arXiv subjects
Publications and source records attributed to A. P. Veselov.
We study the spectral properties of the harmonic modification of the quantum mechanical Kronig-Penney model motivated by the theory of the dressing chain. In particular, we show that in the special cases the corresponding ground states can be given explicitly and all spectral gaps have the same length with the ends expressed as zeros of the corresponding Weber-Hermite functions.
We show that the Markov fractions introduced recently by Springborn coincide with the index of the Cohn matrices defined by Aigner. This provides a simple concatenation rule for the corresponding continued fractions on the Conway topograph. We also discuss the $q$-deformation of Markov fractions in the sense of Morier-Genoud and Ovsienko, as well as their ``metallic" version motivated by the work of Spalding and the author.
We show that the equations of motion of the rigid body about centre of mass in the Newtonian field with a quadratic potential are special reductions of period-one closure of the Darboux dressing chain for the Schrödinger operators with matrix potentials. We show that the corresponding matrix Schrödinger operators are maximally finite-gap (in the sense that for all sufficiently large energies all solutions of the corresponding Schrödinger equation are bounded) and describe their spectrum explicitly. The general $2\times 2$-matrix case of the dressing chain, providing also some exotic matrix versions of the harmonic oscillator, is discussed in more detail.
Markov polynomials are the Laurent-polynomial solutions of the generalised Markov equation $$X^2 + Y^2 + Z^2 = kXYZ, \quad k=\frac{x^2 + y^2 + z^2}{x y z}$$ which are the results of cluster mutations applied to the initial triple $(x, y, z)$. They were first introduced and studied by Itsara, Musiker, Propp and Viana, who proved, in particular, that their coefficients are non-negative integers. We study the coefficients of Markov polynomials as functions on the corresponding Newton polygons, proposing several new conjectures. Some of these conjectures are proved for the special cases of Markov polynomials corresponding to Fibonacci and Pell numbers.
We establish an intriguing relation of the smooth theta divisor $Θ^n$ with permutohedron $Π^n$ and the corresponding toric variety $X_Π^n.$ In particular, we show that the generalised Todd genus of the theta divisor $Θ^n$ coincides with $h$-polynomial of permutohedron $Π^n$ and thus is different from the same genus of $X_Π^n$ only by the sign $(-1)^n.$ As an application we find all the Hodge numbers of the theta divisors in terms of the Eulerian numbers. We reveal also interesting numerical relations between theta-divisors and Tomei manifolds from the theory of the integrable Toda lattice.
We show that the Markov fractions introduced recently by Boris Springborn are precisely the slopes of the exceptional vector bundles on $\mathbb P^2$ studied in 1980s by Drèzet and Le Potier and by Rudakov. In particular, we provide a simpler proof of Rudakov's result claiming that the ranks of the exceptional bundles on $\mathbb P^2$ are Markov numbers.
We use the differential algebra of polytopes to explain the known remarkable relation of the combinatorics of the associahedra and permutohedra with the universal compositional and multiplicative inversion formulas for the formal power series. This approach allows to single out the associahedra and permutohedra among all graph-associahedra and emphasizes the significance of the differential equations for special sequences of simple polytopes derived earlier by one of the authors. We discuss also the link with the geometry of Deligne-Mumford moduli spaces $\bar M_{0,n}$ and the interpretation of the combinatorics of cyclohedra in relation with the classical Faà di Bruno's formula.
A few years ago Morier-Genoud and Ovsienko introduced an interesting quantization of the real numbers as certain power series in a quantization parameter $q.$ It is known now that the golden ratio has minimal radius among all these series. We study the rational numbers having maximal radius of convergence equal to 1, which we call Kronecker fractions. We prove that the corresponding continued fraction expansions must be palindromic and describe all Kronecker fractions with prime denominators. We found several infinite families of Kronecker fractions and all Kronecker fractions with denominator less than 5000. We also comment on the irrational case and on the relation with braids, rational knots and links.
We show that the smooth theta divisors of general principally polarised abelian varieties can be chosen as irreducible algebraic representatives of the coefficients of the Chern-Dold character in complex cobordisms and describe the action of the Landweber-Novikov operations on them. We introduce a quantisation of the complex cobordism theory with the dual Landweber-Novikov algebra as the deformation parameter space and show that the Chern-Dold character can be interpreted as the composition of quantisation and dequantisation maps. Some smooth real-analytic representatives of the cobordism classes of theta divisors are described in terms of the classical Weierstrass elliptic functions. The link with the Milnor-Hirzebruch problem about possible characteristic numbers of irreducible algebraic varieties is discussed.
The geodesic problem on the compact threefolds with the Riemannian metric of Bianchi-VII$_0$ type is studied in both classical and quantum cases. We show that the problem is integrable and describe the eigenfunctions of the corresponding Laplace-Beltrami operators explicitly in terms of the Mathieu functions with parameter depending on the lattice values of some binary quadratic forms. We use the results from number theory to discuss the level spacing statistics in relation with the Berry-Tabor conjecture and compare the situation with Bianchi-VI$_0$ case (Sol-case in Thurston's classification) and with Bianchi-IX case, corresponding to the classical Euler top.
