arXiv subjects

A. P. Veselov

Publications and source records attributed to A. P. Veselov.

At least 73 records · Page 4Linked to original sources

arXiv2005

Spectra of Sol-manifolds: arithmetic and quantum monodromy

The spectral problem of three-dimensional manifolds M_A admitting Sol-geometry in Thurston's sense is investigated. Topologically M_A are torus bundles over a circle with a unimodular hyperbolic gluing map A. The eigenfunctions of the corresponding Laplace-Beltrami operators are described in terms of the modified Mathieu functions. It is shown that the multiplicities of the eigenvalues are the same for generic values of the parameters in the metric and are directly related to the number of representations of an integer by a given indefinite binary quadratic form. As a result the spectral statistics is shown to disagree with the Berry-Tabor conjecture. The topological nature of the monodromy for both classical and quantum systems on Sol-manifolds is demonstrated.

math-ph↗

arXiv2005

Bernoulli numbers and solitons

We present a new formula for the Bernoulli numbers as the following integral $$B_{2m} =\frac{(-1)^{m-1}}{2^{2m+1}} \int_{-\infty}^{+\infty} (\frac{d^{m-1}}{dx^{m-1}} {sech}^2 x)^2dx. $$ This formula is motivated by the results of Fairlie and Veselov, who discovered the relation of Bernoulli polynomials with soliton theory.

math.GM↗

arXiv2004

A remark on rational isochronous potentials

We consider the rational potentials of the one-dimensional mechanical systems, which have a family of periodic solutions with the same period (isochronous potentials). We prove that up to a shift and adding a constant all such potentials have the form $U(x) = 1/2 ω^2 x^2$ or $U(x) = 1/8 ω^2 x ^2 + c^2 x^{-2}.$

math-ph↗

arXiv2004

Coincident root loci and Jack and Macdonald polynomials for special values of the parameters

We consider the coincident root loci consisting of the polynomials with at least two double roots andpresent a linear basis of the corresponding ideal in the algebra of symmetric polynomials in terms of the Jack polynomials with special value of parameter $α= -2.$ As a corollary we present an explicit formula for the Hilbert-Poincarè series of this ideal and the generator of the minimal degree as a special Jack polynomial. A generalization to the case of the symmetric polynomials vanishing on the double shifted diagonals and the Macdonald polynomials specialized at $t^2 q = 1$ is also presented. We also give similar results for the interpolation Jack polynomials.

math.QA↗

arXiv2003

Generalised discriminants, deformed quantum Calogero-Moser system and Jack polynomials

It is shown that the deformed Calogero-Moser-Sutherland (CMS) operators can be described as the restrictions on certain affine subvarieties (called generalised discriminants) of the usual CMS operators for infinite number of particles. The ideals of these varieties are shown to be generated by the Jack symmetric functions related to the Young diagrams with special geometry. A general structure of the ideals which are invariant under the action of the quantum CMS integrals is discussed in this context. The shifted super-Jack polynomials are introduced and combinatorial formulas for them and for super-Jack polynomials are given.

math-ph↗

arXiv2003

Yang-Baxter maps and matrix solitons

New examples of the Yang-Baxter maps (or set-theoretical solutions to the quantum Yang-Baxter equation) on the Grassmannians arising from the theory of the matrix KdV equation are discussed. The Lax pairs for these maps are produced using the relations with the inverse scattering problem for the matrix Schrödinger operator.

math-ph↗

arXiv2003

Quasi-invariants and quantum integrals of the deformed Calogero--Moser systems

The rings of quantum integrals of the generalized Calogero-Moser systems related to the deformed root systems ${\cal A}_n(m)$ and ${\cal C}_n(m,l)$ with integer multiplicities and corresponding algebras of quasi-invariants are investigated. In particular, it is shown that these algebras are finitely generated and free as the modules over certain polynomial subalgebras (Cohen-Macaulay property). The proof follows the scheme proposed by Etingof and Ginzburg in the Coxeter case. For two-dimensional systems the corresponding Poincare series and the deformed $m$-harmonic polynomials are explicitly computed.

math-ph↗

arXiv2003

Deformed quantum Calogero-Moser problems and Lie superalgebras

The deformed quantum Calogero-Moser-Sutherland problems related to the root systems of the contragredient Lie superalgebras are introduced. The construction is based on the notion of the generalized root systems suggested by V. Serganova. For the classical series a recurrent formula for the quantum integrals is found, which implies the integrability of these problems. The corresponding algebras of the quantum integrals are investigated, the explicit formulas for their Poincare series for generic values of the deformation parameter are presented.

math-ph↗

arXiv2003

Discrete Lagrangian systems on the Virasoro group and Camassa-Holm family

We show that the continuous limit of a wide natural class of the right-invariant discrete Lagrangian systems on the Virasoro group gives the family of integrable PDE's containing Camassa-Holm, Hunter-Saxton and Korteweg-de Vries equations. This family has been recently derived by Khesin and Misiolek as Euler equations on the Virasoro algebra for $H^1_{α,β}$-metrics. Our result demonstrates a universal nature of these equations.

