Abstract
The Schrödinger operator with complex decaying potential on a lattice is considered. Trace formulas are derived on the basis of classical results of complex analysis. These formulas are applied to obtain global estimates of all zeros of the Fredholm determinant in terms of the potential.
Article PDF
Avoid common mistakes on your manuscript.
References
A. Borichev, L. Golinskii, and S. Kupin, Bull. London Math. Soc., 41:1 (2009), 117–123.
A. Boutet de Monvel and J. Sahbani, Rev. Math. Phys., 11:9 (1999), 1061–1078.
M. Demuth, M. Hansmann, and G. Katriel, J. Funct. Anal., 257:9 (2009), 2742–2759.
R. L. Frank, Bull. Lond. Math. Soc., 43:4 (2011), 745–750.
R. L. Frank, A. Laptev, E. H. Lieb, and R. Seiringer, Lett. Math. Phys., 77 (2006), 309–316.
R. L. Frank, A. Laptev, and O. Safronov, J. London Math. Soc., 94:2 (2016), 377–390.
R. L. Frank and J. Sabin, http://arxiv.org/abs/1404.2817.
R. L. Frank, A. Laptev, and R. Seiringer, “Spectral theory and analysis,” in: Oper. Theory Adv. Appl., vol. 214, Birkhäuser/Springer Basel AG, Basel, 2011, 39–44.
I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint operators in Hilbert Space, Transl. Math. Monographs, vol. 18, Amer. Math. Soc., Providence RI, 1965.
H. Isozaki and E. Korotyaev, Ann. Henri Poincaré, 13:4 (2012), 751–788.
H. Isozaki and H. Morioka, Inverse Probl. Imaging, 8:2 (2014), 475–489.
P. Koosis, Introduction to H p spaces, Cambridge Tracts in Mathematic, vol. 115, Cambridge University Press, Cambridge, 1998.
E. A. Kopylova, Algebra i Analiz, 21:5 (2009), 87–113; English transl.: St. Petersburg Math. J., 21:5 (2010), 743–760.
E. Korotyaev and N. Saburova, http://arxiv.org/abs/1507.06441.
A. Laptev and O. Safronov, Comm. Math. Phys., 292:1 (2009), 29–54.
E. H. Lieb and W. Thirring, in: Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann, Princeton University Press, Princeton, 1976, 269–303.
M. Malamud and H. Neidhardt, Adv. Math., 274 (2015), 736–832.
G. Rosenblum and M. Solomjak, Problems in Math. Analysis, No. 41, J. Math. Sci. N. Y., 159:2 (2009), 241–263.
O. Safronov, Bull. Lond. Math. Soc., 42:3 (2010), 452–456.
O. Safronov, Proc. Amer. Math. Soc., 138:6 (2010), 2107–2112.
W. Shaban and B. Vainberg, Appl. Anal., 80 (2001), 525–556.
M. Toda, Theory of Nonlinear Lattices, 2nd. ed., Springer-Verlag, Berlin, 1989.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of Professor M. S. Agranovich
Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 51, No. 3, pp. 81–86, 2017
Original Russian Text Copyright © by E. L. Korotyaev and A. Laptev
This work was supported by RSF grant No. 15-11-30007.
Rights and permissions
About this article
Cite this article
Korotyaev, E.L., Laptev, A. Trace formulas for a discrete Schrödinger operator. Funct Anal Its Appl 51, 225–229 (2017). https://doi.org/10.1007/s10688-017-0186-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10688-017-0186-z