Bound states for two delta potentials supported on parallel lines on the plane
DOI:
https://doi.org/10.33910/2687-153X-2022-3-1-37-42Keywords:
spectrum, variational approach, potential supported on lineAbstract
We consider a singular-potential supported on two parallel lines in R2 as a model of two interacting macromolecules. The intensity of the potential is constant, but each line contains a finite segment with a variation. Using variational approach, we study continuous and discrete spectra and estimate the gap between the eigenvalue and the continuous spectrum as a function of shift between the line segments. The existence of bound states for the system is proven by test function with separated variables.
References
Bagmutov, A. S., Popov, I. Yu. (2020) Window-coupled nanolayers: Window shape influence on one-particle and two-particle eigenstates. Nanosystems: Physics, Chemistry, Mathematics, 11 (6), 636–641. https://doi.org/10.17586/2220-8054-2020-11-6-636-641 (In English)
Behrndt, J., Exner, P., Holzmann, M., Lotoreichik, V. (2017a) Approximation of Schroedinger operators with delta-interactions supported on hypersurfaces. Mathematische Nachrichten, 290 (8–9), 1215–1248. https://doi.org/10.1002/mana.201500498 (In English)
Behrndt, J., Frank, R. L., Kuhn, C. et al. (2017b) Spectral theory for schrodinger operators with δ-interactions supported on curves in R3. Annales Henri Poincare, 18 (4), 1305–1347. https://doi.org/10.1007/s00023-016-0532-3 (In English)
Behrndt, J., Langer, M., Lotoreichik, V. (2016) Boundary triples for Schrodinger operators with singular interactions on hypersurfaces. Nanosystems: Physics, Chemistry, Mathematics, 7 (2), 290–302. https://doi.org/10.17586/2220-8054-2016-7-2-290-302 (In English)
Brasche, J., Exner, P., Kuperin, Yu. A., Seba, P. (1994) Schroedinger operator with singular interactions. Journal of Mathematical Analysis and Applications, 184 (1), 112–139. https://doi.org/10.1006/jmaa.1994.1188 (In English)
Brasche, J., Teta, A. (1992) Spectral analysis and scattering theory for Schrodinger operators with an interaction supported by a regular curve. In: Ideas and Methods in Quantum and Statistical Physics. Cambridge: Cambridge University Press, pp. 197–211. (In English)
Exner, P., Ichinose, T. (2001) Geometrically induced spectrum in curved leaky wires. Journal of Physics A: Mathematical and General, 34 (7), 1439–1450. https://doi.org/10.1088/0305-4470/34/7/315 (In English)
Exner, P., Jex, M. (2013) Spectral asymptotics of a strong δ’ interaction on a planar loop. Journal of Physics A: Mathematical and Theoretical, 46 (34), article 345201. https://doi.org/10.1088/1751-8113/46/34/345201 (In English)
Exner, P., Kondej, S. (2002) Curvature-induced bound states for a δ interaction supported by a curve in R3. Annales Henri Poincaré, 3 (5), 967–981. https://doi.org/10.1007/s00023-002-8644-3 (In English)
Exner, P., Kondej, S. (2004) Strong-coupling asymptotic expansion for Schroedinger operators with a singular interaction supported by a curve in R3. Reviews in Mathematical Physics, 16 (5), 559–582. https://doi.org/10.1142/ S0129055X04002084 (In English)
Exner, P., Kondej, S. (2005) Scattering by local deformations of a straight leaky wire. Journal of Physics A: Mathematical and General, 38 (22), 4865–4874. https://doi.org/10.1088/0305-4470/38/22/011 (In English)
Exner, P., Kondej, S. (2015) Gap asymptotics in a weakly bent leaky quantum wire. Journal of Physics A: Mathematical and Theoretical, 48 (49), article 495301. https://doi.org/10.1088/1751-8113/48/49/495301 (In English)
Exner, P., Kondej, S., Lotoreichik, V. (2018) Asymptotics of the bound state induced by δ-interaction supported on a weakly deformed plane. Journal of Mathematical Physics, 59 (1), article 013051. https://doi.org/10.1063/1.5019931 (In English)
Exner, P., Pankrashkin, K. (2014) Strong coupling asymptotics for a singular Schrodinger operator with an interaction supported by an open arc. Communications in Partial Differential Equations, 39 (2), 193–212. https://doi.org/10.1080/03605302.2013.851213 (In English)
Exner, P., Vugalter, S. (2016) On the existence of bound states in asymmetric leaky wires. Journal of Mathematical Physics, 57 (2), article 022104. https://doi.org/10.1063/1.4941139 (In English)
Exner, P., Yoshitomi, K. (2002) Asymptotics of eigenvalues of the Schroedinger operator with a strong delta-interaction on a loop. Journal of Geometry and Physics, 41 (4), 344–358. https://doi.org/10.1016/S0393-0440(01)00071-7 (In English)
Li, F. (2016) Structure, function, and evolution of coronavirus spike proteins. Annual Review of Virology, 3, 237–261. https://doi.org/10.1146/annurev-virology-110615-042301 (In English)
Popov, I. Yu. (1993) The helmholtz resonator and the theory of operator extensions in a space with indefinite metric. Russian Academy of Sciences. Sbornik Mathematics, 75 (2), article 285. https://doi.org/10.1070/SM1993v075n02ABEH003386 (In English)
Popov, I. Yu. (1992a) The extension theory and the opening in semitransparent surface. Journal of Mathematical Physics, 33 (5), 1685–1689. https://doi.org/10.1063/1.529697 (In English)
Popov, I. Yu. (1992b) The resonator with narrow slit and the model based on the operator extensions theory. Journal of Mathematical Physics, 33 (11), 3794–3801. https://doi.org/10.1063/1.529877 (In English)
Posilicano, A. (2001) A Krein-like formula for singular perturbations of self-adjoint operators and applications. Journal of Functional Analysis, 183 (1), 109–147. https://doi.org/10.1006/jfan.2000.3730 (In English)
Posilicano, A. (2004) Boundary triples and weyl functions for singular perturbations of self-adjoint operators. Methods of Functional Analysis and Topology, 10 (2), 57–63. (In English)
Shang, J., Wan, Y., Liu, C. et al. (2020) Structure of mouse coronavirus spike protein complexed with receptor reveals mechanism for viral entry. PLOS Pathogens, 16 (3), article e1008392. https://doi.org/10.1371/journal.ppat.1008392 (In English)
Vorobiev, A. M., Trifanova, E. S., Popov, I. Yu. (2020) Resonance asymptotics for a pair quantum waveguides with common semitransparent perforated wall. Nanosystems: Physics, Chemistry, Mathematics, 11 (6), 619–627. https://doi.org/10.17586/2220-8054-2020-11-6-619-627 (In English)
Downloads
Published
Issue
Section
License
Copyright (c) 2022 Alexander S. Bagmutov, Igor Y. Popov
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
The work is provided under the terms of the Public Offer and of Creative Commons public license Attribution-NonCommercial 4.0 International (CC BY-NC 4.0). This license allows an unlimited number of persons to reproduce and share the Licensed Material in all media and formats. Any use of the Licensed Material shall contain an identification of its Creator(s) and must be for non-commercial purposes only.
