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We explicitly compute the spectrum and eigenfunctions of the magnetic SchrÃƒÂ¶dinger operator H(AÃ¢Â†Â’,V)=(iÃ¢ÂˆÂ‡+AÃ¢Â†Â’)2+V in L2(Ã¢Â„Â2), with Aharonov-Bohm vector potential, AÃ¢Â†Â’(x1,x2)=ÃŽÂ±(Ã¢ÂˆÂ’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 ÃŽÂ± and both greater than unity. In the case of quadratic potential, it turns out that the LT inequality holds for all ÃŽÂ³Ã¢Â‰Â¥1 with the classical constant, as expected from the nonmagnetic system (harmonic oscillator)

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