Geodesic
The spectral function of a~singular differential operator of order~$2m$
Izvestiya. Mathematics , Tome 74 (2010) no. 6, pp. 1205-1224

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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 $\lambda \geqslant 0$.
Keywords: spectral function, eigenvalues, self-adjoint differential operator, regularized traces, singular differential operators, Green's function. 
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     author = {A. I. Kozko and A. S. Pechentsov},
     title = {The spectral function of a~singular differential operator of order~$2m$},
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     url = {http://geodesic.mathdoc.fr/item/IM2_2010_74_6_a3/}
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A. I. Kozko; A. S. Pechentsov. The spectral function of a~singular differential operator of order~$2m$. Izvestiya. Mathematics , Tome 74 (2010) no. 6, pp. 1205-1224. http://geodesic.mathdoc.fr/item/IM2_2010_74_6_a3/