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Puits multiples pour l’opérateur de Dirac. (Multiple wells for the Dirac operator). (French) Zbl 0614.35074

We consider the multiple well problem for the Dirac operator D(h) depending on a small parameter \(h>0:\) \[ D(h)=h\sum^{3}_{j=1}\alpha_ jD_ j+\alpha_ 4VI_ 4, \] defined in \(L^ 2({\mathbb{R}}^ 3;{\mathbb{C}}^ 4)\). Modulo some exponentially small error, measured by Agmon’s distance between the wells, the spectral quantities of the Dirac operators with multiple wells are the direct sum of those Dirac operators corresponding to one well. When the wells are non-degenerate, they can be divided into two families, ”hills” and ”valleys”, which are non-resonant in the sense of Helffer-Sjöstrand. In particular, one proves that for the Dirac operator with one non- degenerate well, the first two positive eigenvalues have the same semi- classical expansion and, in the presence of symmetries for V, are really equal.

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
81Q15 Perturbation theories for operators and differential equations in quantum theory
35P25 Scattering theory for PDEs
35B20 Perturbations in context of PDEs
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References:

[1] R. Abraham , J. Marsden , Foundations of Mechanics , 2 nd Ed., Benjamin , Cumming Publ. Company , 1978 . MR 515141 | Zbl 0393.70001 · Zbl 0393.70001
[2] S. Agmon , Lectures on exponential decay of solutions of second order elliptic equations , Math. Notes , t. 29 , Princeton University Press , 1982 . MR 745286 | Zbl 0503.35001 · Zbl 0503.35001
[3] A.M. Berthier , V. Georgescu , Sur le spectre ponctuel de l’opérateur de Dirac . C. R. Acad. Sci. Paris , t. 297 , 1983 , Série I, p. 335 - 338 . MR 732500 | Zbl 0546.35052 · Zbl 0546.35052
[4] R. Chernoff , Schrödinger and Dirac operators with singular potentials and hyperbolic equations , Pac. J. Math. , t. 72 ( 2 ), 1977 , p. 361 - 383 . Article | MR 510049 | Zbl 0366.35031 · Zbl 0366.35031
[5] J.M. Combes , P. Duclos , R. Seiler , Krein’s formula and one dimensional multiple well , J. Funct. Anal. , t. 52 , 1983 , p. 257 - 301 . MR 707207 | Zbl 0562.47002 · Zbl 0562.47002
[6] J.M. Combes , P. Duclos , R. Seiler , Convergent Expansion for Tunneling , Comm. Math. Phys. , t. 92 , 1982 , p. 229 - 245 . Article | MR 728868 | Zbl 0579.47050 · Zbl 0579.47050
[7] E.M. Harrell , Double Well , Comm. Math. Phys. , t. 75 , 1980 , p. 239 - 261 . Article | MR 581948 | Zbl 0445.35036 · Zbl 0445.35036
[8] E.M. Harrell , M. Klaus , On the double-well problem for Dirac operators , Ann. Inst. Henri Poincaré , t. 38 ( 2 ), 1983 , p. 153 - 166 . Numdam | MR 705337 | Zbl 0529.35062 · Zbl 0529.35062
[9] B. Helffer , D. Robert , Calcul fonctionnel par la transformation de Mellin et applications , J. Funct. Anal. , t. 53 ( 3 ), 1983 , p. 245 - 268 . Zbl 0524.35103 · Zbl 0524.35103
[10] B. Helffer , D. Robert , Puits de potentiel généralisés et asymptotique semi-classique , Ann. Inst. Henri Poincaré , Sect. A, t. 41 , 1984 , p. 291 - 332 . Numdam | MR 776281 | Zbl 0565.35082 · Zbl 0565.35082
[11] B. Helffer , J. Sjöstrand , Multiple wells in the semi-classical limit I , Comm. P. D. E. , t. 9 ( 4 ), 1984 , p. 337 - 408 . MR 740094 | Zbl 0546.35053 · Zbl 0546.35053
[12] B. Helffer , J. Sjöstrand , Puits multiples et limites semi-classique II. Interaction moléculaire, symétries, perturbation , Ann. Inst. Henri Poincaré , section Phys. Théorique , t. 42 , 1985 , p. 127 - 212 . Numdam | MR 798695 | Zbl 0595.35031 · Zbl 0595.35031
[13] B. Helffer , J. Sjöstrand , Multiple wells in the semi-classical limi III . Interaction through non-resonnant wells , à paraître Mathematisch Nachrichte. Zbl 0597.35023 · Zbl 0597.35023
[14] M. Klaus , On the point spectrum of Dirac operators , Helv. Phys. Acta , t. 53 , 1980 , p. 453 - 462 . MR 611769
[15] M. Klaus , R. Wüst , Spectral properties of Dirac operators with singular potentials , J. Math. Anal. and Appl. , t. 72 , 1979 , 206 - 214 . MR 552332 | Zbl 0423.47014 · Zbl 0423.47014
[16] J. Leray , Analyse Lagrangienne , Collège de France , 1976 - 1977 . MR 501198
[17] J. Leray , Solution Asymptotique de l’équation de Dirac , in Trends in Applications of Pure Mathematics to Mechanics , Pitman , 1976 , p. 233 - 240 . Zbl 0346.35092 · Zbl 0346.35092
[18] V.P. Maslov , M.V. Fedoriuk , Semi-classical Approximation in Quantum Mechanics , D. Reidel , 1981 . Zbl 0458.58001 · Zbl 0458.58001
[19] B. Müller , W. Greiner , The two centre Dirac equation ; Z. Naturforsch. , t. 30 ( 1 ), 1976 , p. 1 - 30 .
[20] J.C. Nosmas , Approximation Semi-classique du spectre de systèmes asymptotiques , C. R. Acad. Sci. Paris , t. 295 , 1982 , p. 253 - 256 . Zbl 0535.58040 · Zbl 0535.58040
[21] D. Robert , Calcul fonctionnel sur les opérateurs admissibles et application , J. Funct. Anal. , t. 45 ( 1 ), 1982 , p. 74 - 94 . MR 645646 | Zbl 0482.35069 · Zbl 0482.35069
[22] D. Robert , Autour de l’Approximation Semi-classique , Notas de Curso , N^\circ 21 , Universidada Federal de Pernambuco , Recife , 1983 .
[23] J.P. Serre , Représentations Linéaires des Groupes Finis , Hermann , Paris , 1967 . MR 232867 | Zbl 0189.02603 · Zbl 0189.02603
[24] B. Simon , Semi-classical analysis of low lying eigenvalues, I. Non-degenerate minima: Asymptotic expansions , Ann. Inst. Henri Poincaré , t. 38 , 1983 , p. 295 - 307 . Numdam | MR 708966 | Zbl 0526.35027 · Zbl 0526.35027
[25] B. Simon , Semi-classical analysis of low lying eigenvalues, II. Tunneling , Ann. of Math. , t. 120 , 1984 , p. 89 - 118 . MR 750717 | Zbl 0626.35070 · Zbl 0626.35070
[26] X.P. Wang , Asymptotic behavior of spectral means of pseudo-differential operators , J. of Appr. Theory and Appl. , t. 1 , 1985 , p. 119 - 136 . MR 816606 | Zbl 0595.47036 · Zbl 0595.47036
[27] E. Witten , Fermion Quantum Numbers in Kaluza-Klein Theory , prétirage.
[28] K. Yajima , The quasi-classical approximation to Dirac equation , I. J. Fac. Sci. Univ. Tokyo , t. 29 , 1982 , p. 161 - 194 . Zbl 0486.35075 · Zbl 0486.35075
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