International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 23, Pages 3751-3766

On the spectrum and eigenfunctions of the Schrödinger operator with Aharonov-Bohm magnetic field

Anders M. Hansson

Department of Mathematics, School of Engineering Sciences, Royal Institute of Technology, Stockholm 10044, Sweden

Received 14 June 2005; Revised 6 October 2005

Copyright © 2005 Anders M. Hansson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We explicitly compute the spectrum and eigenfunctions of the magnetic Schrödinger operator H(A,V)=(i+A)2+V in L2(2), with Aharonov-Bohm vector potential, A(x1,x2)=α(x2,x1)/|x|2, and either quadratic or Coulomb scalar potential V. We also determine sharp constants in the CLR inequality, both dependent on the fractional part of α and both greater than unity. In the case of quadratic potential, it turns out that the LT inequality holds for all γ1 with the classical constant, as expected from the nonmagnetic system (harmonic oscillator).