Duke Mathematical Journal

Schrödinger operators with many bound states

Abstract

Consider the Schrödinger operators $H_{\pm}=-d^2/dx^2\pm V(x)$. We present a method for estimating the potential in terms of the negative eigenvalues of these operators. Among the applications are inverse Lieb-Thirring inequalities and several sharp results concerning the spectral properties of $H_{\pm}$

Article information

Source
Duke Math. J., Volume 136, Number 1 (2007), 51-80.

Dates
First available in Project Euclid: 4 December 2006

https://projecteuclid.org/euclid.dmj/1165244879

Digital Object Identifier
doi:10.1215/S0012-7094-07-13612-3

Mathematical Reviews number (MathSciNet)
MR2271295

Zentralblatt MATH identifier
0476.03047

Citation

Damanik, David; Remling, Christian. Schrödinger operators with many bound states. Duke Math. J. 136 (2007), no. 1, 51--80. doi:10.1215/S0012-7094-07-13612-3. https://projecteuclid.org/euclid.dmj/1165244879

References

• M. Christ and A. Kiselev, WKB and spectral analysis of one-dimensional Schrödinger operators with slowly varying potentials, Comm. Math. Phys. 218 (2001), 245--262.
• —, WKB asymptotic behavior of almost all generalized eigenfunctions for one-dimensional Schrödinger operators with slowly decaying potentials, J. Funct. Anal. 179 (2001), 426--447.
• R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, Interscience, New York, 1953.
• D. Damanik, D. Hundertmark, R. Killip, and B. Simon, Variational estimates for discrete Schrödinger operators with potentials of indefinite sign, Comm. Math. Phys. 238 (2003), 545--562.
• D. Damanik and R. Killip, Half-line Schrödinger operators with no bound states, Acta Math. 193 (2004), 31--72.
• D. Damanik, R. Killip, and B. Simon, Schrödinger operators with few bound states, Comm. Math. Phys. 258 (2005), 741--750.
• P. Deift and R. Killip, On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials, Comm. Math. Phys. 203 (1999), 341--347.
• S. A. Denisov, On the application of some of M. G. Krein's results to the spectral analysis of Sturm-Liouville operators, J. Math. Anal. Appl. 261 (2001), 177--191.
• M. S. P. Eastham, The Asympotic Solution of Linear Differential Systems, London Math. Soc. Monogr. (N.S.) 4, Oxford Univ. Press, New York, 1989.
• K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Math. 85, Cambridge Univ. Press, Cambridge, 1986.
• C. L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), 129--206.
• V. Glaser, H. Grosse, and A. Martin, Bounds on the number of eigenvalues of the Schrödinger operator, Comm. Math. Phys. 59 (1978), 197--212.
• C. Jacobi, Zur Theorie der Variations-Rechnung und der Differential-Gleichungen, J. Reine Angew. Math. 17 (1837), 68--82.
• R. Killip and B. Simon, Sum rules for Jacobi matrices and their applications to spectral theory, Ann. of Math. (2) 158 (2003), 253--321.
• A. Kiselev, Y. Last, and B. Simon, Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators, Comm. Math. Phys. 194 (1998), 1--45.
• M. G. KreĭN, Continuous analogues of propositions on polynomials orthogonal on the unit circle (in Russian), Dokl. Akad. Nauk SSSR (N.S.) 105 (1955), 637--640.
• S. Kupin, On a spectral property of Jacobi matrices, Proc. Amer. Math. Soc. 132 (2004), 1377--1383.
• A. Laptev, S. Naboko, and O. Safronov, On new relations between spectral properties of Jacobi matrices and their coefficients, Comm. Math. Phys. 241 (2003), 91--110.
• A. Laptev and T. Weidl, Recent results on Lieb-Thirring inequalities'' in Journées Équations aux Dérivées Partielles'' (La Chapelle sur Erdre, France, 2000), Univ. Nantes, Nantes, 2000, exp. no. 20.
• C. Muscalu, T. Tao, and C. Thiele, A Carleson theorem for a Cantor group model of the scattering transform, Nonlinearity 16 (2003), 219--246.
• —, A counterexample to a multilinear endpoint question of Christ and Kiselev, Math. Res. Lett. 10 (2003), 237--246.
• F. Nazarov, F. Peherstorfer, A. Volberg, and P. Yuditskii, On generalized sum rules for Jacobi matrices, Int. Math. Res. Not. 2005, no. 3, 155--186.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV: Analysis of Operators. Academic Press, New York 1978.
• C. Remling, The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials, Comm. Math. Phys. 193 (1998), 151--170.
• —, Embedded singular continuous spectrum for one-dimensional Schrödinger operators, Trans. Amer. Math. Soc. 351 (1999), 2479--2497.
• —, Bounds on embedded singular spectrum for one-dimensional Schrödinger operators, Proc. Amer. Math. Soc. 128 (2000), 161--171.
• A. Rybkin, On the absolutely continuous and negative discrete spectra of Schrödinger operators on the line with locally integrable globally square summable potentials, J. Math. Phys. 45 (2004), 1418--1425.
• O. Safronov, Multi-dimensional Schrödinger operators with some negative spectrum, preprint, 2006.
• U.-W. Schmincke, On Schrödinger's factorization method for Sturm-Liouville operators, Proc. Roy. Soc. Edinburgh Sect. A 80 (1978), 67--84.
• B. Simon and A. Zlatoš, Sum rules and the Szegö condition for orthogonal polynomials on the real line, Comm. Math. Phys. 242 (2003), 393--423.
• A. Teplyaev, A note on the theorems of M. G. Krein and L. A. Sakhnovich on continuous analogs of orthogonal polynomials on the circle, J. Funct. Anal. 226 (2005), 257--280.
• V. E. Zakharov and L. D. Faddeev, The Korteweg de Vries equation is a fully integrable Hamiltonian system, Functional Anal. Appl. 5 (1971), 280--287.
• A. Zygmund, Trigonometric Series, Vols. I, II, 2nd ed., Cambridge Univ. Press, New York, 1959.