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Uniform Convergence of Spectral Expansions on the Entire Real Line for General Even-Order Differential Operators

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Abstract

The uniform convergence of spectral expansions on the real line \(\mathbb {R} \) is established for a self-adjoint operator \(\mathcal {A} \) generated on \(\mathbb {R} \) by the differential operation \(Au\equiv (-1)^n u^{(2n)}+\sum _{k=0}^{n-1} (q_k(x)u^{(k)})^{(k)}\) with uniformly locally integrable coefficients. The uniform convergence of the derivatives of these expansions is studied as well. The results obtained are based on a uniform estimate for the increment of the spectral function of the operator \(\mathcal {A} \) on the diagonal.

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Notes

  1. Since the operator \(\mathcal {A}\) is semibounded below in the case under consideration, we can assume that \(\mathcal {A} \) is strictly positive; one only needs to add an appropriate constant to the coefficient \(q_0(x)\) in (1).

  2. Since the operator \(\mathcal {A}\) is strictly positive, we have \(d\rho (t)\equiv 0\) with \(t<\lambda _0\) for some positive \(\lambda _0 \). Consequently, the integrals in the definition of \(E_\lambda f(x) \) and below, in Parseval’s identity, are actually taken over the set \(\lambda \ge \lambda _0 \).

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Funding

This work was supported by the Scientific Committee of the Ministry of Education and Science of the Republic of Kazakhstan, project no. AP 05131225.

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  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    L. V. Kritskov

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  1. L. V. Kritskov
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Translated by V. Potapchouck

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Kritskov, L.V. Uniform Convergence of Spectral Expansions on the Entire Real Line for General Even-Order Differential Operators. Diff Equat 56, 426–437 (2020). https://doi.org/10.1134/S0012266120040035

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