arXiv subjects

Dmitry Zakharov

Publications and source records attributed to Dmitry Zakharov.

27 records · Page 2Linked to original sources

arXiv2018

On symmetric primitive potentials

The concept of a primitive potential for the Schroedinger operator on the line was introduced in [2,3,4]. Such a potential is determined by a pair of positive functions on a finite interval, called the dressing functions, which are not uniquely determined by the potential. The potential is constructed by solving a contour problem on the complex plane. In this paper, we consider a reduction where the dressing functions are equal. We show that in this case, the resulting potential is symmetric, and describe how to analytically compute the potential as a power series. In addition, we establish that if the dressing functions are both equal to one, then the resulting primitive potential is the elliptic one-gap potential.

nlin.SI↗

arXiv2015

Closed symmetric 3-differentials on complex surfaces

We give a necessary and sufficient condition for a non-degenerate symmetric 3-differential with nonzero Blaschke curvature on a complex surface to be locally representable as a product of three closed holomorphic 1-forms. We give two versions of this condition corresponding to different choices of coordinates, one of which defines a coordinate-free differential operator, answering a question of Bogomolov and de Oliveira.

math.AG↗

arXiv2015

Non-periodic one-gap potentials in quantum mechanics

We describe a broad class of bounded non-periodic potentials in one-dimensional stationary quantum mechanics having the same spectral properties as periodic potentials. The spectrum of the corresponding Schroedinger operator consists of a finite or infinite number of allowed bands separated by gaps. In this letter we consider the simplest class of potentials, whose spectra consist of an interval on the negative semiaxis and the entire positive axis. The potentials are reflectionless, and a particle with positive energy moves freely in both directions. The potential is constructed as a limit of Bargmann potentials and is determined by a Riemann-Hilbert problem, which is equivalent to a pair of singular integral equations that can be efficiently solved using numerical techniques.

nlin.SI↗

arXiv2015

Dense PGL-orbits in products of Grassmannians

In this paper, we find some necessary and sufficient conditions on the dimension vector $\underline{\bf{d}} = (d_1,..., d_k; n)$ so that the diagonal action of $\mathbb{P}GL(n)$ on $\prod_{i=1}^k Gr(d_i;n)$ has a dense orbit. Consequently, we obtain some algorithms for finding dense and sparse dimension vectors and classify dense dimension vectors with small length or size. We also characterize the dense dimension vectors of the form $(d,d,..., d; n)$.

math.AG↗

arXiv2013

The zero section of the universal semiabelian variety, and the double ramification cycle

We study the Chow ring of the boundary of the partial compactification of the universal family of principally polarized abelian varieties (ppav). We describe the subring generated by divisor classes, and compute the class of the partial compactification of the universal zero section, which turns out to lie in this subring. Our formula extends the results for the zero section of the universal uncompactified family. The partial compactification of the universal family of ppav can be thought of as the first two boundary strata in any toroidal compactification of the moduli space of ppav. Our formula provides a first step in a program to understand the Chow groups of toroidal compactifications of the moduli of ppav, especially of the perfect cone compactification, by induction on genus. By restricting to the locus of Jacobians of curves, our results extend the results of Hain on the double ramification (two-branch-point) cycle.

math.AG↗

arXiv2012

The double ramification cycle and the theta divisor

We compute the classes of universal theta divisors of degrees zero and g-1 over the Deligne-Mumford compactification of the moduli space of curves, with various integer weights on the points, in particular reproving a recent result of Müller. We also obtain a formula for the class in the Chow ring of the moduli space of curves of compact type of the double ramification locus, given by the condition that a fixed linear combination of the marked points is a principal divisor, reproving a recent result of Hain. Our approach for computing the theta divisor is more direct, via test curves and the geometry of the theta divisor, and works easily over the entire Deligne-Mumford compactification. We use our extended result in another paper to study the partial compactification of the double ramification cycle.

math.AG↗

arXiv2012

A discrete analogue of the modified Novikov-Veselov hierarchy

We construct a discrete analogue of the integrable two-dimensional Dirac operator and describe the spectral properties of its eigenfunctions. We construct an integrable discrete analogue of the modified Novikov-Veselov hierarchy. We derive the first two equations of the hierarchy and give explicit formulas for the eigenfunctions in terms of the theta-functions of the associated spectral curve.

nlin.SI↗

arXiv2010

A note on critical points of integrals of soliton equations

We consider the problem of extending the integrals of motion of soliton equations to the space of all finite-gap solutions. We consider the critical points of these integrals on the moduli space of Riemann surfaces with marked points and jets of local coordinates. We show that the solutions of the corresponding variational problem have an explicit description in terms of real-normalized differentials on the spectral curve. Such conditions have previously appeared in a number of problems of mathematical physics.

math.AG↗

arXiv2009

The Weierstrass representation of discrete isotropic surfaces in $R^{2,1}$, $R^{3,1}$ and $R^{2,2}$

Using an integrable discrete Dirac operator, we construct a discrete version of the Weierstrass representation of time-like surfaces parametrized along isotropic directions in $R^{2,1}$, $R^{3,1}$ and $R^{2,2}$. The corresponding discrete surfaces have isotropic edges. We show that any discrete surface satisfying a general monotonicity condition and having isotropic edges admits such a representation.

math.DG↗