On the spectrum of a differential operator of high-order. (English. Russian original) Zbl 1088.47035

Gimadislamov, M. G.

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.

Reviewer: Vyacheslav Pivovarchik (Odessa)

MSC:

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

Keywords:

differential operator of high order; discrete spectrum; piecewise continuous real function; Dirichlet boundary conditions; Courant variational principle

References:

[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