

Algebraic Solutions of the Multicomponent KP Hierarchy

It is shown that it is possible to write down tau functions for the $n$-component KP hierarchy in terms of non-abelian theta functions. This is a generalization of the rank 1 situation; that is, the relation of theta functions of Jacobians and tau functions for the KP hierarchy.Comment: 19 page

Integrable Dynamics of Charges Related to Bilinear Hypergeometric Equation

A family of systems related to a linear and bilinear evolution of roots of polynomials in the complex plane is considered. Restricted to the line, the evolution induces dynamics of the Coulomb charges in external potentials, while its fixed points correspond to equilibria of charges (or point vortices in hydrodynamics) in the plane. The construction reveals a direct connection with the theories of the Calogero-Moser systems and Lie-algebraic differential operators. A study of the equilibrium configurations amounts in a construction (bilinear hypergeometric equation) for which the classical orthogonal and the Adler-Moser polynomials represent some particular casesComment: 27 pages, Latex, A new corrected version of older submissio

The Heat Kernel Coefficients to the Matrix Schr\"odinger Operator

The heat kernel coefficients $H_k$ to the Schr\"odinger operator with a matrix potential are investigated. We present algorithms and explicit expressions for the Taylor coefficients of the $H_k$. Special terms are discussed, and for the one-dimensional case some improved algorithms are derived.Comment: 16 pages, Plain TeX, 33 KB, no figure

Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies

We propose a method for computing any Gelfand-Dickey tau function living in Segal-Wilson Grassmannian as the asymptotics of block Toeplitz determinant associated to a certain class of symbols. Also truncated block Toeplitz determinants associated to the same symbols are shown to be tau function for rational reductions of KP. Connection with Riemann-Hilbert problems is investigated both from the point of view of integrable systems and block Toeplitz operator theory. Examples of applications to algebro-geometric solutions are given.Comment: 35 pages. Typos corrected, some changes in the introductio

The inverse resonance problem for perturbations of algebro-geometric potentials

We prove that a compactly supported perturbation of a rational or simply periodic algebro-geometric potential of the one-dimensional Schr\"odinger equation on the half line is uniquely determined by the location of its Dirichlet eigenvalues and resonances.Comment: 14 page

An Alternative Approach to Algebro-Geometric Solutions of the AKNS Hierarchy

We develop an alternative systematic approach to the AKNS hierarchy based on elementary algebraic methods. In particular, we recursively construct Lax pairs for the entire AKNS hierarchy by introducing a fundamental polynomial formalism and establish the basic algebro-geometric setting including associated Burchnall-Chaundy curves, Baker-Akhiezer functions, trace formulas, Dubrovin-type equations for analogs of Dirichlet and Neumann divisors, and theta function representations for algebro-geometric solutions.Comment: LaTeX, submitted to Reviews in Mathematical Physic

Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains

A chain of one-dimensional Schr\"odinger operators connected by successive Darboux transformations is called the ``Darboux chain'' or ``dressing chain''. The periodic dressing chain with period $N$ has a control parameter $\alpha$. If $\alpha \not= 0$, the $N$-periodic dressing chain may be thought of as a generalization of the fourth or fifth (depending on the parity of $N$) Painlev\'e equations . The $N$-periodic dressing chain has two different Lax representations due to Adler and to Noumi and Yamada. Adler's $2 \times 2$ Lax pair can be used to construct a transition matrix around the periodic lattice. One can thereby define an associated ``spectral curve'' and a set of Darboux coordinates called ``spectral Darboux coordinates''. The equations of motion of the dressing chain can be converted to a Hamiltonian system in these Darboux coordinates. The symplectic structure of this Hamiltonian formalism turns out to be consistent with a Poisson structure previously studied by Veselov, Shabat, Noumi and Yamada.Comment: latex2e, 41 pages, no figure; (v2) some minor errors are corrected; (v3) fully revised and shortend, and some results are improve

On Darboux-Treibich-Verdier potentials

It is shown that the four-parameter family of elliptic functions $$u_D(z)=m_0(m_0+1)\wp(z)+\sum_{i=1}^3 m_i(m_i+1)\wp(z-\omega_i)$$ introduced by Darboux and rediscovered a hundred years later by Treibich and Verdier, is the most general meromorphic family containing infinitely many finite-gap potentials.Comment: 8 page