Self-adjoint and skew-symmetric extensions of the Laplacian with singular Robin boundary condition

Nazarov, Sergei A.; Popoff, Nicolas

Comptes Rendus. Mathématique, Tome 356 (2018) no. 9, pp. 927-932.

Nous étudions le laplacien dans un domaine borné, avec une condition à la frontière de type Robin, variable et singulière en un point. La forme quadratique associée n'est pas bornée inférieurement, et le laplacien correspondant n'est pas self-adjoint ; son spectre résiduel couvre entièrement le plan complexe. Nous décrivons ses extensions self-adjointes et nous en montrons une anti-symétrique, pertinente en physique. Nous approchons la condition de frontière à l'aide d'une famille d'opérateurs self-adjoints et nous décrivons son spectre par la méthode d'appariement des développements asymptotiques. Une partie du spectre adopte un comportement étrange quand le paramètre ε>0 de petite perturbation tend vers zéro ; précisément, il devient presque périodique en échelle logarithmique |log⁡(ε)|, et ainsi « erre » le long de l'axe réel à une vitesse O(ε−1).

We study the Laplacian in a bounded domain, with a varying Robin boundary condition singular at one point. The associated quadratic form is not semi-bounded from below, and the corresponding Laplacian is not self-adjoint, it has a residual spectrum covering the whole complex plane. We describe its self-adjoint extensions and exhibit a physically relevant skew-symmetric one. We approximate the boundary condition, giving rise to a family of self-adjoint operators, and we describe its spectrum by the method of matched asymptotic expansions. A part of the spectrum acquires a strange behavior when the small perturbation parameter ε>0 tends to zero, namely it becomes almost periodic in the logarithmic scale |ln⁡ε|, and in this way “wanders” along the real axis at a speed O(ε−1).

[1] Berry, M.V.; Dennis, M. Boundary-condition-varying circle billiards and gratings: the Dirichlet singularity, J. Phys. A, Volume 41 (2008) no. 13

[2] Claeys, X.; Chesnel, L.; Nazarov, S. Oscillating behaviour of the spectrum for a plasmonic problem in a domain with a rounded corner, ESAIM: Math. Model. Numer. Anal. (2016) (in press) | DOI

[3] Exner, P.; Seba, P. A simple model of thin-film point contact in two and three dimensions, Czechoslov. J. Phys., Volume 38 (1988) no. 10, pp. 1095-1110

[4] Kondrat'ev, V.A. Boundary value problems for elliptic equations in domains with conical or angular points, Tr. Mosk. Mat. Obŝ., Volume 16 (1967), pp. 209-292

[5] Levitin, M.; Parnovski, L. On the principal eigenvalue of a Robin problem with a large parameter, Math. Nachr., Volume 281 (2008) no. 2, pp. 272-281

[6] Marlettta, M.; Rozenblum, G. A Laplace operator with boundary conditions singular at one point, J. Phys. A, Volume 42 (2009) no. 12

[7] Nazarov, S.A. Selfadjoint extensions of the Dirichlet problem operator in weighted function spaces, Math. USSR Sb., Volume 65 (1990) no. 1, p. 229

[8] Nazarov, S.A. Asymptotic conditions at a point, selfadjoint extensions of operators, and the method of matched asymptotic expansions, Proceedings of the St. Petersburg Mathematical Society, Vol. V, Amer. Math. Soc. Transl. Ser. 2, vol. 193, American Mathematical Society, Providence, RI, USA, 1999, pp. 77-125

[9] Nazarov, S.A.; Plamenevsky, B. Elliptic Problems in Domains with Piecewise Smooth Boundaries, De Gruyter Expositions in Mathematics, vol. 13, Walter de Gruyter & Co., Berlin, 1994

[10] Rofe-Beketov, F.S. Selfadjoint extensions of differential operators in a space of vector-valued functions, Dokl. Akad. Nauk SSSR, Volume 184 (1969), pp. 1034-1037