A priori estimates for the Hill and Dirac operators

Details A priori estimates for the Hill and Dirac operators A priori estimates for the Hill and Dirac operators

2007-01-16

arxiv.org/abs/math/0701437v1

Mathematics

Spectral Theory

Scientific paper

10.1134/S1061920808030038

Consider the Hill operator $Ty=-y''+q'(t)y$ in $L^2(\R)$, where $q\in L^2(0,1)$ is a 1-periodic real potential. The spectrum of $T$ is is absolutely continuous and consists of bands separated by gaps $\g_n,n\ge 1$ with length $|\g_n|\ge 0$. We obtain a priori estimates of the gap lengths, effective masses, action variables for the KDV. For example, if $\m_n^\pm$ are the effective masses associated with the gap $\g_n=(\l_n^-,\l_n^+)$, then $|\m_n^-+\m_n^+|\le C|\g_n|^2n^{-4}$ for some constant $C=C(q)$ and any $n\ge 1$. In order prove these results we use the analysis of a conformal mapping corresponding to quasimomentum of the Hill operator. That makes possible to reformulate the problems for the differential operator as the problems of the conformal mapping theory. Then the proof is based on the analysis of the conformal mapping and the identities. Moreover, we obtain the similar estimates for the Dirac operator.

Affiliated with

Also associated with

No associations

Rating

A priori estimates for the Hill and Dirac operators does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with A priori estimates for the Hill and Dirac operators, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A priori estimates for the Hill and Dirac operators will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-189043