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On the asymptotics of spectral function for ordinary differential selfadjoint operator. (Russian. English summary) Zbl 1050.47044
Let \(T\) be a selfadjoint ordinary differential operator of order \(n\) defined by regular boundary conditions on the segment \([0,1]\) with eigenvalues \(\lambda_i\), \(\lambda_1\leq \lambda_2\leq \ldots\) and eigenfunctions \(v_i(x)\) orthonormal in \(L_2[0,1]\). Let \(P\) be the operator of multiplication with the real essentially bounded on \([0,1]\) function \(p(x)\). Suppose that \(\mu_i\), \(\mu_1\leq \mu_2\leq \ldots\) are the eigenvalues of \(T+P\). The asymptotics of the spectral function for \(T+P\) are investigated. As a corollary, it is obtained that if \(n\geq 2 \), then \(\sum_{i=1}^N \mu_i =\sum_{i=1}^N \lambda_i +\sum_{i=1}^N (pv_i,v_i)+o(N^{-n+2})\), where \((f,g)=\int_0^1 f(x) \overline g(x)\,dx\).
MSC:
47E05 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX)
47A75 Eigenvalue problems for linear operators
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