Skip to main content
×
×
Home

REDUCTION OF DIMENSION AS A CONSEQUENCE OF NORM-RESOLVENT CONVERGENCE AND APPLICATIONS

  • D. Krejčiřík (a1), N. Raymond (a2), J. Royer (a3) and P. Siegl (a4) (a5)
Abstract

This paper is devoted to dimensional reductions via the norm-resolvent convergence. We derive explicit bounds on the resolvent difference as well as spectral asymptotics. The efficiency of our abstract tool is demonstrated by its application on seemingly different partial differential equation problems from various areas of mathematical physics; all are analysed in a unified manner, known results are recovered and new ones established.

Copyright
References
Hide All
1. Borisov, D. and Krejčiřík, D., O𝓣-symmetric waveguides. Integral Equations Operator Theory 62 2008, 489515.
2. Borisov, D. and Krejčiřík, D., The effective Hamiltonian for thin layers with non-Hermitian Robin-type boundary conditions. Asymptot. Anal. 76(1) 2012, 4959.
3. Duclos, P. and Exner, P., Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7(1) 1995, 73102.
4. Fournais, S. and Helffer, B., Spectral Methods in Surface Superconductivity (Progress in Nonlinear Differential Equations and their Applications 77 ), Birkhäuser (Boston, 2010).
5. Grieser, D., Thin tubes in mathematical physics, global analysis and spectral geometry. In Analysis on Graphs and its Applications (Proceedings of Symposia in Pure Mathematics 77 ), American Mathematical Society (Providence, RI, 2008), 565593.
6. Helffer, B. and Kachmar, A., Eigenvalues for the Robin Laplacian in domains with variable curvature. Trans. Amer. Math. Soc. 369(5) 2017, 32533287.
7. Helffer, B., Kachmar, A. and Raymond, N., Tunneling for the Robin Laplacian in smooth planar domains. Commun. Contemp. Math. 19(1) 2017, 1650030, 38.
8. Jecko, T., On the mathematical treatment of the Born–Oppenheimer approximation. J. Math. Phys. 55(5) 2014, 053504, 26.
9. Kachmar, A., Keraval, P. and Raymond, N., Weyl formulae for the Robin Laplacian in the semiclassical limit. Confluentes Math. 8(2) 2016, 3957.
10. Krejčiřík, D. and Raymond, N., Magnetic effects in curved quantum waveguides. Ann. Henri Poincaré 15(10) 2014, 19932024.
11. Krejčiřík, D., Raymond, N. and Tušek, M., The magnetic Laplacian in shrinking tubular neighborhoods of hypersurfaces. J. Geom. Anal. 25(4) 2015, 25462564.
12. Krejčiřík, D. and Šediváková, H., The effective Hamiltonian in curved quantum waveguides under mild regularity assumptions. Rev. Math. Phys. 24(7) 2012, 1250018, 39.
13. Lampart, J. and Teufel, S., The adiabatic limit of the Laplacian on thin fibre bundles. In Microlocal Methods in Mathematical Physics and Global Analysis (Trends in Mathematics), Birkhäuser/Springer (Basel, 2013), 3336.
14. Lampart, J. and Teufel, S., The adiabatic limit of Schrödinger operators on fibre bundles. Math. Ann. 367 2016, 16471683.
15. Mazya, V., Nazarov, S. and Plamenevskij, B., Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains (Operator Theory: Advances and Applications 111 ), Birkhäuser (Basel, 2000). Translated from the German by Georg Heinig and Christian Posthoff.
16. Oliveira, C. R. d. and Rossini, A. F., Effective operators for Robin Laplacian in thin two- and three-dimensional curved waveguides. Submitted for publication.
17. Pankrashkin, K. and Popoff, N., An effective Hamiltonian for the eigenvalue asymptotics of the Robin Laplacian with a large parameter. J. Math. Pures Appl. (9) 106(4) 2016, 615650.
18. Raymond, N., Bound States of the Magnetic Schrödinger Operator (EMS Tracts 27 ), European Mathematical Society (Zurich, 2017).
19. Wachsmuth, J. and Teufel, S., Effective Hamiltonians for constrained quantum systems. Mem. Amer. Math. Soc. 230(1083) 2014.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed