3,400 research outputs found
Norm-inflation for periodic NLS equations in negative Sobolev spaces
In this paper we consider Schr{\"o}dinger equations with nonlinearities of
odd order 2 + 1 on T^d. We prove that for d2, they are
strongly illposed in the Sobolev space H^s for any s \textless{} 0, exhibiting
norm-inflation with infinite loss of regularity. In the case of the
one-dimensional cubic nonlinear Schr{\"o}dinger equation and its renormalized
version we prove such a result for H^s with s \textless{} --2/3.Comment: 18 page
A Canonical Quadratic Form on the Determinant Line of a Flat Vector Bundle
We introduce and study a canonical quadratic form, called the torsion
quadratic form, of the determinant line of a flat vector bundle over a closed
oriented odd-dimensional manifold. This quadratic form caries less information
than the refined analytic torsion, introduced in our previous work, but is
easier to construct and closer related to the combinatorial Farber-Turaev
torsion. In fact, the torsion quadratic form can be viewed as an analytic
analogue of the Poincare-Reidemeister scalar product, introduced by Farber and
Turaev. Moreover, it is also closely related to the complex analytic torsion
defined by Cappell and Miller and we establish the precise relationship between
the two. In addition, we show that up to an explicit factor, which depends on
the Euler structure, and a sign the Burghelea-Haller complex analytic torsion,
whenever it is defined, is equal to our quadratic form. We conjecture a formula
for the value of the torsion quadratic form at the Farber-Turaev torsion and
prove some weak version of this conjecture. As an application we establish a
relationship between the Cappell-Miller and the combinatorial torsions.Comment: 13 page
A Refinement of the Ray-Singer Torsion
This is a short version of math.DG/0505537. For an acyclic representation of
the fundamental group of a compact oriented odd-dimensional manifold, which is
close enough to a unitary representation, we define a refinement of the
Ray-Singer torsion associated to this representation. This new invariant can be
viewed as an analytic counterpart of the refined combinatorial torsion
introduced by Turaev.
The refined analytic torsion is a holomorphic function of the representation
of the fundamental group. When the representation is unitary, the absolute
value of the refined analytic torsion is equal to the Ray-Singer torsion, while
its phase is determined by the eta-invariant. The fact that the Ray-Singer
torsion and the eta-invariant can be combined into one holomorphic function
allows to use methods of complex analysis to study both invariants.Comment: 6 pages, to apper in Comptes rendus Acad. Sci. Pari
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