114,841 research outputs found
Perturbation theory for spectral gap edges of 2D periodic Schr\"odinger operators
We consider a two-dimensional periodic Schr\"odinger operator
with being the lattice of periods. We investigate the structure of the
edges of open gaps in the spectrum of . We show that under arbitrary small
perturbation periodic with respect to where is some
integer, all edges of the gaps in the spectrum of which are perturbation
of the gaps of become non-degenerate, i.e. are attained at finitely many
points by one band function only and have non-degenerate quadratic
minimum/maximum. We also discuss this problem in the discrete setting and show
that changing the lattice of periods may indeed be unavoidable to achieve the
non-degeneracy.Comment: 25 pages; several typos are fixed and comments are added; subsection
3.2 is expanded to include more detailed proof of Theorem 3.
The non-Abelian gauge theory of matrix big bangs
We study at the classical and quantum mechanical level the time-dependent
Yang-Mills theory that one obtains via the generalisation of discrete
light-cone quantisation to singular homogeneous plane waves. The non-Abelian
nature of this theory is known to be important for physics near the
singularity, at least as far as the number of degrees of freedom is concerned.
We will show that the quartic interaction is always subleading as one
approaches the singularity and that close enough to t=0 the evolution is driven
by the diverging tachyonic mass term. The evolution towards asymptotically flat
space-time also reveals some surprising features.Comment: 29 pages, 8 eps figures, v2: minor changes, references added: v3
small typographical changes
Perestroikas of Shocks and Singularities of Minimum Functions
The shock discontinuities, generically present in inviscid solutions of the
forced Burgers equation, and their bifurcations happening in the course of time
(perestroikas) are classified in two and three dimensions -- the
one-dimensional case is well known. This classification is a result of
selecting among all the perestroikas occurring for minimum functions depending
generically on time, the ones permitted by the convexity of the Hamiltonian of
the Burgers dynamics. Topological restrictions on the admissible perestroikas
of shocks are obtained. The resulting classification can be extended to the
so-called viscosity solutions of a Hamilton--Jacobi equation, provided the
Hamiltonian is convex.Comment: 20 pages, 8 figures, 3 tables; my e-mail: [email protected]
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