Stability of the magnetic Schrödinger operator in a waveguide

Communications in Partial Differential Equations

Volumn 30, Issue 4-6, 2005, Pages 539-565

  Ekholm, Tomas ^a   Kovařík, Hynek ^b,c  

^a ROYAL INSTITUTE OF TECHNOLOGY   (Sweden)

^b UNIVERSITY OF STUTTGART   (Germany)

^c NUCLEAR PHYSICS INSTITUTE   (Czech Republic)

Author keywords

Hardy inequality; Magnetic field; Schr dinger operator

Indexed keywords

EID: 23444462495     PISSN: 03605302     EISSN: None     Source Type: Journal    
DOI: 10.1081/PDE-200050113     Document Type: Article

References (22)

1. Avron, J., Herbst, J., Simon, B. (1978). Schrödinger operators with magnetic fields. I. General interactions. Duke Math. J. 45:847-883.
2. Balinsky, A. A. (2003). Hardy type inequalities for Aharonov-Bohm magnetic potentials with multiple singularities. Math. Res. Lett. 10:169-176.
3. Balinsky, A. A., Evans, W. D., Lewis, R. T. (in press). On the number of negative eigenvalues of Schrödinger operators with an Aharonov-Bohm magnetic field. mp_arc 00-405.
4. Balinsky, A. A., Laptev, A., Sobolev, A. V. (2004). Generalized Hardy inequality for the magnetic Dirichlet forms. J. Stat. Phys. 116(1/4):507-521.
5. Birman, M. S., Solomyak, M. Z. (1987). Spectral Theory of Self-Adjoint Operators in Hilbert space. Dordrecht: D. Reidel.
6. Birman, M. S., Solomyak, M. Z. (1992). Schrödinger operator. Estimates for number of bound states as function-theoretical problem. Amer. Math. Soc. Transi. (2) 150:1-54.
7. Blankenbelcer, R., Goldberger, M. L., Simon, B. (1977). The bound states of weakly coupled long-range one-dimensional quantum Hamiltonians. Ann. Phys. 108:69-78.
8. Borisov, D., Exner, P., Gadyl'shin, R. R., Krejčiří k, D. (2001). Bound states in weakly deformed strips and layers. Ann. Henri Poincaré 2:553-572.
9. Bulla, W., Renger, W. (1995). Existence of bound states in quantum waveguides under weak conditions. Lett. Math. Phys. 35(1):1-12.
10. Bulla, W., Gesztesy, F., Renger, W., Simon, B. (1997). Weakly coupled bound states in quantum waveguides. Proc. Amer. Math. Soc. 125(5):1487-1495.
11. Duclos, P., Exner, P. (1995). Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7:73-102.
12. Evans, W. D., Lewis, R. T. (2004). On the Rellich inequality with magnetic potentials. mp_arc 04-93.
13. Exner, P., Šeba, P. (1989). Bound states in curved quantum waveguides. J. Math. Phys. 30:2574-2580.
14. Goldstone, J., Jaffe, R. L. (1992). Bound states in twisting tubes. Phys. Rev. B45:14100-14107.
15. Hardy, G. H. (1920). Note on a theorem of Hilbert. Math. Zeit. 6:314-317.
16. Hundertmark, D., Simon, B. (2004). A diamagnetic inequality for semigroup differences. J. Reine Angew. Math. 571:107-130.
17. Kato, T. (1973). Schrödinger operators with singular potentials. Israel J. Math. 13:135-148.
18. Laptev, A., Weidl, T. (1999). Hardy inequalities for magnetic Dirichlet forms. Oper. Theory Adv. Appl. 108:299-305.
19. Mantoiu, M., Richard, S. (2000). Absence of singular spectrum for Schrödinger operators with anisotropic potentials and magnetic fields. J. Math. Phys. 41:2732-2740.
20. Melgaard, M., Ouhabaz, E.-M., Rozenblum, G. (2004). Spectral properties of perturbed multivortex Aharonov-Bohm Hamiltonians. Ann. Henri Poincaré 5:979-1012.
21. Simon, B. (1979). Maximal and minimal Schrödinger forms. J. Operator Theory 1:37-47.
22. Weidl, T. (1997). Remarks on virtual bound states for semi-bounded operators. Comm. PDE 24(12):25-60.