Recently Valentin Ovsienko introduced a ``shadow" version of the celebrated Markov triples as the solutions of certain version of Markov equation over dual numbers. We will discuss similar question for the Mordell Diophantine equation $$ X^2+Y^2+Z^2=2XYZ+1. $$ We will see that the shadows of the special solution $(1,1,1)$ can be described using the original Conway topograph of the values of binary quadratic forms. The shadows of other Mordell triples will be written explicitly in terms of the corresponding solutions of Pell's equation. Their growth along the paths on the Conway topograph is described in terms of the Lyapunov function of the Euclid tree.
We introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let $Γ$ be a finite index subgroup of $\mathrm{SL}(2,\mathbb{Z})$ with an action on a complex simple Lie algebra $\mathfrak g$, which can be extended to $\mathrm{SL}(2,\mathbb{C})$. We show that the Lie algebra of the corresponding $\mathfrak{g}$-valued modular forms is isomorphic to the extension of $\mathfrak{g}$ over the usual modular forms. This establishes a modular analogue of a well-known result by Kac on twisted loop algebras. The case of principal congruence subgroups $Γ(N), \, N\leq 6$ are considered in more details in relation to the classical results of Klein and Fricke and the celebrated Markov Diophantine equation. We finish with a brief discussion of the extensions and representations of these Lie algebras.
Lassalle and Nekrasov discovered in the 1990s a surprising correspondence between the rational Calogero-Moser system with a harmonic term and its trigonometric version. We present a conceptual explanation of this correspondence using the rational Cherednik algebra and establish its quasi-invariant extension. More specifically, we consider configurations $\mathcal A$ of real hyperplanes with multiplicities admitting the rational Baker-Akhiezer function and use this to introduce a new class of non-symmetric polynomials, which we call $\mathcal A$-Hermite polynomials. These polynomials form a linear basis in the space of $\mathcal A$-quasi-invariants, which is an eigenbasis for the corresponding generalised rational Calogero-Moser operator with harmonic term. In the case of the Coxeter configuration of type $A_N$ this leads to a quasi-invariant version of the Lassalle-Nekrasov correspondence and its higher order analogues.
We classify certain integrable (both classical and quantum) generalisations of Dirac magnetic monopole on topological sphere $S^2$ with constant magnetic field, completing the previous local results by Ferapontov, Sayles and Veselov. We show that there are two integrable families of such generalisations with integrals, which are quadratic in momenta. The first family corresponds to the classical Clebsch systems, which can be interpreted as Dirac magnetic monopole in harmonic electric field. The second family is new and can be written in terms of elliptic functions on sphere $S^2$ with very special metrics.
The integrability of the geodesic flow on the three-folds $\mathcal M^3$ admitting $SL(2,\mathbb R)$-geometry in Thurston's sense is investigated. The main examples are the quotients $\mathcal M^3_Γ=Γ\backslash PSL(2,\mathbb R)$, where $Γ\subset PSL(2,\mathbb R)$ is a cofinite Fuchsian group. We show that the corresponding phase space $T^*M_Γ^3$ contains two open regions with integrable and chaotic behaviour with zero and positive topological entropy respectively. As a concrete example we consider the case of modular 3-fold with the modular group $Γ=PSL({2,\mathbb Z})$, when $\mathcal M^3_Γ$ is known to be homeomorphic to the complement of a trefoil knot $\mathcal K$ in 3-sphere. Ghys proved a remarkable fact that the lifts of the periodic geodesics to the modular surface to $\mathcal M^3_Γ$ produce the same isotopy class of knots, which appeared in the chaotic version of the celebrated Lorenz system and were extensively studied by Birman and Williams. We show that in the integrable limit of the geodesic system on $\mathcal M^3_Γ$ they are replaced by the simple class of cable knots of trefoil.
We study the Lyapunov exponents $Λ(x)$ for Markov dynamics as a function of path determined by $x\in \mathbb RP^1$ on a binary planar tree, describing the Markov triples and their "tropical" version - Euclid triples. We show that the corresponding Lyapunov spectrum is $[0, \ln φ]$, where $φ$ is the golden ratio, and prove that on the Markov-Hurwitz set $\mathbb{X}$ of the most irrational numbers the corresponding function $Λ_\mathbb{X}$ is monotonically increasing and in the Farey parametrization is convex.
We study the growth of the values of binary quadratic forms $Q$ on a binary planar tree as it was described by Conway. We show that the corresponding Lyapunov exponents $Λ_Q(x)$ as a function of the path determined by $x\in \mathbb RP^1$ are twice the values of the corresponding exponents for the growth of Markov numbers \cite{SV}, except for the paths corresponding to the Conway rivers, when $Λ_Q(x)=0.$ The relation with Galois results about continued fraction expansions for quadratic irrationals is explained and interpreted geometrically.
We present a new family of integrable versions of the Euler two-centre problem on two-dimensional sphere in the presence of the Dirac magnetic monopole of arbitrary charge. The new systems have very special algebraic potential and additional integral quadratic in momenta, both in classical and quantum versions.