Mild singular potentials as effective Laplacians in narrow strips

  • César R. de Oliveira
  • Alessandra A. Verri

Abstract

We propose to obtain information on one-dimensional Schrödinger operators on bounded intervals by approaching them as effective operators of the Laplacian in thin planar strips.  Here we develop this idea to get spectral knowledge of some mild singular potentials with Dirichlet boundary conditions.  Special preparations, including a result on perturbations of quadratic forms, are included as well.

References

Bedoya, R., de Oliveira, C. R., and Verri, A. A., Complex Γ-convergence and magnetic Dirichlet Laplacian in bounded thin tubes, J. Spectr. Theory 4 (2014), no. 3, 621–642. http://dx.doi.org/10.4171/JST/81

Duclos, P. and Exner, P., Curvature-induced bound states in quantum waveguides in two and three dimensions, Rev. Math. Phys. 7 (1995), no. 1, 73–102. http://dx.doi.org/10.1142/S0129055X95000062

Faris, W. G., Self-adjoint operators, Lecture Notes in Mathematics, Vol. 433, Springer-Verlag, Berlin-New York, 1975.

Frank, W. M., Land, D. J., and Spector, R. M., Singular potentials, Rev. Modern Phys. 43 (1971), no. 1, 36–98.

Freitas, P. and Krejčiř'ık, D., Location of the nodal set for thin curved tubes, Indiana Univ. Math. J. 57 (2008), no. 1, 343–375. http://dx.doi.org/10.1512/iumj.2008.57.3170

Friedlander, L. and Solomyak, M., On the spectrum of the Dirichlet Laplacian in a narrow infinite strip, Spectral theory of differential operators, Amer. Math. Soc. Transl. Ser. 2, vol. 225, Amer. Math. Soc., Providence, RI, 2008, pp. 103--116. http://dx.doi.org/10.1090/trans2/225/07

Gesztesy, F., On the one-dimensional Coulomb Hamiltonian, J. Phys. A 13 (1980), no. 3, 867–875.

Gesztesy, F. and Zinchenko, M., On spectral theory for Schrödinger operators with strongly singular potentials, Math. Nachr. 279 (2006), no. 9-10, 1041–1082. http://dx.doi.org/10.1002/mana.200510410

Kato, T., Perturbation theory for linear operators, second ed., Grundlehren der Mathematischen Wissenschaften, vol. 132, Springer-Verlag, Berlin-New York, 1976.

Klaus, M., Removing cut-offs from one-dimensional Schrödinger operators, J. Phys. A 13 (1980), no. 9, L295–L298.

Krejčiř'ık, D. and Kř'ız, J., On the spectrum of curved planar waveguides, Publ. Res. Inst. Math. Sci. 41 (2005), no. 3, 757–791.

Krejčiř'ık, D. and Šediváková, H., The effective Hamiltonian in curved quantum waveguides under mild regularity assumptions, Rev. Math. Phys. 24 (2012), no. 7, 1250018, 39 p. http://dx.doi.org/10.1142/S0129055X12500183

Naĭmark, M. A., Linear differential operators. Part II: Linear differential operators in Hilbert space, Frederick Ungar Publishing Co., New York, 1968.

de Oliveira, C. R., Intermediate spectral theory and quantum dynamics, Progress in Mathematical Physics, vol. 54, Birkhäuser Verlag, Basel, 2009. http://dx.doi.org/10.1007/978-3-7643-8795-2

de Oliveira, C. R. and Verri, A. A., On the spectrum and weakly effective operator for Dirichlet Laplacian in thin deformed tubes, J. Math. Anal. Appl. 381 (2011), no. 1, 454–468. http://dx.doi.org/10.1016/j.jmaa.2011.03.022

de Oliveira, C. R. and Verri, A. A., Mathematical predominance of Dirichlet condition for the one-dimensional Coulomb potential, J. Math. Phys. 53 (2012), no. 5, 052104, 20 p. http://dx.doi.org/10.1063/1.4719976

Oseguera, U. and de Llano, M., Two singular potentials: the space-splitting effect, J. Math. Phys. 34 (1993), no. 10, 4575–4589. http://dx.doi.org/10.1063/1.530358

Pressley, A., Elementary differential geometry, second ed., Springer Undergraduate Mathematics Series, Springer-Verlag London, Ltd., London, 2010. http://dx.doi.org/10.1007/978-1-84882-891-9

Weidmann, J., Spectral theory of ordinary differential operators, Lecture Notes in Mathematics, vol. 1258, Springer-Verlag, Berlin, 1987. http://dx.doi.org/10.1007/BFb0077960

Weidmann, J., Spectral theory of Sturm-Liouville operators approximation by regular problems, Sturm-Liouville theory, Birkhäuser, Basel, 2005, pp. 75--98. http://dx.doi.org/10.1007/3-7643-7359-8_4

Zettl, A., Sturm-Liouville theory, Mathematical Surveys and Monographs, vol. 121, American Mathematical Society, Providence, RI, 2005.

Published
2017-02-23
How to Cite
de Oliveira, C., & Verri, A. (2017). Mild singular potentials as effective Laplacians in narrow strips. MATHEMATICA SCANDINAVICA, 120(1), 145-160. https://doi.org/10.7146/math.scand.a-25510
Section
Articles