Opuscula Math. 36, no. 6 (2016), 769-786
http://dx.doi.org/10.7494/OpMath.2016.36.6.769

Opuscula Mathematica

# Dispersion estimates for spherical Schrödinger equations: the effect of boundary conditions

Markus Holzleitner
Aleksey Kostenko
Gerald Teschl

Abstract. We investigate the dependence of the $$L^1\to L^{\infty}$$ dispersive estimates for one-dimensional radial Schrödinger operators on boundary conditions at $$0$$. In contrast to the case of additive perturbations, we show that the change of a boundary condition at zero results in the change of the dispersive decay estimates if the angular momentum is positive, $$l\in (0,1/2)$$. However, for nonpositive angular momenta, $$l\in (-1/2,0]$$, the standard $$O(|t|^{-1/2})$$ decay remains true for all self-adjoint realizations.

Keywords: Schrödinger equation, dispersive estimates, scattering.

Mathematics Subject Classification: 35Q41, 34L25, 81U30, 81Q15.

Full text (pdf)

1. A. Ananieva, V. Budika, To the spectral theory of the Bessel operator on a finite interval and half-line, Ukrainian Mat. Visnyk 12 (2015) 2, 160-199 [in Russian]; English transl.: J. Math. Sci. 211 (2015), 624-645.
2. V.I. Bogachev, Measure Theory. I, Springer-Verlag, Berlin, Heidelberg, 2007.
3. A.V. Bukhvalov, Application of methods of the theory of order-bounded operators to the theory of operators in $$L^p$$-spaces, Russ. Math. Surveys 38 (1983), 43-98.
4. J. Dereziński, S. Richard, On almost homogeneous Schrödinger operators, arXiv:1604.03340.
5. I. Egorova, E. Kopylova, V. Marchenko, G. Teschl, Dispersion estimates for one-dimensional Schrödinger and Klein-Gordon equations revisited, Russ. Math. Surveys 71 (2016), 391-415.
6. A. Erdelyi, Tables of Integral Transforms, vol. 1, McGraw-Hill, New York, 1954.
7. W.N. Everitt, H. Kalf, The Bessel differential equation and the Hankel transform, J. Comp. Appl. Math. 208 (2007), 3-19.
8. F. Gesztesy, B. Simon, G. Teschl, Zeros of the Wronskian and renormalized oscillation theory, Amer. J. Math. 118 (1996), 571-594.
9. M. Goldberg, W. Schlag, Dispersive estimates for Schrödinger operators in dimensions one and three, Comm. Math. Phys. 251 (2004), 157-178.
10. E. Kopylova, Dispersion estimates for Schrödinger and Klein-Gordon equation, Russ. Math. Surveys 65 (2010) 1, 95-142.
11. A. Kostenko, A. Sakhnovich, G. Teschl, Weyl-Titchmarsh theory for Schrödinger operators with strongly singular potentials, Int. Math. Res. Not. 2012 (2012), 1699-1747.
12. A. Kostenko, G. Teschl, Spectral asymptotics for perturbed spherical Schrödinger operators and applications to quantum scattering, Comm. Math. Phys. 322 (2013), 255-275.
13. A. Kostenko, G. Teschl, J.H. Toloza, Dispersion estimates for spherical Schrödinger equations, Ann. Henri Poincaré 17 (2016), 3147-3176.
14. H. Kovařík, F. Truc, Schrödinger operators on a half-line with inverse square potentials, Math. Model. Nat. Phenom. 9 (2014), 170-176.
15. E. Liflyand, S. Samko, R. Trigub, The Wiener algebra of absolutely convergent Fourier integrals: an overview, Anal. Math. Phys. 2 (2012), 1-68.
16. E. Liflyand, R. Trigub, Conditions for the absolute convergence of Fourier integrals, J. Approx. Theory 163 (2011), 438-459.
17. F.W.J. Olver et al. (eds), NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010.
18. W. Schlag, Dispersive estimates for Schrödinger operators: a survey, [in:] Mathematical aspects of nonlinear dispersive equations, Ann. Math. Stud. 163, Princeton Univ. Press, Princeton, NJ, 2007, 255-285.
19. E.M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Math. Series 43, Princeton University Press, Princeton, NJ, 1993.
20. G. Teschl, Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators, 2nd ed., Amer. Math. Soc., Rhode Island, 2014.
21. G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, 1944.
22. R. Weder, $$L^p-L^{\dot{p}}$$ estimates for the Schrödinger equation on the line and inverse scattering for the nonlinear Schrödinger equation with a potential, J. Funct. Anal. 170 (2000), 37-68.
23. R. Weder, The $$L^p-L^{\dot{p}}$$ estimates for the Schrödinger equation on the half-line, J. Math. Anal. Appl. 281 (2003), 233-243.
• Markus Holzleitner
• University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
• Aleksey Kostenko
• University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
• Gerald Teschl
• University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
• International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria
• Communicated by S.N. Naboko.
• Received: 2016-01-07.
• Revised: 2016-05-12.
• Accepted: 2016-05-12.
• Published online: 2016-10-29.

Cite this article as:
Markus Holzleitner, Aleksey Kostenko, Gerald Teschl, Dispersion estimates for spherical Schrödinger equations: the effect of boundary conditions, Opuscula Math. 36, no. 6 (2016), 769-786, http://dx.doi.org/10.7494/OpMath.2016.36.6.769

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.