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We consider the Schrödinger operator    H  =  −  Δ  +  V  (  |  x  |  )    with radial potential   V   which may have singularity at 0 and a quadratic decay at infinity. First, we study the structure of positive harmonic functions of   H   and give their precise behavior. Second, under quite general conditions we prove an upper bound for the correspond heat kernel    p  (  x  ,  y  ,  t  )    of the type    0  <  p   (  x  ,  y  ,  t  )   ⩽  Ct   −  N  /  2U  (  min  {  |  x  |  ,    t    }  )  U  (  min  {  |  y  |  ,    t    }  )U    (   t   )   2exp   −|  x  −  y  |   2    C  tfor all   x  ,    y  ∈    R   Nand    t  >  0   , where   U   is a positive harmonic function of   H  . Third, if    U  2    is an    A  2    weight onR   N   , then we prove a lower bound of a similar type.

The heat kernel of a Schrödinger operator with inverse square potential