arXiv subjects

E. Horozov

Publications and source records attributed to E. Horozov.

10 recordsLinked to original sources

arXiv2001

Fuchsian bispectral operators

The aim of this paper is to classify the bispectral operators of any rank with regular singular points (the infinite point is the most important one). We characterise them in several ways. Probably the most important result is that they are all Darboux transformations of powers of generalised Bessel operators (in the terminology of q-alg/9602011). For this reason they can be effectively parametrised by the points of a certain (infinite) family of algebraic manifolds as pointed out in q-alg/9602011.

math.DS↗

arXiv1999

The Pfaff lattice and skew-orthogonal polynomials

Consider a semi-infinite skew-symmetric moment matrix, $m_{\iy}$ evolving according to the vector fields $\pl m / \pl t_k=\Lb^k m+m \Lb^{\top k} ,$ where $\Lb$ is the shift matrix. Then the skew-Borel decomposition $ m_{\iy}:= Q^{-1} J Q^{\top -1} $ leads to the so-called Pfaff Lattice, which is integrable, by virtue of the AKS theorem, for a splitting involving the affine symplectic algebra. The tau-functions for the system are shown to be pfaffians and the wave vectors skew-orthogonal polynomials; we give their explicit form in terms of moments. This system plays an important role in symmetric and symplectic matrix models and in the theory of random matrices (beta=1 or 4).

solv-int↗

arXiv1997

The solution to the q-KdV equation

Let KdV stand for the Nth Gelfand-Dickey reduction of the KP hierarchy. The purpose of this paper is to show that any KdV solution leads effectively to a solution of the q-approximation of KdV. Two different q-KdV approximations were proposed, one by Frenkel and a variation by Khesin et al. We show there is a dictionary between the solutions of q-KP and the 1-Toda lattice equations, obeying some special requirement; this is based on an algebra isomorphism between difference operators and D-operators, where $Df(x)=f(qx)$. Therefore, every notion about the 1-Toda lattice can be transcribed into q-language.

solv-int↗

arXiv1997

Bispectral algebras of commuting ordinary differential operators

We develop a systematic way for constructing bispectral algebras of commuting ordinary differential operators of any rank $N$. It combines and unifies the ideas of Duistermaat-Grünbaum and Wilson. Our construction is completely algorithmic and enables us to obtain all previously known classes or individual examples of bispectral operators. The method also provides new broad families of bispectral algebras which may help to penetrate deeper into the problem.

q-alg↗

arXiv1997

Automorphisms of the Weyl algebra and bispectral operators

In our previous paper q-alg/9605011 we proposed several algebraic methods for constructing new solutions to the bispectral problem. In the present note the corresponding eigenfunctions are explicitly constructed as multiple Laplace integrals.

q-alg↗

arXiv1996

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.

q-alg↗

arXiv1996

Highest weight modules of W_{1+infty}, Darboux transformations and the bispectral problem

We announce a systematic way for constructing bispectral algebras of commuting differential operators of any rank N. It enables us to obtain all previously known classes and examples of bispectral operators. Moreover, we give a representation-theoretic explanation of the results including those of Duistermaat and Grünbaum. The manifold of bispectral operators of any order is preserved by an hierarchy of symmetries. We point out that our methods provide a completely algorithmic procedure for obtaining bispectral algebras. We conjecture that the class built in the present paper exhausts all bispectral scalar operators. The proofs and details appeared in our preprints hep-th/9510211, q-alg/9602010, q-alg/9602011, q-alg/9602012.

q-alg↗

arXiv1996

Bäcklund--Darboux transformations in Sato's Grassmannian

We define Bäcklund--Darboux transformations in Sato's Grassmannian. They can be regarded as Darboux transformations on maximal algebras of commuting ordinary differential operators. We describe the action of these transformations on related objects: wave functions, tau-functions and spectral algebras. This paper is the second of a series of papers (hep-th/9510211, q-alg/9602011, q-alg/9602012) on the bispectral problem.

q-alg↗

arXiv1996

Highest weight modules over W_{1+infty} algebra and the bispectral problem

The present paper establishes a connection between the Lie algebra W_{1+infty} and the bispectral problem. We show that the manifolds of bispectral operators obtained by Darboux transformations on powers of Bessel operators are in one to one correspondence with the manifolds of tau-functions lying in the W_{1+infty}-modules M_beta introduced in our previous paper hep-th/9510211. An immediate corollary is that they are preserved by hierarchies of symmetries generated by subalgebras of W_{1+infty}. This paper is the last of a series of papers (hep-th/9510211, q-alg/9602010, q-alg/9602011) on the bispectral problem.

q-alg↗

arXiv1996

Tau-functions as highest weight vectors for W_{1+infty} algebra

For each r = (r_1, r_2,...,r_N) we construct a highest weight module M_r of the Lie algebra W_{1+infty}. The highest weight vectors are specific tau-functions of the N-th Gelfand--Dickey hierarchy. We show that these modules are quasifinite and we give a complete description of the reducible ones together with a formula for the singular vectors. This paper is the first of a series of papers (q-alg/9602010, q-alg/9602011, q-alg/9602012) on the bispectral problem.

hep-th↗