20 research outputs found
The spectral curve of a quaternionic holomorphic line bundle over a 2-torus
A conformal immersion of a 2-torus into the 4-sphere is characterized by an
auxiliary Riemann surface, its spectral curve. This complex curve encodes the
monodromies of a certain Dirac type operator on a quaternionic line bundle
associated to the immersion. The paper provides a detailed description of the
geometry and asymptotic behavior of the spectral curve. If this curve has
finite genus the Dirichlet energy of a map from a 2-torus to the 2-sphere or
the Willmore energy of an immersion from a 2-torus into the 4-sphere is given
by the residue of a specific meromorphic differential on the curve. Also, the
kernel bundle of the Dirac type operator evaluated over points on the 2-torus
linearizes in the Jacobian of the spectral curve. Those results are presented
in a geometric and self contained manner.Comment: 36 page
Generalized Weierstrass Relations and Frobenius reciprocity
This article investigates local properties of the further generalized
Weierstrass relations for a spin manifold immersed in a higher dimensional
spin manifold from viewpoint of study of submanifold quantum mechanics. We
show that kernel of a certain Dirac operator defined over , which we call
submanifold Dirac operator, gives the data of the immersion. In the derivation,
the simple Frobenius reciprocity of Clifford algebras and plays
important roles.Comment: 17pages. to be published in Mathematical Physics, Analysis and
Geometr
The type numbers of closed geodesics
A short survey on the type numbers of closed geodesics, on applications of
the Morse theory to proving the existence of closed geodesics and on the recent
progress in applying variational methods to the periodic problem for Finsler
and magnetic geodesicsComment: 29 pages, an appendix to the Russian translation of "The calculus of
variations in the large" by M. Mors
On the Evolution Equation for Magnetic Geodesics
In this paper we prove the existence of long time solutions for the parabolic
equation for closed magnetic geodesics.Comment: In this paper we prove the existence of long time solutions for the
parabolic equation for closed magnetic geodesic
On Darboux-Treibich-Verdier potentials
It is shown that the four-parameter family of elliptic functions
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
On numerical study of the discrete spectrum of a two-dimensional Schrödinger operator with soliton potential
Abstract The discrete spectra of certain two-dimensional Schrödinger operators are numerically calculated. These operators are obtained by the Moutard transformation and have interesting spectral properties: their kernels are multi-dimensional and the deformations of potentials via the Novikov–Veselov equation (a two-dimensional generalization of the Korteweg–de Vries equation) lead to blowups. The calculations supply the numerical evidence for some statements about the integrable systems related to a 2D Schrödinger operator. The numerical scheme is applicable to a general 2D Schrödinger operator with fast decaying potential