Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions

 Access to full text

 Full (PDF)

1. D. Borisov and P. Freitas, Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin planar domains. Ann. Inst. H. Poincaré Anal. Non Linéaire (2008) doi: .  DOI10.1016/j.anihpc.2007.12.001
2. G. Bouchitté, M.L. Mascarenhas and L. Trabucho, On the curvature and torsion effects in one dimensional waveguides. ESAIM: COCV13 (2007) 793–808.  
3. G. Carron, P. Exner and D. Krejčiřík, Topologically nontrivial quantum layers. J. Math. Phys.45 (2004) 774–784.  
4. E.B. Davies, Spectral theory and differential operators. Camb. Univ. Press, Cambridge (1995).  
5. J. Dittrich and J. Kříž, Curved planar quantum wires with Dirichlet and Neumann boundary conditions. J. Phys. A35 (2002) L269–L275.  
6. P. Duclos and P. Exner, Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys.7 (1995) 73–102.  
7. P. Duclos, P. Exner and D. Krejčiřík, Bound states in curved quantum layers. Commun. Math. Phys.223 (2001) 13–28.  
8. P. Exner and P. Šeba, Bound states in curved quantum waveguides. J. Math. Phys.30 (1989) 2574–2580.  
9. P. Freitas, Precise bounds and asymptotics for the first Dirichlet eigenvalue of triangles and rhombi. J. Funct. Anal.251 (2007) 376–398.  
10. P. Freitas and D. Krejčiřík, Instability results for the damped wave equation in unbounded domains. J. Diff. Eq.211 (2005) 168–186.  
11. P. Freitas and D. Krejčiřík, Waveguides with combined Dirichlet and Robin boundary conditions. Math. Phys. Anal. Geom.9 (2006) 335–352.  
12. P. Freitas and D. Krejčiřík, Location of the nodal set for thin curved tubes. Indiana Univ. Math. J.57 (2008) 343–376.  
13. L. Friedlander and M. Solomyak, On the spectrum of the Dirichlet Laplacian in a narrow strip, I. Israel J. Math. (to appear).  
14. L. Friedlander and M. Solomyak, On the spectrum of the Dirichlet Laplacian in a narrow strip, II. Amer. Math. Soc. (to appear).  
15. J. Goldstone and R.L. Jaffe, Bound states in twisting tubes. Phys. Rev. B45 (1992) 14100–14107.  
16. D. Grieser, Thin tubes in mathematical physics, global analysis and spectral geometry, in Analysis on Graphs and its Applications (Cambridge, 2007), Proceedings of Symposia in Pure Mathematics, Amer. Math. Soc. (to appear).  
17. E.R. Johnson, M. Levitin and L. Parnovski, Existence of eigenvalues of a linear operator pencil in a curved waveguide – localized shelf waves on a curved coast. SIAM J. Math. Anal.37 (2006) 1465–1481.  
18. L. Karp and M. Pinsky, First-order asymptotics of the principal eigenvalue of tubular neighborhoods, in Geometry of random motion (Ithaca, N.Y., 1987), Contemp. Math.73, Amer. Math. Soc., Providence, RI (1988) 105–119.  
19. D. Krejčiřík, Quantum strips on surfaces. J. Geom. Phys.45 (2003) 203–217.  
20. D. Krejčiřík, Hardy inequalities in strips on ruled surfaces. J. Inequal. Appl.2006 (2006) 46409.  
21. D. Krejčiřík and J. Kříž, On the spectrum of curved quantum waveguides. Publ. RIMS, Kyoto University41 (2005) 757–791.  
22. Ch. Lin and Z. Lu, Existence of bound states for layers built over hypersurfaces in ${ℝ}^{n+1}$. J. Funct. Anal.244 (2007) 1–25.  
23. Ch. Lin and Z. Lu, Quantum layers over surfaces ruled outside a compact set. J. Math. Phys.48 (2007) 053522.  
24. O. Olendski and L. Mikhailovska, Localized-mode evolution in a curved planar waveguide with combined Dirichlet and Neumann boundary conditions. Phys. Rev. E67 (2003) 056625.  
25. M. Reed and B. Simon, Methods of modern mathematical physics, I. Functional analysis. Academic Press, New York (1972).  
26. M. Schatzman, On the eigenvalues of the Laplace operator on a thin set with Neumann boundary conditions. Appl. Anal.61 (1996) 293–306.