zbMATH — the first resource for mathematics

On the Lieb-Thirring estimates for the Pauli operator. (English) Zbl 0882.47056
The aim of this paper is to establish some spectral properties of the Pauli operator, that is, of the operator describing the motion of a particle with spin in a magnetic field. We confine ourselves to the case when the spin is allowed to take one of the values \(+1/2\) or \(-1/2\).

47N50 Applications of operator theory in the physical sciences
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
Full Text: DOI
[1] Y. Aharonov and A. Casher, Ground state of a spin-\(1\over 2\$charged particle in a two-dimensional magnetic field , Phys. Rev. A (3) 19 (1979), no. 6, 2461-2462.\) · doi:10.1103/PhysRevA.19.2461
[2] J. Avron, I. Herbst, and B. Simon, Schrödinger operators with magnetic fields. I. General interactions , Duke Math. J. 45 (1978), no. 4, 847-883. · Zbl 0399.35029 · doi:10.1215/S0012-7094-78-04540-4
[3] M. Š. Birman and M. Z. Solomjak, Spectral theory of selfadjoint operators in Hilbert space , Mathematics and its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht, 1987. · Zbl 0744.47017
[4] H. L. Cycon, R. G. Fröese, W. Kirsch, and B. Simon, Schrödinger operators with application to quantum mechanics and global geometry , Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. · Zbl 0619.47005
[5] B. A. Dubrovin and S. P. Novikov, Ground states in a periodic field. Magnetic Bloch functions and vector bundles , Soviet Math. Dokl. 22 (1980), 240-244. · Zbl 0489.46055
[6] N. Dunford and J. T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space , With the assistance of William G. Bade and Robert G. Bartle, Interscience Publishers John Wiley & Sons New York-London, 1963. · Zbl 0128.34803
[7] D. E. Edmunds and A. A. Ilyin, On some multiplicative inequalities and approximation numbers , Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 178, 159-179. · Zbl 0818.46027 · doi:10.1093/qmath/45.2.159
[8] L. Erdős, Ground-state density of the Pauli operator in the large field limit , Lett. Math. Phys. 29 (1993), no. 3, 219-240. · Zbl 0850.81030 · doi:10.1007/BF00761110
[9] L. Erdős, Magnetic Lieb-Thirring inequalities , Comm. Math. Phys. 170 (1995), no. 3, 629-668. · Zbl 0843.47040 · doi:10.1007/BF02099152
[10] L. Erdős, Magnetic Lieb-Thirring inequalities and estimates on stochastic oscillatory integrals , Ph.D. thesis, Princeton University, 1994.
[11] L. Hörmander, The analysis of linear partial differential operators. I , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. · Zbl 0521.35001
[12] O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural’ceva, Linear and quasilinear equations of parabolic type , Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1967. · Zbl 0174.15403
[13] E. H. Lieb, J. P. Solovej, and J. Yngvason, Ground states of large quantum dots in magnetic fields , Phys. Rev. B51 (1995), 10646-10665. · Zbl 0929.35126
[14] E. H. Lieb, J. P. Solovej, and J. yngvason, Quantum dots , in Proceedings of the Conference on Partial Differential Equations and Mathematical Physics, Birmingham, Alabama, 1994, International Press, · Zbl 0929.35126
[15] E. H. Lieb, J. P. Solovej, and J. Yngvason, Asymptotics of heavy atoms in high magnetic fields. II. Semiclassical regions , Comm. Math. Phys. 161 (1994), no. 1, 77-124. · Zbl 0807.47058 · doi:10.1007/BF02099414
[16] E. H. Lieb and W. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities , Studies in Mathematical Physics: Essays in Honor of V. Bargman eds. E. Lieb, B. Simon, and A. S. Wightman, Princeton University Press, Princeton, New Jersey, 1976, pp. 269-303. · Zbl 0342.35044
[17] A. Pietsch, Eigenvalues and \(s\)-numbers , Cambridge Studies in Advanced Mathematics, vol. 13, Cambridge University Press, Cambridge, 1987. · Zbl 0615.47019
[18] M. Reed and B. Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness , Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1975. · Zbl 0308.47002
[19] M. Reed and B. Simon, Methods of modern mathematical physics. III , Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1979. · Zbl 0405.47007
[20] M. Reed and B. Simon, Methods of modern mathematical physics. IV. Analysis of operators , Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978. · Zbl 0401.47001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.