ABSTRACT
The free Laplacian L 0 = − Δ in L 2 ( R n ) defines a self-adjoint operator with domain D ( L 0 ) = H 2 ( R n ) . Its spectrum is continuous and fills the nonnegative half line [ 0 , ∞ ) . Let R 0 ( ζ ) , ζ ∈ C \ [ 0 , ∞ ) , be the resolvent and { E 0 ( λ ) ; λ > 0 } be the spectral measure of L 0. Then the Stieltjes inversion formula and the Parseval equality give ( E 0 ( λ ) f , g ) = lim ϵ ↓ 0 1 2 π i ∫ 0 λ ( { R 0 ( τ + i ϵ ) − R 0 ( τ − i ϵ ) } f , g ) d τ = lim ϵ ↓ 0 1 2 π i ∫ 0 λ d τ ∫ R n 2 ϵ i ( | ξ | 2 − τ ) 2 + ϵ 2 f ^ ( ξ ) g ^ ( ξ ) ¯ d ξ = ∫ | ξ | 2 ≤ λ f ^ ( ξ ) g ^ ( ξ ) ¯ d ξ ,
