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 $L_2[0,\infty)$ and obtain the formulae for the spectral function of the operator $(-1)^{m}y^{(2m)}(x)$ with general boundary conditions at the zero. In particular, for the boundary conditions $y(0)=y'(0)=\dots=y^{(m-1)}(0)=0$ we find the explicit form of the spectral function $\Theta_{mB'}(x,x,\lambda)$ on the diagonal $x=y$ for $\lambda \geqslant 0$.