TY - JOUR
T1 - A Schauder and Riesz basis criterion for non-self-adjoint Schrödinger operators with periodic and antiperiodic boundary conditions
AU - Gesztesy, Fritz
AU - Tkachenko, Vadim
N1 - Funding Information:
We are very grateful to Anton Lunyov and Mark Malamud for an exceptionally careful and critical reading of our manuscript, which led to a number of improvements in our presentation. In particular, they suggested a modified version of the principal estimate in Lemma 4.2. We are also indebted to Yura Lyubarskii for helpful comments. V. Tkachenko gratefully acknowledges the award of a Miller Scholarship from the Department of Mathematics of the University of Missouri, Columbia, USA. He is indebted to the department for its great hospitality during his stay in the month of October 2010. He also gratefully acknowledges partial support by the Israel Science Foundation under grant 125/09.
PY - 2012/7/15
Y1 - 2012/7/15
N2 - Under the assumption that V∈L 2([0, π];dx), we derive necessary and sufficient conditions in terms of spectral data for (non-self-adjoint) Schrödinger operators -d 2/dx 2+V in L 2([0, π];dx) with periodic and antiperiodic boundary conditions to possess a Riesz basis of root vectors (i.e., eigenvectors and generalized eigenvectors spanning the range of the Riesz projection associated with the corresponding periodic and antiperiodic eigenvalues).We also discuss the case of a Schauder basis for periodic and antiperiodic Schrödinger operators -d 2/dx 2+V in L p([0, π];dx), p∈(1, ∞).
AB - Under the assumption that V∈L 2([0, π];dx), we derive necessary and sufficient conditions in terms of spectral data for (non-self-adjoint) Schrödinger operators -d 2/dx 2+V in L 2([0, π];dx) with periodic and antiperiodic boundary conditions to possess a Riesz basis of root vectors (i.e., eigenvectors and generalized eigenvectors spanning the range of the Riesz projection associated with the corresponding periodic and antiperiodic eigenvalues).We also discuss the case of a Schauder basis for periodic and antiperiodic Schrödinger operators -d 2/dx 2+V in L p([0, π];dx), p∈(1, ∞).
KW - Non-self-adjoint Hill operators
KW - Periodic and antiperiodic boundary conditions
KW - Primary
KW - Riesz basis
KW - Secondary
UR - http://www.scopus.com/inward/record.url?scp=84860642586&partnerID=8YFLogxK
U2 - 10.1016/j.jde.2012.04.002
DO - 10.1016/j.jde.2012.04.002
M3 - Article
AN - SCOPUS:84860642586
SN - 0022-0396
VL - 253
SP - 400
EP - 437
JO - Journal of Differential Equations
JF - Journal of Differential Equations
IS - 2
ER -