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On the spectrum of a differential operator of high-order. (English. Russian original) Zbl 1088.47035
Math. Notes 77, No. 2, 172-176 (2005); translation from Mat. Zametki 77, No. 2, 188-193 (2005).
Let \(L\) be a self-adjoint extension of the operator defined by the operation
\[ Ly=(-1)^ry^{(2r)}+qy, \;\;0\leq x<\infty \] on functions which satisfy
\[ y^{(k)}(0)=0, \;\;k=0,1,\dots,r-1 \]
and vanish for sufficiently large values of \(x\). Take points \(\{a_k\}\), \(k\geq 1\), satisfying the conditions
\[ a_1<a_2<...<a_k<\dots, \;\;a_k\to +\infty, \]
\[ a_k-a_{k-1}\to 0, \;\;k\to \infty. \] Choose an interval \(\Delta_k=(a_k-h_k, a_k+h_k)\), where the numbers \(h_k\) are positive and small enough for these intervals to be pairwise disjoint. The authors prove that, if \(q(x)\) is a peacewise continuous real function on \((0,\infty)\) with the following properties: \( (1) \;q(x)\leq 0 \;\text{ for} \;x\in \mathop{\cup}\limits_{k=1}^{\infty}\Delta_k \;\text{ and} \;q(x)\geq 0 \) otherwise; \( (2) \;h_k^{2r-1}\int_{\Delta_k}| q(x)| dx\rightarrow +\infty; \) \( (3) \;\int_{\Delta_k}| x-a_k| | q(x)| dx<{1\over 3}, \) then the spectrum of \(L\) is discrete and not bounded below.
47E05 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX)
47A10 Spectrum, resolvent
34L05 General spectral theory of ordinary differential operators
Full Text: DOI
[1] I. M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators [in Russian], Fizmatgiz, Moscow, 1963. · Zbl 0143.36505
[2] E. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Oxford, 1958; Russian translation: Inostr. Lit., Moscow, 1960. · Zbl 0097.27601
[3] R. A. Ismagilov, ”On the spectrum of the Sturm-Liouville equation with oscillating potential,” Mat. Zametki [Math. Notes], 37 (1985), no. 6, 869–879. · Zbl 0597.34017
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