Multiple Aharonov–Bohm eigenvalues: The case of the first eigenvalue on the disk
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Abstract:
It is known that the first eigenvalue for Aharonov–Bohm operators with half-integer circulation in the unit disk is double if the potential’s pole is located at the origin. We prove that in fact it is simple as the pole $a\neq 0$.References
- L. Abatangelo, Sharp asymptotics for the eigenvalue function of Aharonov–Bohm operators with a moving pole, Rend. Sem. Mat. Univ. Politec. Torino Bruxelles-Torino Talks in PDE’s Turin, May 2–5, 2016, Vol. 74, 2: 19–29, 2016.
- Laura Abatangelo and Veronica Felli, Sharp asymptotic estimates for eigenvalues of Aharonov-Bohm operators with varying poles, Calc. Var. Partial Differential Equations 54 (2015), no. 4, 3857–3903. MR 3426097, DOI 10.1007/s00526-015-0924-0
- Laura Abatangelo and Veronica Felli, On the leading term of the eigenvalue variation for Aharonov-Bohm operators with a moving pole, SIAM J. Math. Anal. 48 (2016), no. 4, 2843–2868. MR 3542002, DOI 10.1137/15M1044898
- Laura Abatangelo, Veronica Felli, and Corentin Léna, On Aharonov-Bohm operators with two colliding poles, Adv. Nonlinear Stud. 17 (2017), no. 2, 283–296. MR 3641642, DOI 10.1515/ans-2017-0004
- Laura Abatangelo, Veronica Felli, Benedetta Noris, and Manon Nys, Sharp boundary behavior of eigenvalues for Aharonov-Bohm operators with varying poles, J. Funct. Anal. 273 (2017), no. 7, 2428–2487. MR 3677830, DOI 10.1016/j.jfa.2017.06.023
- Laura Abatangelo and Manon Nys, On multiple eigenvalues for Aharonov-Bohm operators in planar domains, Nonlinear Anal. 169 (2018), 1–37. MR 3761092, DOI 10.1016/j.na.2017.11.010
- R. Adami and A. Teta, On the Aharonov-Bohm Hamiltonian, Lett. Math. Phys. 43 (1998), no. 1, 43–53. MR 1607550, DOI 10.1023/A:1007330512611
- Y. Aharonov and D. Bohm, Significance of electromagnetic potentials in the quantum theory, Phys. Rev. (2) 115 (1959), 485–491. MR 110458
- A. A. Balinsky, Hardy type inequalities for Aharonov-Bohm magnetic potentials with multiple singularities, Math. Res. Lett. 10 (2003), no. 2-3, 169–176. MR 1981894, DOI 10.4310/MRL.2003.v10.n2.a4
- Virginie Bonnaillie-Noël and Bernard Helffer, On spectral minimal partitions: the disk revisited, Ann. Univ. Buchar. Math. Ser. 4(LXII) (2013), no. 1, 321–342. MR 3093548
- V. Bonnaillie-Noël, B. Helffer, and T. Hoffmann-Ostenhof, Aharonov-Bohm Hamiltonians, isospectrality and minimal partitions, J. Phys. A 42 (2009), no. 18, 185203, 20. MR 2591197, DOI 10.1088/1751-8113/42/18/185203
- V. Bonnaillie-Noël and B. Helffer, Numerical analysis of nodal sets for eigenvalues of Aharonov-Bohm Hamiltonians on the square with application to minimal partitions, Exp. Math. 20 (2011), no. 3, 304–322. MR 2836255, DOI 10.1080/10586458.2011.565240
- Virginie Bonnaillie-Noël and Corentin Léna, Spectral minimal partitions of a sector, Discrete Contin. Dyn. Syst. Ser. B 19 (2014), no. 1, 27–53. MR 3245081, DOI 10.3934/dcdsb.2014.19.27
- Virginie Bonnaillie-Noël, Benedetta Noris, Manon Nys, and Susanna Terracini, On the eigenvalues of Aharonov-Bohm operators with varying poles, Anal. PDE 7 (2014), no. 6, 1365–1395. MR 3270167, DOI 10.2140/apde.2014.7.1365
- Monique Dauge and Bernard Helffer, Eigenvalues variation. II. Multidimensional problems, J. Differential Equations 104 (1993), no. 2, 263–297. MR 1231469, DOI 10.1006/jdeq.1993.1072
- Veronica Felli, Alberto Ferrero, and Susanna Terracini, Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 1, 119–174. MR 2735078, DOI 10.4171/JEMS/246
- B. Helffer, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and M. P. Owen, Nodal sets for groundstates of Schrödinger operators with zero magnetic field in non-simply connected domains, Comm. Math. Phys. 202 (1999), no. 3, 629–649. MR 1690957, DOI 10.1007/s002200050599
