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

We explicitly compute the spectrum and eigenfunctions of the magnetic Schrödinger operator H(A→,V)=(i∇+A→)2+V in L2(ℝ2), 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 γ≥1 with the classical constant, as expected from the nonmagnetic system (harmonic oscillator)