The spectral function of a singular differential operator of order 2m

We study the spectral function of a self-adjoint semibounded below differential operator on a Hilbert space L2[0,∞) and obtain the formulae for the spectral function of the operator (-1)my(2m)(x) with general boundary conditions at the zero. In particular, for the boundary conditions y(0)=y'(0)=...=y(m-1)(0)=0 we find the explicit form of the spectral function ΘmB'(x,x,λ) on the diagonal x=y for λ≥0.