Asymptotic Behavior of Spectral Functions for Schrodinger Forms with Signed Measures

Let {X-t}(t >= 0) be the rotationally invariant a-stable process and define the Schrodinger forms by two methods. In one method, the perturbation is given by -(mu(0) + lambda nu) (lambda >= 0), where both mu(0) and nu are positive and mu(0) is critical. In the other method, the perturbation is given by lambda mu (lambda is an element of R), where mu is a critical signed measure. In this paper we consider the asymptotic behavior of the spectral functions defined from these Schrodinger forms. The results are consistent with the differentiability of the spectral functions given in Nishimori (Tohoku Math J 65:467-494, 2013, [5]) or Takeda and Tsuchida (Trans Amer Math 359:4031-4054, 2007, [7]).