math-ph↗

arXiv2002

Yang-Baxter maps and integrable dynamics

The hierarchy of commuting maps related to a set-theoretical solution of the quantum Yang-Baxter equation (Yang-Baxter map) is introduced. They can be considered as dynamical analogues of the monodromy and/or transfer-matrices. The general scheme of producing Yang-Baxter maps based on matrix factorisation is discussed in the context of the integrability problem for the corresponding dynamical systems. Some examples of birational Yang-Baxter maps coming from the theory of the periodic dressing chain and matrix KdV equation are discussed.

math.QA↗

arXiv2002

On the real zeroes of the Hurwitz zeta-function and Bernoulli polynomials

The behaviour of real zeroes of the Hurwitz zeta function $$ζ(s,a)=\sum_{r=0}^{\infty}(a+r)^{-s}\qquad\qquad a > 0$$ is investigated. It is shown that $ζ(s,a)$ has no real zeroes $(s=σ,a)$ in the region $a >\frac{-σ}{2πe}+\frac{1}{4πe}\log (-σ) +1$ for large negative $σ$. In the region $0 < a < \frac{-σ}{2πe}$ the zeroes are asymptotically located at the lines $σ+ 4a + 2m =0$ with integer $m$. If $N(p)$ is the number of real zeroes of $ζ(-p,a)$ with given $p$ then $$\lim_{p\to\infty}\frac{N(p)}{p}=\frac{1}{πe}.$$ As a corollary we have a simple proof of Inkeri's result that the number of real roots of the classical Bernoulli polynomials $B_n(x)$ for large $n$ is asymptotically equal to $\frac{2n}{πe}$.

math.GM↗

arXiv2002

On generalisations of Calogero-Moser-Sutherland quantum problem and WDVV equations

It is proved that if the Schrödinger equation $L ψ= λψ$ of Calogero-Moser-Sutherland type with $$L = -Δ+ \sum\limits_{α\in{\cal A}_{+}} \frac{m_α(m_α+1) (α,α)}{\sin^{2}(α,x)}$$ has a solution of the product form $ψ_0 = \prod_{α\in {\cal {A}_+}} \sin^{-m_α}(α,x),$ then the function $F(x) =\sum\limits_{α\in \cal {A}_{+}} m_α (α,x)^2 {\rm log} (α,x)^2$ satisfies the generalised WDVV equations.

math-ph↗

arXiv2001

Quasiinvariants of Coxeter groups and m-harmonic polynomials

The space of m-harmonic polynomials related to a Coxeter group G and a multiplicity function m on its root system is defined as the joint kernel of the properly gauged invariant integrals of the corresponding generalised quantum Calogero-Moser problem. The relation between this space and the ring of all quantum integrals of this system (which is isomorphic to the ring of corresponding quasiinvariants) is investigated.

math-ph↗

arXiv2001

On geometry of a special class of solutions to generalised WDVV equations

A special class of solutions to the generalised WDVV equations related to a finite set of covectors is considered. We describe the geometric conditions ($\vee$-conditions) on such a set which are necessary and sufficient for the corresponding function to satisfy the generalised WDVV equations. These conditions are satisfied for all Coxeter systems but there are also other examples discovered in the theory of the generalised Calogero-Moser systems. As a result some new solutions for the generalized WDVV equations are found.

hep-th↗

arXiv2001

Locus configurations and $\vee$-systems

We present a new family of the locus configurations which is not related to $\vee$-systems thus giving the answer to one of the questions raised in \cite{V1} about the relation between the generalised quantum Calogero-Moser systems and special solutions of the generalised WDVV equations. As a by-product we have new examples of the hyperbolic equations satisfying the Huygens' principle in the narrow Hadamard's sense. Another result is new multiparameter families of $\vee$-systems which gives new solutions of the generalised WDVV equation.

math-ph↗

arXiv2000

On Stieltjes relations, Painlevé-IV hierarchy and complex monodromy

A generalisation of the Stieltjes relations for the Painlevé-IV transcendents and their higher analogues determined by the dressing chains is proposed. It is proven that if a rational function from a certain class satisfies these relations it must be a solution of some higher Painlevé-IV equation. The approach is based on the interpretation of the Stieltjes relations as local trivial monodromy conditions for certain Schrödinger equations in the complex domain. As a corollary a new class of the Schrödinger operators with trivial monodromy is constructed in terms of the Painlevé-IV transcendents.

math-ph↗

arXiv2000

On algebro-geometric Poisson brackets for the Volterra lattice

A generalization of the theory of algebro-geometric Poisson brackets on the space of finite-gap Schroedinger operators, developped by S. P. Novikov and A. P. Veselov, to the case of periodic zero-diagonal difference operators of second order is proposed. A necessary and sufficient condition for such a bracket to be compatible with higher Volterra flows is found.

math-ph↗

arXiv2000

Integrable Schrödinger operators with magnetic fields: factorisation method on curved surfaces

The factorisation method for Schrödinger operators with magnetic fields on a two-dimensional surface $M^2$ with non-trivial metric is investigated. This leads to the new integrable examples of such operators and brings a new look at some classical problems such as Dirac magnetic monopole and Landau problem. The global geometric aspects and related spectral properties of the operators from the factorisation chains are discussed in details. We also consider the Laplace transformations on a curved surface and extend the class of Schrödinger operators with two integrable levels introduced in the flat case by S.P.Novikov and one of the authors.

math-ph↗