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Friedrichs extension and min-max principle for operators with a gap. (English) Zbl 1432.49065

Summary: Semibounded symmetric operators have a distinguished self-adjoint extension, the Friedrichs extension. The eigenvalues of the Friedrichs extension are given by a variational principle that involves only the domain of the symmetric operator. Although Dirac operators describing relativistic particles are not semibounded, the Dirac operator with Coulomb potential is known to have a distinguished extension. Similarly, for Dirac-type operators on manifolds with a boundary a distinguished self-adjoint extension is characterised by the Atiyah-Patodi-Singer boundary condition. In this paper, we relate these extensions to a generalisation of the Friedrichs extension to the setting of operators satisfying a gap condition. In addition, we prove, in the general setting, that the eigenvalues of this extension are also given by a variational principle that involves only the domain of the symmetric operator. We also clarify what we believe to be inaccuracies in the existing literature.

MSC:

49R05 Variational methods for eigenvalues of operators
49S05 Variational principles of physics
47B25 Linear symmetric and selfadjoint operators (unbounded)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
49J35 Existence of solutions for minimax problems
47A75 Eigenvalue problems for linear operators
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[1] Atiyah, Mf; Patodi, Vk; Singer, Im, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Camb. Philos. Soc., 77, 43 (1975) · Zbl 0297.58008 · doi:10.1017/S0305004100049410
[2] Balinsky, Aa; Evans, Wd, Spectral Analysis of Relativistic Operators (2011), London: Imperial College Press, London · Zbl 1215.81001
[3] Birman, M.S., Solomjak, M.Z.: Spectral Theory of Selfadjoint Operators in Hilbert Space. Translated from the 1980 Russian original by S. Khrush and V. Peller. Mathematics and its Applications (Soviet Series). D. Reidel Publishing Co, Dordrecht (1987) · Zbl 0744.47017
[4] Brasche, Jg; Neidhardt, H., Some remarks on Kreĭn’s extension theory, Math. Nachr., 165, 159-181 (1994) · Zbl 0826.47005 · doi:10.1002/mana.19941650111
[5] Brown, Ge; Ravenhall, Dg, On the interaction of two electrons, Proc. Roy. Soc. Lond. Ser. A., 208, 552-559 (1951) · Zbl 0044.23209 · doi:10.1098/rspa.1951.0181
[6] Datta, Sn; Deviah, G., The minimax technique in relativistic Hartree-Fock calculations, Pramana, 30, 5, 387-405 (1988) · doi:10.1007/BF02935594
[7] Davies, Eb, Spectral Theory and Differential Operators (1995), Cambridge: Cambridge University Press, Cambridge · Zbl 0893.47004
[8] Deckert, Da; Oelker, M., Distinguished self-adjoint extension of the two-body Dirac operator with Coulomb interaction, Ann. Henri Poincaré, 20, 7, 2407-2445 (2019) · Zbl 1416.81055 · doi:10.1007/s00023-019-00802-6
[9] Dolbeault, J.; Esteban, Mj; Loss, M.; Vega, L., An analytical proof of Hardy-like inequalities related to the Dirac operator, J. Funct. Anal., 216, 1, 1-21 (2004) · Zbl 1060.35120 · doi:10.1016/j.jfa.2003.09.010
[10] Dolbeault, J.; Esteban, Mj; Séré, E., On the eigenvalues of operators with gaps. Application to Dirac operators, J. Funct. Anal., 174, 208-226 (2000) · Zbl 0982.47006 · doi:10.1006/jfan.1999.3542
[11] Douglas, Rg; Wojciechowski, Kp, Adiabatic limit of the \(\eta \)-invariants. The odd-dimensional Atiyah-Patodi-Singer problem, Commun. Math. Phys., 142, 139-168 (1991) · Zbl 0746.58074 · doi:10.1007/BF02099174
[12] Esteban, Mj; Lewin, M.; Séré, E., Domains for Dirac-Coulomb min-max levels, Rev. Mat. Iberoam., 35, 3, 877-924 (2019) · Zbl 1450.81039 · doi:10.4171/rmi/1074
[13] Esteban, Mj; Loss, M., Self-adjointness for Dirac operators via Hardy-Dirac inequalities, J. Math. Phys., 48, 11, 112107 (2007) · Zbl 1152.81423 · doi:10.1063/1.2811950
