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We establish sharp pointwise estimates for the ground states of some singular fractional Schrödinger operators on relatively compact Euclidean subsets. The considered operators are of the type (−∆) α/2 | Ω − c|x| −α , where (−∆) α/2 | Ω is the fraction-Laplacien on an open subset Ω in R d with zero exterior condition and 0 < c ≤ (2 ) 2 . The intrinsic ultracontractivity property for such operators is discussed as well and a sharp large time asymptotic for their heat kernels is derived.

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