Djakov, Plamen Borissov and Mityagin, Boris Samuel (2012) Unconditional convergence of spectral decompositions of 1D Dirac operators with regular boundary conditions. Indiana University Mathematics Journal, 61 (1). pp. 359398. ISSN 00222518
This is the latest version of this item.
Official URL: http://dx.doi.org/10.1512/iumj.2012.61.4531
Abstract
Onedimensional Dirac operators
Lbc(v)y = i(1 0 0 1) dy/dx + v(x)y, y = (y(1)y(2)), x is an element of [0, pi],
considered with L2potentials v (x) = (0 Q(x) P(x)0) and subject to regular boundary conditions (bc), have discrete spectrum.
For strictly regular bc, it is shown that every eigenvalue of the free operator Lbc(0) is simple and has the form lambda(0)(k,alpha) = k + tau(alpha), where alpha is an element of {1, 2}, k is an element of 2Z and tau(alpha) = tau(alpha)(bc); if vertical bar k vertical bar > N(v, bc), each of the discs Dk(alpha) = {z : vertical bar z  lambda(0)(k,alpha) vertical bar < rho = rho(bc)}, alpha is an element of {1, 2}, contains exactly one simple eigenvalue lambda(k,alpha)< of Lbc (v) and (lambda(k,alpha)  lambda(0)(k,alpha))(k is an element of 2Z) is an l(2)sequence. Moreover, it is proven that the root projections Pn,Palpha = 1/(2 pi i) integral(partial derivative Dn)alpha (z  Lbc (v))(1) dz satisfy the BariMarkus condition
Sigma(vertical bar n vertical bar>N) parallel to Pn,Palpha  Pn,alpha(0)parallel to(2) <infinity, n is an element of 2Z,
where Pn,alpha(0) are the root projections of the free operator Lbc(0). Hence, for strictly regular bc, there is a Riesz basis consisting of root functions (all but finitely many being eigenfunctions). Similar results are obtained for regular but not strictly regular bcthen in general there is no Riesz basis consisting of root functions, but we prove that the corresponding system of twodimensional root projections is a Riesz basis of projections.
Item Type:  Article 

Uncontrolled Keywords:  Dirac operators; Riesz bases; regular boundary conditions 
Subjects:  Q Science > QA Mathematics > QA299.6433 Analysis 
Divisions:  Faculty of Engineering and Natural Sciences > Basic Sciences > Mathematics Faculty of Engineering and Natural Sciences 
Depositing User:  Plamen Borissov Djakov 
Date Deposited:  29 Apr 2013 14:52 
Last Modified:  01 Aug 2019 10:25 
URI:  https://research.sabanciuniv.edu/id/eprint/21549 
Available Versions of this Item

Unconditional convergence of spectral decompositions of 1D dirac operators with regular boundary conditions. (deposited 05 Dec 2011 15:23)
 Unconditional convergence of spectral decompositions of 1D Dirac operators with regular boundary conditions. (deposited 29 Apr 2013 14:52) [Currently Displayed]