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Hearing the zero locus of a magnetic field. (English) Zbl 0827.58076
This is a study of a two-dimensional particle in a magmetic field that vanishes nondegenerately on a closed curve. The author shows that the ground state concentrates on this curve in the limit $$e/\hbar \to \infty$$ where $$e$$ is the charge and that the ground state energy grows as $$(e/\hbar)^{2/3}$$. If the gradient of the magnetic field is of constant magnitude $$b_0$$ along its zero locus, then every energy level has asymptotes $$(e/\hbar)^{2/3} (b_0)^{2/3}E_* + O(1)$$ where $$E_* \approx 0.5698$$ is the infimum of the ground state energies of the anharmonic oscillator family $$- {d^2\over dy^2} + ({1\over 2} y^2 - \beta)^2$$ where $$\beta$$ is a parameter.

##### MSC:
 58Z05 Applications of global analysis to the sciences 82D40 Statistical mechanical studies of magnetic materials
##### Keywords:
quantum particle; magmetic field; ground state
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##### References:
 [1] Bismut, J-M.: Large Deviations and the Malliavin Calculus. Basel-Boston: Birkhäuser, 1984 · Zbl 0537.35003 [2] Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators, Berlin, Heidelberg, New York: Springer 1992 · Zbl 0744.58001 [3] Bryant, R., Hsu, L.: Rigidity of Integral Curves of Rank Two Distributions. Invent. Math.114, 435–461 (1993) · Zbl 0807.58007 · doi:10.1007/BF01232676 [4] Chern, S-S.: Complex Manifolds without Potential Theory. Berlin, Heidelberg, New York: Springer, 2nd ed., 1979 [5] Copson, E.T.: Asymptotic Expansions Cambridge Tracts in Mathematics and Mathematical Physics, Cambridge. Cambridge Univ. Press, 1967 [6] Cycon, H., Froese, R., Kirsch, W., Simon, B.: Schrödinger Operators. Texts and monographs in physics. Berlin, Heidelberg, New York: Springer, 1987 · Zbl 0619.47005 [7] Demailly, J.P.: Champs magnétique et inégalité de Morse pur la d”-cohomologie. C.R.A.S., ser. I,301, no. 4, 119–122 (1985) · Zbl 0595.58014 [8] Demailly, J.P.: Champs magnétique et inégalités de Morse pur la d”-cohomologie. Ann. Inst Fourier,35, fasc. 4, (1985) [9] Eastham, M.S.P.: The Asymptotic solution of Linear Differential Systems. Oxford: Oxford Science Pub., 1989 · Zbl 0674.34045 [10] Erdelyi, A.: Asymptotic Expansions. New York, London, Dover, 1956 [11] Gromov, M.: Carnot-Caratheodory spaces seen from within. IHES preprint M/94/6, (221 pages), 1994 [12] Guillemin, V., Uribe, A.: The Trace Formula for Vector Bundles. Bulletin of the A.M.S.15 no. 2, 222–224 (1986) · Zbl 0626.58018 · doi:10.1090/S0273-0979-1986-15482-0 [13] Guillemin, V., Uribe, A.: Some Spectral Properties of Periodic Potentials. In: Lect. Notes in Math., No.1256, Berlin, Heidelberg, New York: Springer, 1987, pp. 192–214 · Zbl 0651.35067 [14] Guillemin, V., Uribe, A.: The Laplace Operator on then th Tensor Power of a Line Bundle: Eigenvalues Which are Uniformly Bounded inn. Asymptotic Analysis,1, 105–113 (1988) · Zbl 0649.53026 [15] Helffer, B.: Conditions nécessaires d’hypoellipticité. J. Diff. Eq.44, 460–481 (1982) · Zbl 0477.35032 · doi:10.1016/0022-0396(82)90008-0 [16] Helffer, B.: Survey on linear PDE on Nilpotent Groups. In: Lect. Notes Math.1077, Berlin, Heidelberg, New York: Springer 1984, 210–252, esp. pp. 219–223 [17] Helffer, B., Nourrigat, J.: Hypoellipticité Maximale pour des Opérateurs Polynomes de Champs de Vecteurs, Boston: Birkhäuser, 1985 · Zbl 0568.35003 [18] Helffer, B.: Semi-classical analysis for the Schrödinger Operator and Applictions. Lect. Notes in Math1336, Berlin, Heidelberg, New York: Springer 1980 [19] Hormander, L.: Hypoelliptic Second Order Differential Equations. Acta. Math.119, 147–171 (1967) · Zbl 0156.10701 · doi:10.1007/BF02392081 [20] Landau, L.D., Lifshitz, E.M.: Quantum Mechanics, v.3 Course of Theoretical Physics, Reading, MA, USA, Pergamon Press, 1958 · Zbl 0081.22207 [21] Liouville, J.: Sur le développment des fonctions ou parties de fonctions en séries..., J. Math. Pure Appl.2, 16–35 (1837) [22] Liu, W-S., Sussmann, H.J.: Shortest paths for subRiemannian metrics on rank two distributions. Preprint, to appear, Trans. A.M.S., 1994 [23] Martinet, J.: Sur les Singularites Des Formes Differentialles. Ana. Insp. (Grenoble), vol.20, no. 1, 95–198 (1970) [24] Mitchell, J.: On Carnot-Caratheodory Metrics. J. Diff. Geom. v.21, 34–45 (1985) · Zbl 0554.53023 [25] Montgomery, R.: Abnormal Minimizers. To appear, SIAM J. Control and Opt., 1994 · Zbl 0816.49019 [26] Rothschild, L.P., Stein, E.M.: Hypoelliptic differential operators and nilpotent groups. Acta Math.137, 248–315 (1977) · Zbl 0346.35030 [27] Simon, B.: Semiclassical Analysis of Low Lying Eigenvalues I. Non-degenerate Minima: Asymptotic Expansions. Ann. Inst. Henri Poincaré, Physique Théorique38, np. 3, 295–307 (1983) [28] Strichartz, R.: Sub-Riemannian Geometry. J. Diff. Geom.24, 221–263 (1983) · Zbl 0609.53021 [29] Treves, F.: Analytic hypoellipticity of a class of pseudodifferential operators with double characteristics and applications to the $$\bar \partial$$ -Neumann problem. Comm. P.D.E.3, 475–642, esp. p. 478 and 501 (1978) · Zbl 0384.35055 · doi:10.1080/03605307808820074 [30] Vershik, A.M., Ya Gerhskovich, V.: Nonholonomic Dynamical Systems, Geometry of Distributions and Variational Problems. In: Dynamical Systems VII, ed. V.I. Arnol’d and S.P. Novikov, vol.16 of the Encyclopaedia of Mathematical Sciences series, NY: Springer, 1994 (Russian original 1987)
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