najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2016-7-2-315-323najo-830Research ArticleINVITED SPEAKERSINVITED SPEAKERSLaplacians with singular perturbations supported on hypersurfacesLaplacians with singular perturbations supported on hypersurfacesMantileA.MantileA.

EA4535 URCA

FR 3399 CNRS

EA4535 URCA

FR 3399 CNRS

andrea.mantile@univ-reims.frPosilicanoA.PosilicanoA.

via Valleggio 11, I-22100 Como

via Valleggio 11, I-22100 Como

Laboratoire de Math´ematiques de Reims; F´ed´eration de Recherche ARC Math´ematiquesLaboratoire de Math´ematiques de Reims; F´ed´eration de Recherche ARC Math´ematiquesDiSAT, Sezione di Matematica, Universit`a dell’InsubriaDiSAT, Sezione di Matematica, Universit`a dell’Insubria20161308202572Special Issue315323Copyright © Mantile A., Posilicano A., 20252025Mantile A., Posilicano A.Mantile A., Posilicano A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/830

We review the main results of our recent work on singular perturbations supported on bounded hypersurfaces. Our approach consists in using the theory of self-adjoint extensions of restrictions to build self-adjoint realizations of the n-dimensional Laplacian with linear boundary conditions on (a relatively open part of) a compact hypersurface. This allows one to obtain Kreın-like resolvent formulae where the reference operator coincides with the free selfadjoint Laplacian in Rn, providing in this way with an useful tool for the scattering problem from a hypersurface. As examples of this construction, we consider the cases of Dirichlet and Neumann boundary conditions assigned on an unclosed hypersurface.

We review the main results of our recent work on singular perturbations supported on bounded hypersurfaces. Our approach consists in using the theory of self-adjoint extensions of restrictions to build self-adjoint realizations of the n-dimensional Laplacian with linear boundary conditions on (a relatively open part of) a compact hypersurface. This allows one to obtain Kreın-like resolvent formulae where the reference operator coincides with the free selfadjoint Laplacian in Rn, providing in this way with an useful tool for the scattering problem from a hypersurface. As examples of this construction, we consider the cases of Dirichlet and Neumann boundary conditions assigned on an unclosed hypersurface.

