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 L 2 ( ℝ 2 ) , with Aharonov-Bohm vector potential, A → ( x 1 , x 2 ) = α ( − x 2 , x 1 ) / | 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).