Waveguides with combined Dirichlet and Robin boundary conditions

Mathematical Physics Analysis and Geometry

Volumn 9, Issue 4, 2006, Pages 335-352

  Freitas, P ^a   Krejcirik D ^b  

^a UNIVERSITY OF LISBON   (Portugal)

^b NUCLEAR PHYSICS INSTITUTE   (Czech Republic)

Author keywords

Dirichlet and Robin boundary conditions; Eigenvalues in strips and annuli; Hardy inequality; Laplacian; Waveguides

Indexed keywords

EID: 34249713084     PISSN: 13850172     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11040-007-9015-6     Document Type: Article

References (34)

1. Adams, R.A.: Sobolev Spaces. Academic, New York (1975)
2. Ashbaugh, M.S., Exner, P.: Lower bounds to bound state energies in bent tubes. Phys. Lett. A 150(3-4), 183-186 (1990)
3. Borisov, D., Ekholm, T., Kovařík, H.: Spectrum of the magnetic Schrödinger operator in a waveguide with combined boundary conditions. Ann. Henri Poincaré 6, 327-342 (2005)
4. Chenaud, B., Duclos, P., Freitas, P., Krejčiřík, D.: Geometrically induced discrete spectrum in curved tubes. Differential Geom. Appl. 23(2), 95-105 (2005)
5. Dancer, E.N., Daners, D.: Domain perturbation for elliptic equations subject to Robin boundary conditions. J. Differential Equations 138(1), 86-132 (1997)
6. Daners, D.: Robin boundary value problems on arbitrary domains. Trans. Amer. Math. Soc. 352(9), 4207-4236 (2000)
7. Daners, D.: Dirichlet problems on varying domains. J. Differential Equations 188, 591-624 (2003)
8. Davies, E.B., Parnovski, L.: Trapped modes in acoustic waveguides. Quart. J. Mech. Appl. Math. 51, 477-492 (1998)
9. Dittrich J., Kříž, J.: Curved planar quantum wires with Dirichlet and Neumann boundary conditions. J. Phys. A 35, L269-275 (2002)
10. Duclos, P., Exner, P.: Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7, 73-102 (1995)
11. Ekholm, T., Kovařík, H.: Stability of the magnetic Schrödinger operator in a waveguide. Coram. Partial Differential Equations 30(4), 539-565 (2005)
12. Ekholm, T., Kovařík, H., Krejčiřík, D.: A Hardy inequality in twisted waveguides. Arch. Rational Mech. Anal, (preprint on math-ph/0512050 (2005)) (in press)
13. Evans, D.V., Levitin, M., Vassiliev, D.: Existence theorems for trapped modes. J. Fluid Mech. 261, 21-31 (1994)
14. Exner, P., Freitas, P., Krejčiřík, D.: A lower bound to the spectral threshold in curved tubes. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 460(2052), 3457-3467 (2004)
15. Exner, P., Krejčiřík, D.: Quantum waveguides with a lateral semitransparent barrier: spectral and scattering properties. J. Phys. A 32, 4475-4494 (1999)
16. Exner, P., Krejčiřík, D.: Waveguides coupled through a semitransparent barrier: a Birman-Schwinger analysis. Rev. Math. Phys. 13(3), 307-334 (2001)
17. Exner, P., Šeba, P.: Bound states in curved quantum waveguides. J. Math. Phys. 30, 2574-2580 (1989)
18. Flucher, M.: Approximation of Dirichlet eigenvalues on domains with small holes. J. Math. Anal. Appl. 193(1), 169-199 (1995)
19. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin Heidelberg New York (1983)
20. Goldstone, J., Jaffe, R.L.: Bound states in twisting tubes. Phys. Rev. B 45, 14100-14107 (1992)
21. Johnson, E.R., Levitin, M., Parnovski, L.: Existence of eigenvalues of a linear operator pencil in a curved waveguide - localized shelf waves on a curved coast. SIAM J. Math. Anal. 37(5), 1465-1481 (2006)
22. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin Heidelberg New York (1966)
23. Klingenberg, W.: A Course in Differential Geometry. Springer, Berlin Heidelberg New York (1978)
24. Krejčiřík, D.: Hardy inequalities for strips on ruled surfaces. J. Inequal. Appl. 2006, Article ID 46409, 10 (2006)
25. Krejčiřík, D., Kříž, J.: On the spectrum of curved quantum waveguides. Publ. RIMS, Kyoto University 41(3), 757-791 (2005)
26. Kreyszig, E.: Differential Geometry. University of Toronto Press, Toronto (1959)
27. Kuchment, P., Zeng, H.: Convergence of spectra of mesoscopic systems collapsing onto a graph. J. Math. Anal. Appl. 258, 671-700 (2001)
28. Kuchment, P., Zeng, H.: Asymptotics of spectra of Neumann Laplacians in thin domains. In: Advances in Differential Equations and Mathematical Physics, Birmingham, AL, 2002. Contemporary Mathematics, vol. 327, pp. 199-213. American Mathematical Society, Providence, RI (2003)
29. Londergan, J.T., Carini, J.P., Murdock, D.P.: Binding and Scattering in Two-dimensional Systems, LNP, vol. m60. Springer, Berlin Heidelberg New York (1999)
30. Olendski, O., Mikhailovska, L.: Localized-mode evolution in a curved planar waveguide with combined Dirichlet and Neumann boundary conditions. Phys. Rev. E 67, art. 056625 (2003)
31. Rauch, J., Taylor, M.: Potential and scattering theory on wildly perturbed domains. J. Funct. Anal. 18, 27-59 (1975)
32. Renger, W., Bulla, W.: Existence of bound states in quantum waveguides under weak conditions. Lett. Math. Phys. 35, 1-12 (1995)
33. Stollmann, P.: A convergence theorem for Dirichlet forms with applications to boundary value problems with varying domains. Math. Z. 219, 275-287 (1995)
34. Wang, Z.X., Guo, D.R.: Special Functions. World Scientific, Teaneck, NJ (1989)