Kreın’s resolvent formulaboundary conditionsself-adjoint extensionsKreın’s resolvent formulaboundary conditionsself-adjoint extensionsReferencesMantile A., Posilicano A., Sini M. Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces. Preprint arXiv:1505.07236, 2015.Mantile A., Posilicano A., Sini M. Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces. Preprint arXiv:1505.07236, 2015.Posilicano A. A Kr˘eın-like formula for singular perturbations of self-adjoint operators and applications. J. Funct. Anal., 2001, 183, P. 109–147.Posilicano A. A Kr˘eın-like formula for singular perturbations of self-adjoint operators and applications. J. Funct. Anal., 2001, 183, P. 109–147.Posilicano A. Self-adjoint extensions by additive perturbations. Ann. Sc. Norm. Super. Pisa Cl. Sci., 2003, 2(5), P. 1–20.Posilicano A. Self-adjoint extensions by additive perturbations. Ann. Sc. Norm. Super. Pisa Cl. Sci., 2003, 2(5), P. 1–20.Posilicano A. Boundary triples and Weyl functions for singular perturbations of self-adjoint operators. Methods Funct. Anal. Topology, 2004, 10, P. 57–63.Posilicano A. Boundary triples and Weyl functions for singular perturbations of self-adjoint operators. Methods Funct. Anal. Topology, 2004, 10, P. 57–63.Posilicano A. Self-adjoint extensions of restrictions. Oper. Matrices, 2008, 2, P. 483–506.Posilicano A. Self-adjoint extensions of restrictions. Oper. Matrices, 2008, 2, P. 483–506.Albeverio S., Brasche J., R¨ockner M. Dirichlet forms and generalized Schr¨odinger operators. In: Schr¨odinger operators, Lecture Notes in Phys., 345, P. 1–42, Springer, Berlin, 1989.Albeverio S., Brasche J., R¨ockner M. Dirichlet forms and generalized Schr¨odinger operators. In: Schr¨odinger operators, Lecture Notes in Phys., 345, P. 1–42, Springer, Berlin, 1989.Albeverio S., Fenstad J.E., et al. Perturbations of the Laplacian supported by null sets, with applications to polymer measures and quantum fields. Phys. Lett. A, 1984, 104, P. 396–400.Albeverio S., Fenstad J.E., et al. Perturbations of the Laplacian supported by null sets, with applications to polymer measures and quantum fields. Phys. Lett. A, 1984, 104, P. 396–400.Albeverio S., Fenstad J.E., et al. Schr¨odinger operators with potentials supported by null sets, in Ideas and Methods in Mathematical Analysis (Albeverio S., Fenstad J.R., Holden H., and LindstrøT. m, Eds.), Vol. II, Cambridge Univ. Press, New York/Cambridge, MA, 1991.Albeverio S., Fenstad J.E., et al. Schr¨odinger operators with potentials supported by null sets, in Ideas and Methods in Mathematical Analysis (Albeverio S., Fenstad J.R., Holden H., and LindstrøT. m, Eds.), Vol. II, Cambridge Univ. Press, New York/Cambridge, MA, 1991.Antoine J.-P., Gesztesy F., Shabani J. Exactly solvable models of sphere interactions in quantum mechanics. J. Phys. A, 1987, 20, P. 3687–3712.Antoine J.-P., Gesztesy F., Shabani J. Exactly solvable models of sphere interactions in quantum mechanics. J. Phys. A, 1987, 20, P. 3687–3712.Behrndt J., Exner P., Lotoreichik V. Schr¨odinger operators with δ-interactions supported on conical surfaces. J. Phys. A, 2014, 47(35), 355202, 16 pp.Behrndt J., Exner P., Lotoreichik V. Schr¨odinger operators with δ-interactions supported on conical surfaces. J. Phys. A, 2014, 47(35), 355202, 16 pp.Behrndt J., Exner P., Lotoreichik V. Schr¨odinger operators with δ- and δ′-interactions on Lipschitz surfaces and chromatic numbers of associated partitions. Rev. Math. Phys., 2014, 26, 1450015, 43 pp.Behrndt J., Exner P., Lotoreichik V. Schr¨odinger operators with δ- and δ′-interactions on Lipschitz surfaces and chromatic numbers of associated partitions. Rev. Math. Phys., 2014, 26, 1450015, 43 pp.Behrndt J., Grubb G., Langer M., Lotoreichik V. Spectral asymptotics for resolvent differences of elliptic operators with δ and δ′-interactions on hypersurfaces. J. Spectr. Theory, 2015, 5, P. 697–729.Behrndt J., Grubb G., Langer M., Lotoreichik V. Spectral asymptotics for resolvent differences of elliptic operators with δ and δ′-interactions on hypersurfaces. J. Spectr. Theory, 2015, 5, P. 697–729.Behrndt J., Langer M., Lotoreichik V. Spectral Estimates for Resolvent Differences of Self-Adjoint Elliptic Operators. Integr. Equ. Oper. Theory, 2013, 77, P. 1–37.Behrndt J., Langer M., Lotoreichik V. Spectral Estimates for Resolvent