najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМОnajo-1265Research ArticleMATHEMATICSМАТЕМАТИКАNote on 2D Schrödinger operators with δ-interactions on angles and crossing linesLotoreichikV.

Steyrergasse 30, 8010 Graz

lotoreichik@math.tugraz.atTechnische Universität Graz, Institut für Numerische MathematikAustria20132208202542166172Copyright © Lotoreichik V., 20252025Lotoreichik V.Lotoreichik V.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/1265

In this note we sharpen the lower bound previously obtained by Lobanov et al [LLP10] for the spectrum of the 2D Schrödinger operator with a δ-interaction supported on a planar angle. Using the same method we obtain the lower bound on the spectrum of the 2D Schrödinger operator with a δ-interaction supported on crossing straight lines. The latter operators arise in the three-body quantum problem with δ-interactions between particles.

Schrödinger operatorsδ-interactionsspectral estimatesThe work was supported by Austrian Science Fund (FWF): project P 25162-N26 and partially supported by the Ministry of Education and Science of Russian Federation: project 14.B37.21.0457.References[BLL13] J. Behrndt, M. Langer, and V. Lotoreichik, Schr¨odinger operators with δ and δ′-potentials supported on hypersurfaces, Ann. Henri Poincar ´e, 14, P. 385–423 (2013).[BLL13] J. Behrndt, M. Langer, and V. Lotoreichik, Schr¨odinger operators with δ and δ′-potentials supported on hypersurfaces, Ann. Henri Poincar ´e, 14, P. 385–423 (2013).[BEKS94] J.F. Brasche, P. Exner, Yu.A. Kuperin, and P. ˇ Seba, Schr¨odinger operators with singular inter actions, J. Math. Anal. Appl., 184, P. 112–139 (1994).[BEKS94] J.F. Brasche, P. Exner, Yu.A. Kuperin, and P. ˇ Seba, Schr¨odinger operators with singular inter actions, J. Math. Anal. Appl., 184, P. 112–139 (1994).[BEW08] B.M. Brown, M. S. P. Eastham, and I. Wood, An example on the discrete spectrum of a star graph, in Analysis on Graphs and Its Applications, Proceedings of Symposia in Pure Mathematics, American Mathematical Society (2008).[BEW08] B.M. Brown, M. S. P. Eastham, and I. Wood, An example on the discrete spectrum of a star graph, in Analysis on Graphs and Its Applications, Proceedings of Symposia in Pure Mathematics, American Mathematical Society (2008).[BEW09] B.M. Brown, M. S. P. Eastham, and I. Wood, Estimates for the lowest eigenvalue of a star graph, J. Math. Anal. Appl., 354, P. 24–30 (2009).[BEW09] B.M. Brown, M. S. P. Eastham, and I. Wood, Estimates for the lowest eigenvalue of a star graph, J. Math. Anal. Appl., 354, P. 24–30 (2009).[CDR06] H. Cornean, P. Duclos, and B. Ricaud, On critical stability of three quantum charges interacting through delta potentials, Few-Body Systems, 38, P. 125–131 (2006).[CDR06] H. Cornean, P. Duclos, and B. Ricaud, On critical stability of three quantum charges interacting through delta potentials, Few-Body Systems, 38, P. 125–131 (2006).[CDR08] H. Cornean, P. Duclos, and B. Ricaud, On the skeleton method and an application to a quantum scissor, in Analysis on graphs and its applications, Proc. Sympos. Pure Math. Amer. Math. Soc. Providence (2008).[CDR08] H. Cornean, P. Duclos, and B. Ricaud, On the skeleton method and an application to a quantum scissor, in Analysis on graphs and its applications, Proc. Sympos. Pure Math. Amer. Math. Soc. Providence (2008).[E08] P. Exner, Leaky quantum graphs: a review, in Analysis on graphs and its applications, Proc. Sympos. Pure Math. Amer. Math. Soc. Providence (2008).[E08] P. Exner, Leaky quantum graphs: a review, in Analysis on graphs and its applications, Proc. Sympos. Pure Math. Amer. Math. Soc. Providence (2008).[EI01] P. Exner and I. Ichinose, Geometrically induced spectrum in curved leaky wires, J. Phys. A, 34, P. 1439–1450 (2001).[EI01] P. Exner and I. Ichinose, Geometrically induced spectrum in curved leaky wires, J. Phys. A, 34, P. 1439–1450 (2001).[EN03] P. Exner and K. Nˇemcov´a, Leaky quantum graphs: approximations by point-interaction Hamiltoni ans, J. Phys. A, 36, P. 10173–10193 (2003).[EN03] P. Exner and K. Nˇemcov´a, Leaky quantum graphs: approximations by point-interaction Hamiltoni ans, J. Phys. A, 36, P. 10173–10193 (2003).[LP08] M. Levitin and L. Parnovski, On the principal eigenvalue of a Robin problem with a large parameter, Math. Nachr., 281, P. 272–281 (2008).[LP08] M. Levitin and L. Parnovski, On the principal eigenvalue of a Robin problem with a large parameter, Math. Nachr., 281, P. 272–281 (2008).[LLP10] I. Lobanov, V. Lotoreichik, and I.Yu. Popov, Lower bound on the spectrum of the two-dimensional Schr¨odinger operator with a delta-perturbation on a curve, Theor. Math. Phys., 162, P. 332–340 (2010).[LLP10] I. Lobanov, V. Lotoreichik, and I.Yu. Popov, Lower bound on the spectrum of the two-dimensional Schr¨odinger operator with a delta-perturbation on a curve, Theor. Math. Phys., 162, P. 332–340 (2010).[M87] J. Marschall, The trace of Sobolev-Slobodeckij spaces on Lipschitz domains, Manuscripta Math., 58, P. 47–65 (1987).[M87] J. Marschall, The trace of Sobolev-Slobodeckij spaces on Lipschitz domains, Manuscripta Math., 58, P. 47–65 (1987).[McL] W. McLean, Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000)[McL] W. McLean, Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000)

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