37 research outputs found
Discrete -nets and Guichard nets
We provide a convincing discretisation of Demoulin's -surfaces along
with their specialisations to Guichard and isothermic surfaces with no loss of
integrable structure.Comment: 39 A4 page
Constructing solutions to the Bj\"orling problem for isothermic surfaces by structure preserving discretization
In this article, we study an analog of the Bj\"orling problem for isothermic
surfaces (that are more general than minimal surfaces): given a real analytic
curve in , and two analytic non-vanishing orthogonal
vector fields and along , find an isothermic surface that is
tangent to and that has and as principal directions of
curvature. We prove that solutions to that problem can be obtained by
constructing a family of discrete isothermic surfaces (in the sense of Bobenko
and Pinkall) from data that is sampled along , and passing to the limit
of vanishing mesh size. The proof relies on a rephrasing of the
Gauss-Codazzi-system as analytic Cauchy problem and an in-depth-analysis of its
discretization which is induced from the geometry of discrete isothermic
surfaces. The discrete-to-continuous limit is carried out for the Christoffel
and the Darboux transformations as well.Comment: 29 pages, some figure
Generalized isothermic lattices
We study multidimensional quadrilateral lattices satisfying simultaneously
two integrable constraints: a quadratic constraint and the projective Moutard
constraint. When the lattice is two dimensional and the quadric under
consideration is the Moebius sphere one obtains, after the stereographic
projection, the discrete isothermic surfaces defined by Bobenko and Pinkall by
an algebraic constraint imposed on the (complex) cross-ratio of the circular
lattice. We derive the analogous condition for our generalized isthermic
lattices using Steiner's projective structure of conics and we present basic
geometric constructions which encode integrability of the lattice. In
particular, we introduce the Darboux transformation of the generalized
isothermic lattice and we derive the corresponding Bianchi permutability
principle. Finally, we study two dimensional generalized isothermic lattices,
in particular geometry of their initial boundary value problem.Comment: 19 pages, 11 figures; v2. some typos corrected; v3. new references
added, higlighted similarities and differences with recent papers on the
subjec
The PT-symmetric brachistochrone problem, Lorentz boosts and non-unitary operator equivalence classes
The PT-symmetric (PTS) quantum brachistochrone problem is reanalyzed as
quantum system consisting of a non-Hermitian PTS component and a purely
Hermitian component simultaneously. Interpreting this specific setup as
subsystem of a larger Hermitian system, we find non-unitary operator
equivalence classes (conjugacy classes) as natural ingredient which contain at
least one Dirac-Hermitian representative. With the help of a geometric analysis
the compatibility of the vanishing passage time solution of a PTS
brachistochrone with the Anandan-Aharonov lower bound for passage times of
Hermitian brachistochrones is demonstrated.Comment: 12 pages, 2 figures, strongly extended versio