Differences of Self-Adjoint Elliptic Operators. Integr. Equ. Oper. Theory, 2013, 77, P. 1–37.Behrndt J., Langer M., Lotoreichik V. Schr¨odinger operators with δ- and δ′-potentials supported on hypersurfaces. Ann. Henri Poincar´e, 2013, 14, P. 385–423.Behrndt J., Langer M., Lotoreichik V. Schr¨odinger operators with δ- and δ′-potentials supported on hypersurfaces. Ann. Henri Poincar´e, 2013, 14, P. 385–423.Exner P. Spectral properties of Schr¨odinger operators with a strongly attractive δ interaction supported by a surface. In: Proc. NSF Summer Research Conference (Mt. Holyoke 2002), Contemp. Math., Amer. Math. Providence, R.I., 2003.Exner P. Spectral properties of Schr¨odinger operators with a strongly attractive δ interaction supported by a surface. In: Proc. NSF Summer Research Conference (Mt. Holyoke 2002), Contemp. Math., Amer. Math. Providence, R.I., 2003.Exner P. Leaky quantum graphs: a review. Proc. Symp. Pure Math., 2008, 77, P. 523–564.Exner P. Leaky quantum graphs: a review. Proc. Symp. Pure Math., 2008, 77, P. 523–564.Exner P., Praas M. On geometric perturbations of critical Schr¨odinger operators with a surface interaction. J. Math. Phys., 2009, 50, 112101.Exner P., Praas M. On geometric perturbations of critical Schr¨odinger operators with a surface interaction. J. Math. Phys., 2009, 50, 112101.Exner P., Ichinose I. Geometrically induced spectrum in curved leaky wires. J. Phys. A, 2001, 34, P. 1439–1450.Exner P., Ichinose I. Geometrically induced spectrum in curved leaky wires. J. Phys. A, 2001, 34, P. 1439–1450.Exner P., Kondej S. Curvature-induced bound states for a δ interaction supported by a curve in R3. Ann. Henri Poincar´e, 2002, 3, P. 967–981.Exner P., Kondej S. Curvature-induced bound states for a δ interaction supported by a curve in R3. Ann. Henri Poincar´e, 2002, 3, P. 967–981.Exner P., Kondej S. Bound states due to a strong δ interaction supported by a curved surface. J. Phys. A, 2003, 36, P. 443–457.Exner P., Kondej S. Bound states due to a strong δ interaction supported by a curved surface. J. Phys. A, 2003, 36, P. 443–457.Exner P., Kondej S. Scattering by local deformations of a straight leaky wire. J. Phys. A, 2005, 38, P. 4865–4874.Exner P., Kondej S. Scattering by local deformations of a straight leaky wire. J. Phys. A, 2005, 38, P. 4865–4874.Exner P., Pankrashkin K. Strong coupling asymptotics for a singular Schr¨odinger operator with an interaction supported by an open arc. Comm. Partial Diff. Eq., 2014, 39, P. 193–212.Exner P., Pankrashkin K. Strong coupling asymptotics for a singular Schr¨odinger operator with an interaction supported by an open arc. Comm. Partial Diff. Eq., 2014, 39, P. 193–212.Exner P., Yoshitomi K. Band gap of the Schr¨odinger operator with a strong δ-interaction on a periodic curve. Ann. Henri Poincar´e, 2001, 2, P. 1139–1158.Exner P., Yoshitomi K. Band gap of the Schr¨odinger operator with a strong δ-interaction on a periodic curve. Ann. Henri Poincar´e, 2001, 2, P. 1139–1158.Figari R., Teta A. Effective potential and fluctuations for a boundary value problem on a randomly perforated domain. Lett. Math. Phys., 1992, 26, P. 295–305.Figari R., Teta A. Effective potential and fluctuations for a boundary value problem on a randomly perforated domain. Lett. Math. Phys., 1992, 26, P. 295–305.Figari R., Teta A. A boundary value problem of mixed type on perforated domains. Asymptotic Anal., 1993, 6, P. 271–284.Figari R., Teta A. A boundary value problem of mixed type on perforated domains. Asymptotic Anal., 1993, 6, P. 271–284.Lions J.-L., Magenes E. Non-homogeneous Boundary Value Problems and Applications. Springer, New-York, 1972.Lions J.-L., Magenes E. Non-homogeneous Boundary Value Problems and Applications. Springer, New-York, 1972.McLean W. Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambrdige, 2000.McLean W. Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambrdige, 2000.Dias N.C., Posilicano A., Prata J.N. Self-adjoint, globally defined Hamiltonian operators for systems with boundaries. Commun. Pure Appl. Anal., 2011, 10, P. 1687–1706.Dias N.C., Posilicano A., Prata J.N. Self-adjoint, globally defined Hamiltonian operators for systems with boundaries. Commun. Pure Appl. Anal., 2011, 10, P. 1687–1706.

The authors declare that there are no conflicts of interest present.