- B. Helffer, On spectral minimal partitions: a survey, Milan J. Math. 78 (2010), no. 2, 575–590. MR 2781853, DOI 10.1007/s00032-010-0129-0
- Bernard Helffer and Thomas Hoffmann-Ostenhof, On minimal partitions: new properties and applications to the disk, Spectrum and dynamics, CRM Proc. Lecture Notes, vol. 52, Amer. Math. Soc., Providence, RI, 2010, pp. 119–135. MR 2743435, DOI 10.1090/crmp/052/07
- Bernard Helffer and Thomas Hoffmann-Ostenhof, On a magnetic characterization of spectral minimal partitions, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 6, 2081–2092. MR 3120736, DOI 10.4171/JEMS/415
- B. Helffer, T. Hoffmann-Ostenhof, and S. Terracini, Nodal domains and spectral minimal partitions, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 1, 101–138. MR 2483815, DOI 10.1016/j.anihpc.2007.07.004
- Bernard Helffer, Thomas Hoffmann-Ostenhof, and Susanna Terracini, Nodal minimal partitions in dimension 3, Discrete Contin. Dyn. Syst. 28 (2010), no. 2, 617–635. MR 2644760, DOI 10.3934/dcds.2010.28.617
- Bernard Helffer, Thomas Hoffmann-Ostenhof, and Susanna Terracini, On spectral minimal partitions: the case of the sphere, Around the research of Vladimir Maz’ya. III, Int. Math. Ser. (N. Y.), vol. 13, Springer, New York, 2010, pp. 153–178. MR 2664708, DOI 10.1007/978-1-4419-1345-6_{6}
- Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
- Ari Laptev and Timo Weidl, Hardy inequalities for magnetic Dirichlet forms, Mathematical results in quantum mechanics (Prague, 1998) Oper. Theory Adv. Appl., vol. 108, Birkhäuser, Basel, 1999, pp. 299–305. MR 1708811
- Corentin Léna, Eigenvalues variations for Aharonov-Bohm operators, J. Math. Phys. 56 (2015), no. 1, 011502, 18. MR 3390812, DOI 10.1063/1.4905647
- M. Melgaard, E.-M. Ouhabaz, and G. Rozenblum, Negative discrete spectrum of perturbed multivortex Aharonov-Bohm Hamiltonians, Ann. Henri Poincaré 5 (2004), no. 5, 979–1012. MR 2091985, DOI 10.1007/s00023-004-0187-3
- Anna Maria Micheletti, Perturbazione dello spettro dell’operatore di Laplace, in relazione ad una variazione del campo, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 26 (1972), 151–169 (Italian). MR 367480
- Benedetta Noris, Manon Nys, and Susanna Terracini, On the Aharonov-Bohm operators with varying poles: the boundary behavior of eigenvalues, Comm. Math. Phys. 339 (2015), no. 3, 1101–1146. MR 3385993, DOI 10.1007/s00220-015-2423-8
- Benedetta Noris and Susanna Terracini, Nodal sets of magnetic Schrödinger operators of Aharonov-Bohm type and energy minimizing partitions, Indiana Univ. Math. J. 59 (2010), no. 4, 1361–1403. MR 2815036, DOI 10.1512/iumj.2010.59.3964
- Mikhail Teytel, How rare are multiple eigenvalues?, Comm. Pure Appl. Math. 52 (1999), no. 8, 917–934. MR 1686977, DOI 10.1002/(SICI)1097-0312(199908)52:8<917::AID-CPA1>3.3.CO;2-J
- G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
Additional Information
- Laura Abatangelo
- Affiliation: Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Via Cozzi 55, 20125 Milano, Italy
- MR Author ID: 924694
- Email: laura.abatangelo@unimib.it
- Received by editor(s): December 22, 2017
- Received by editor(s) in revised form: February 26, 2018, and March 3, 2018
- Published electronically: October 18, 2018
- Additional Notes: The author was partially supported by the project ERC Advanced Grant 2013 n. 339958: “Complex Patterns for Strongly Interacting Dynamical Systems – COMPAT”, by the PRIN2015 grant “Variational methods, with applications to problems in mathematical physics and geometry” and by the 2017-GNAMPA project “Stabilità e analisi spettrale per problemi alle derivate parziali”.
- Communicated by: Joachim Krieger
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 179-190
- MSC (2010): Primary 35J10, 35J75, 35P99, 35Q40, 35Q60
- DOI: https://doi.org/10.1090/proc/14149
- MathSciNet review: 3876741