Effective masses for zigzag nanotubes in magnetic fields

We consider the Schr\"odinger operator with a periodic potential on quasi-1D models of zigzag single-wall carbon nanotubes in magnetic field. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We obtain identities and a priori estimates in terms of effective masses and gap lengths

Inverse problems generated by conformal mappings on complex plane with parallel slits

We study the properties of a conformal mapping z(k; h) from K(h) = C n [\Gamma n where \Gamma n = [u n \Gamma ijh n j; u n + ijh n j]; n 2 Z is a vertical slit and h = fh n g 2 ` 2 R , onto the complex plane with horizontal slits fl n ae R; n 2 Z, with the asymptotics z(iv; h) = iv + (iQ 0 (h) + o(1))=v; v ! +1. Here u n+1 \Gamma u n ? 1; n 2 Z, and the Dirichlet integral Q 0 (h) = RR C jz 0 (k; h) \Gamma 1j 2 dudv=(2) ! 1; k = u + iv. Introduce the sequences l = fl n g; J = fJ n g; where l n = jfl n j sign h n , and J n = jJ n j sign h n ; J 2 n = R \Gamma n j Im z(k; h)jjdkj=. The following results are obtain: 1) an analytic continuation of the function z(\Delta; \Delta) : K(h) \Theta ff : kf \Gamma hk ! rg ! C onto the domain K(h) \Theta ff : kf \Gamma hkC ! rg for h 2 ` 2 R and some r ? 0, and the Lowner equation for z(k; h) when the height of some slit h n is changed, 2) an analytic continuation of the functional Q 0 : ` 2 R ! R+ in the domain ff : k Im fk ! r..

Marchenko-Ostrovski mappings for periodic Jacobi matrices

We consider the 1D periodic Jacobi matrices. The spectrum of this operator is purely absolutely continuous and consists of intervals separated by gaps. We solve the inverse problem (including characterization) in terms of vertical slits on the quasimomentum domain . Furthermore, we obtain a priori two-sided estimates for vertical slits in terms of Jacoby matrices

A priori estimates for the Hill and Dirac operators

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

Inverse problems generated by conformal mappings on complex plane with parallel slits

SIGLEAvailable from TIB Hannover: RR 1596(458) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman