2D Schrödinger operators with singular potentials concentrated near curves

References

• Green IM, Moszkowski SA. Nuclear coupling schemes with a surface delta interaction. Phys Rev. 1965;139(B4):B790.
   Google Scholar
• Lloyd P. Pseudo-potential models in the theory of band structure. Proc Phys Soc. 1965;86(4):825.
   Google Scholar
• Faessler A, Plastino A. The surface delta interaction in the transuranic nuclei. Z Phys. 1967;203(4):333–345.
   Google Scholar
• Blinder SM. Modified delta-function potential for hyperfine interactions. Phys Rev A. 1978;18(3):853.
   Web of Science ®Google Scholar
• Antoine JP, Gesztesy F, Shabani J. Exactly solvable models of sphere interactions in quantum mechanics. J Phys A. 1987;20(12):3687.
   Google Scholar
• Shimada SI. The approximation of the Schrödinger operators with penetrable wall potentials in terms of short range Hamiltonians. J Math Kyoto Univ. 1992;32(3):583–592.
   Google Scholar
• Behrndt J, Exner P, Holzmann M, et al. Approximation of Schrödinger operators with δ-interactions supported on hypersurfaces. Math Nachr. 2017;290(8–9):1215–1248.
   Web of Science ®Google Scholar
• Shimada SI. Low energy scattering with a penetrable wall interaction. J Math Kyoto Univ. 1994;34(1):95–147.
   Google Scholar
• Shimada SI. The analytic continuation of the scattering kernel associated with the Schrödinger operator with a penetrable wall interaction. J Math Kyoto Univ. 1994;34(1):171–190.
   Google Scholar
• Exner P, Fraas M. On the dense point and absolutely continuous spectrum for Hamiltonians with concentric δ shells. Lett Math Phys. 2007;82(1):25–37.
   Web of Science ®Google Scholar
• Albeverio S, Kostenko A, Malamud M, et al. Spherical Schrödinger operators with δ-type interactions. J Math Phys. 2013;54(5):052103.
   Google Scholar
• Exner P, Fraas M. Interlaced dense point and absolutely continuous spectra for Hamiltonians with concentric-shell singular interactions. In: Mathematical results in quantum mechanics; 2008. pp. 48–65.
   Google Scholar
• Exner P, Fraas M. On geometric perturbations of critical Schrödinger operators with a surface interaction. J Math Phys. 2009;50(11):112101.
   Google Scholar
• Dittrich J, Exner P, Kühn C, et al. On eigenvalue asymptotics for strong δ-interactions supported by surfaces with boundaries. Asymptot Anal. 2016;97(1–2):1–25.
   Web of Science ®Google Scholar
• Mantile A, Posilicano A, Sini M. Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces. J Differ Equ. 2016;261(1):1–55.
   Web of Science ®Google Scholar
• Behrndt J, Langer M, Lotoreichik V. Schrödinger operators with δ- and δ′-potentials supported on hypersurfaces. Ann Henri Poincaré. 2013;14(2):385–423.
   Web of Science ®Google Scholar
• Exner P, Jex M. Spectral asymptotics of a strong δ′ interaction supported by a surface. Phys Lett A. 2014;378(30–31):2091–2095.
   Web of Science ®Google Scholar
• Behrndt J, Exner P, Lotoreichik V. Schrödinger operators with δ- and δ′-interactions on Lipschitz surfaces and chromatic numbers of associated partitions. Rev Math Phys. 2014;26(08):1450015.
   Google Scholar
• Lotoreichik V. Spectral isoperimetric inequalities for singular interactions on open arcs. Appl Anal. 2019;98(8):1451–1460.
   Web of Science ®Google Scholar
• Lotoreichik V, Rohleder J. An eigenvalue inequality for Schrödinger operators with δ- and δ′-interactions supported on hypersurfaces. In: Operator algebras and mathematical physics. Birkhäuser, Cham; 2015. p. 173–184.
   Google Scholar
• Behrndt J, Exner P, Lotoreichik V. Schrödinger operators with δ-interactions supported on conical surfaces. J Phys A. 2014;47(35):355202.
   Google Scholar
• Ourmières-Bonafos T, Pankrashkin K. Discrete spectrum of interactions concentrated near conical surfaces. Appl Anal. 2018;97(9):1628–1649.
   Web of Science ®Google Scholar
• Exner P, Rohleder J. Generalized interactions supported on hypersurfaces. J Math Phys. 2016;57(4):041507.
   Google Scholar
• Exner P, Khrabustovskyi A. On the spectrum of narrow Neumann waveguide with periodically distributed traps. J Phys A. 2015;48(31):315301.
   Google Scholar
• Jex M. Spectral asymptotics for a δ′ interaction supported by an infinite curve. In: Mathematical Results in Quantum Mechanics: Proceedings of the QMath12 Conference; 2015. p. 259–265.
   Google Scholar
• Jex M, Lotoreichik V. On absence of bound states for weakly attractive δ′-interactions supported on non-closed curves in R2. J Math Phys. 2016;57(2):022101.
   Google Scholar
• Golovaty Yu, Man'ko S. Solvable models for the Schrödinger operators with δ′-like potentials. Ukr Math Bull. 2009;6(2):169–203. Available from: arXiv:0909.1034v2 [math.SP]
   Google Scholar
• Golovaty YuD, Hryniv RO. On norm resolvent convergence of Schrödinger operators with δ′-like potentials. J Phys A. 2010;43:155204 (14pp) (A Corrigendum: 2011 J. Phys. A: Math. Theor. 44 049802).
   Google Scholar
• Golovaty Yu. Schrödinger operators with (αδ′+βδ)-like potentials: norm resolvent convergence and solvable models. Methods Funct Anal Topol. 2012;18(3):243–255.
   Google Scholar
• Golovaty Yu. D, Hryniv RO. Norm resolvent convergence of singularly scaled Schrödinger operators and δ′-potentials. Proc R Soc Edinb A Math. 2013;143:791–816.
   Web of Science ®Google Scholar
• Golovaty Yu. 1D Schrödinger operators with short range interactions: two-scale regularization of distributional potentials. Integral Equ Oper Theory. 2013;75(3):341–362.
   Web of Science ®Google Scholar
• Christiansen PL, Arnbak HC, Zolotaryuk AV, et al. On the existence of resonances in the transmission probability for interactions arising from derivatives of Dirac's delta function. J Phys A. 2003;36:7589–7600.
   Google Scholar
• Zolotaryuk AV. Two-parametric resonant tunneling across the δ′(x) potential. Adv Sci Lett. 2008;1:187–191.
   Google Scholar
• Zolotaryuk AV. Point interactions of the dipole type defined through a three-parametric power regularization. J Phys A. 2010;43:105302.
   Google Scholar
• Unverdi SO, Tryggvason G. A front-tracking method for viscous, incompressible, multi-fluid flows. J Comput Phys. 1992;100:25–37.
   Web of Science ®Google Scholar
• Juric D, Tryggvason G. A front-tracking method for dendritic solidification. J Comput Phys. 1996;123(1):127–148.
   Web of Science ®Google Scholar
• Uddin E, Sung HJ. Simulation of flow-flexible body interactions with large deformation. Int J Numer Methods Fluids. 2012;70(9):1089–1102.
   Web of Science ®Google Scholar
• Lange RJ. Potential theory, path integrals and the Laplacian of the indicator. J High Energy Phys. 2012(11):32.
   Google Scholar
• Manasse FK, Misner CW. Fermi normal coordinates and some basic concepts in differential geometry. J Math Phys. 1963;4(6):735–745.
   Web of Science ®Google Scholar
• Berezin FA, Shubin MA. The Schrödinger equation. Vol. 66. Springer Science & Business Media; 2012.
   Google Scholar
• Golovaty YD, Lavrenyuk AS. Asymptotic expansions of local eigenvibrations for plate with density perturbed in neighbourhood of one-dimensional manifold. Mat Stud. 2000;13(1):51–62.
   Google Scholar
• Golovaty YD, Gómez D, Lobo M, et al. On vibrating membranes with very heavy thin inclusions. Math Models Methods Appl Sci. 2004;14(07):987–1034.
   Google Scholar
• Gómez D, Nazarov SA, Pérez-Martínez M. A Dirichlet spectral problem in domains surrounded by thin stiff and heavy bands. In: Constanda C. editor. Computational and analytic methods in science and engineering. Birkhäuser, Cham, 2020.
   Google Scholar
• Fedoruyk MV, Babich VM, Lazutkin VF, et al. Partial differential equations V: asymptotic methods for partial differential equations. Vol. 5. Springer Science & Business Media; 1999.
   Google Scholar