Abstract
We establish a sharp uniform estimate on the size of the spectral clusters of the Landau Hamiltonian with (possibly complex-valued) \(L^p\) potentials as the cluster index tends to infinity.
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Cuenin, J.-C.: Eigenvalue bounds for Dirac and fractional Schrödinger operators with complex potentials. J. Funct. Anal. 272(7), 2987–3018 (2017)
Cuenin, J.-C., Kenig, C.E.: \(L^p\) resolvent estimates for magnetic Schrödinger operators with unbounded background fields. Commun. Partial Differ. Equ. 42(2), 235–260 (2017)
Davies, E.B.: Non-self-adjoint differential operators. Bull. Lond. Math. Soc. 34(5), 513–532 (2002)
Davies, E.B.: Linear Operators and Their Spectra, Volume 106 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2007)
Demuth, M., Hansmann, M., Katriel, G.: On the discrete spectrum of non-selfadjoint operators. J. Funct. Anal. 257(9), 2742–2759 (2009)
Demuth, M., Hansmann, M., Katriel, G.: Eigenvalues of non-selfadjoint operators: a comparison of two approaches. In: Mathematical Physics, Spectral Theory and Stochastic Analysis, Volume 232 of Oper. Theory Adv. Appl., pp. 107–163. Birkhäuser/Springer Basel AG, Basel (2013)
Dos Santos Ferreira, D., Kenig, C.E., Salo, M.: Determining an unbounded potential from Cauchy data in admissible geometries. Commun. Partial Differ. Equ. 38(1), 50–68 (2013)
Frank, R.L., Sabin, J.: Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates. ArXiv e-prints, April (2014)
Frank, R.L.: Eigenvalue bounds for Schrödinger operators with complex potentials. Bull. Lond. Math. Soc. 43(4), 745–750 (2011)
Frank, R.L., Laptev, A., Lieb, E.H., Seiringer, R.: Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials. Lett. Math. Phys. 77(3), 309–316 (2006)
Gohberg, I., Goldberg, S., Kaashoek, M. A.: Classes of linear operators. Vol. I, volume 49 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel (1990)
Jerison, D.: Carleman inequalities for the Dirac and Laplace operators and unique continuation. Adv. Math. 62(2), 118–134 (1986)
Jerison, D., Kenig, C.E.: Unique continuation and absence of positive eigenvalues for Schrödinger operators. Ann. Math. (2) 121(3), 463–494 (1985). With an appendix by E. M. Stein
Kenig, C.E., Ruiz, A., Sogge, C.D.: Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators. Duke Math. J. 55(2), 329–347 (1987)
Kenig, C.E.: Restriction theorems, Carleman estimates, uniform Sobolev inequalities and unique continuation. In: Harmonic analysis and partial differential equations (El Escorial, 1987), volume 1384 of Lecture Notes in Math., pp. 69–90. Springer, Berlin (1989)
Koch, H., Ricci, F.: Spectral projections for the twisted Laplacian. Stud. Math. 180(2), 103–110 (2007)
Koch, H., Tataru, D.: Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients. Commun. Pure Appl. Math. 54(3), 339–360 (2001)
Koch, H., Tataru, D.: Dispersive estimates for principally normal pseudodifferential operators. Commun. Pure Appl. Math. 58(2), 217–284 (2005)
Koch, H., Tataru, D.: Carleman estimates and absence of embedded eigenvalues. Commun. Math. Phys. 267(2), 419–449 (2006)
Korotyaev, E., Pushnitski, A.: A trace formula and high-energy spectral asymptotics for the perturbed Landau Hamiltonian. J. Funct. Anal. 217(1), 221–248 (2004)
Laptev, A., Safronov, O.: Eigenvalue estimates for Schrödinger operators with complex potentials. Commun. Math. Phys. 292(1), 29–54 (2009)
Lungenstrass, T., Raikov, G.: A trace formula for long-range perturbations of the Landau Hamiltonian. Ann. Henri Poincaré 15(8), 1523–1548 (2014)
Pushnitski, A., Raikov, G., Villegas-Blas, C.: Asymptotic density of eigenvalue clusters for the perturbed Landau Hamiltonian. Commun. Math. Phys. 320(2), 425–453 (2013)
Raĭkov, G.D.: Eigenvalue asymptotics for the Schrödinger operator with homogeneous magnetic potential and decreasing electric potential. I. Behaviour near the essential spectrum tips. Commun. Partial Differ. Equ. 15(3), 407–434 (1990)
Sambou, D.: Lieb-Thirring type inequalities for non-self-adjoint perturbations of magnetic Schrödinger operators. J. Funct. Anal. 266(8), 5016–5044 (2014)
Sogge, C.D.: Concerning the \(L^p\) norm of spectral clusters for second-order elliptic operators on compact manifolds. J. Funct. Anal. 77(1), 123–138 (1988)
Tao, T.: Nonlinear dispersive equations, volume 106 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis
Trefethen, L.N., Embree, M.: Spectra and pseudospectra. In: The behavior of nonnormal matrices and operators. Princeton University Press, Princeton (2005)
Wolff, T.H.: A property of measures in \({ R}^N\) and an application to unique continuation. Geom. Funct. Anal. 2(2), 225–284 (1992)
Yajima, K.: Schrödinger evolution equations with magnetic fields. J. Anal. Math. 56, 29–76 (1991)
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Cuenin, JC. Sharp Spectral Estimates for the Perturbed Landau Hamiltonian with \(\varvec{L^{p}}\) Potentials. Integr. Equ. Oper. Theory 88, 127–141 (2017). https://doi.org/10.1007/s00020-017-2367-9
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DOI: https://doi.org/10.1007/s00020-017-2367-9