[14] Esteban, M.J., Loss, M.: Self-adjointness via partial Hardy-like inequalities. In: Mathematical Results in Quantum Mechanics, pp. 41-47. World Scientific Publication, Hackensack (2008) · Zbl 1156.81369
[15] Evans, L.C.: Partial differential equations, second edition. In: Graduate Studies in Mathematics, Vol. 19. American Mathematical Society, Providence RI · Zbl 1194.35001
[16] Furutani, K., Atiyah-Patodi-Singer boundary condition and a splitting formula of a spectral flow, J. Geom. Phys., 56, 310-321 (2006) · Zbl 1091.58016 · doi:10.1016/j.geomphys.2005.02.003
[17] Friedrichs, K., Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren, Math. Ann., 109, 1, 465-487 (1934) · Zbl 0008.39203 · doi:10.1007/BF01449150
[18] Gallone, M.; Michelangeli, A., Self-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei, Anal. Math. Phys., 9, 1, 585-616 (2019) · Zbl 07074241 · doi:10.1007/s13324-018-0219-7
[19] Gallone, M.; Michelangeli, A., Discrete spectra for critical Dirac-Coulomb Hamiltonians, J. Math. Phys., 59, 6, 062108 (2018) · Zbl 1391.81076 · doi:10.1063/1.5011305
[20] Griesemer, M.; Siedentop, H., A minimax principle for the eigenvalues in spectral gaps, J. Lond. Math. Soc. (2), 60, 2, 490-500 (1999) · Zbl 0651.34030 · doi:10.1112/S0024610799007930
[21] Klaus, M., Wüst, R.: Characterization and uniqueness of distinguished selfadjoint extensions of Dirac operators. Commun. Math. Phys. 64(2), 171-176 (1978/1979) · Zbl 0408.47022
[22] Kraus, M.; Langer, M.; Tretter, C., Variational principles and eigenvalue estimates for unbounded block operator matrices and applications, J. Comput. Appl. Math., 171, 1-2, 311-334 (2004) · Zbl 1066.49030 · doi:10.1016/j.cam.2004.01.024
[23] Krein, Mg, The theory of self-adjoint extensions of semibounded Hermitian operators and its applications. I, Mat. Sbornik, 20, 3, 431-495 (1947) · Zbl 0029.14103
[24] Krejčířik, D.; Lu, Z., Location of the essential spectrum in curved quantum layers, J. Math. Phys., 55, 8, 083520 (2014) · Zbl 1301.82073 · doi:10.1063/1.4893035
[25] Morozov, S.; Müller, D., On the minimax principle for Coulomb-Dirac operators, Math. Z., 280, 733-747 (2015) · Zbl 1320.49032 · doi:10.1007/s00209-015-1445-4
[26] Müller, D., Minimax principles, Hardy-Dirac inequalities and operator cores for two and three dimensional Coulomb-Dirac operators, Doc. Math., 21, 1151-1169 (2016) · Zbl 1350.49074
[27] Nenciu, G., Self-adjointness and invariance of the essential spectrum for Dirac operators defined as quadratic forms, Commun. Math. Phys., 48, 3, 235-247 (1976) · Zbl 0349.47014 · doi:10.1007/BF01617872
[28] Reed, Mike; Simon, Barry, Preface, Methods of Modern Mathematical Physics, vii (1972) · Zbl 0242.46001
[29] Reed, M., Simon, B.: Methods of modern mathematical physics. II. In: Fourier Analysis, Self-adjointness. Academic Press, New York (1975) · Zbl 0308.47002
[30] Schmincke, U-W, Essential selfadjointness of Dirac operators with a strongly singular potential, Math. Z., 126, 71-81 (1972) · Zbl 0248.35091 · doi:10.1007/BF01580357
[31] Talman, Jd, Minimax principle for the Dirac equation, Phys. Rev. Lett., 57, 9, 1091-1094 (1986) · doi:10.1103/PhysRevLett.57.1091
[32] Tix, C.: Self-adjointness and spectral properties of a pseudo-relativistic Hamiltonian due to Brown and Ravenhall. Preprint, mp-arc, pp. 97-441 (1997)
[33] Tretter, C., Spectral Theory of Block Operator Matrices and Applications (2008), London: Imperial College Press, London · Zbl 1173.47003
[34] Wüst, R., Distinguished self-adjoint extensions of Dirac operators constructed by means of cut-off potentials, Math. Z., 141, 93-98 (1975) · Zbl 0311.47020 · doi:10.1007/BF01236987
[35] Wüst, R., Dirac operations with strongly singular potentials. Distinguished self-adjoint extensions constructed with a spectral gap theorem and cut-off potentials, Math. Z., 152, 3, 259-271 (1977) · Zbl 0361.35051 · doi:10.1007/BF01488968
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