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В работе получены асимптотические формулы при $x\to \infty$ для фундаментальной системы решений уравнения вида \begin{equation*} l(y): = (-1)^n(p(x)y^{(n)})^{(n)}+q(x)y=\lambda y, \qquad x\in [1,\infty), \end{equation*} где локально суммируемая функция $p$ допускает представление $$ p(x) = (1+r(x))^{-1},\qquad r\in L^1(1,\infty), $$ а $q$ - обобщенная функция, представимая при некотором фиксированном $k$, $0\leqslant k\leqslant n$, в виде $q= \sigma^{(k)}$, где $$ \begin{aligned} \sigma &amp;\in L^1(1,\infty), \qquad если\quad k <n, |\sigma|(1+|r|) (1+ |\sigma|) &amp;\in L^1(1,\infty), \qquad если\quad k = n. \end{aligned} $$ Аналогичные результаты получены для функций, допускающих при некотором фиксированном $k$, $0\leqslant k\leqslant n$, представление $$ p(x) = x^{2n+\nu}(1+ r(x))^{-1},\qquad q= \sigma^{(k)},\qquad \sigma(x)=x^{k+\nu} (\beta +s(x)), $$ где функции $r$ и $s$ удовлетворяют некоторым условиям интегрального убывания. Получены также теоремы об индексах дефекта минимального симметрического оператора, порожденного дифференциальным выражением $l(y)$ (при условии вещественности функций $p$ и $q$), и теоремы о спектрах соответствующих самосопряженных расширений. Полные доказательства даны только для случая $n=1$. Библиография: 18 названий.

Abstract. In the paper we find the leading term of the asymptotics at infinity for some fundamental system of solutions to a class of linear differential equations of arbitrary order = , where is a fixed complex number. At that we consider a special class of ShinZettl type and is a quasi-differential expression generated by the matrix in this class. The conditions we assume for the primitives of the coefficients of the quasi-differential expression , that is, for the entries of the corresponding matrix, are not related with their smoothness but just ensures a certain power growth of these primitives at infinity. Thus, the coefficients of the expression can also oscillate. In particular, this includes a wide class of differential equations of arbitrary even or odd order with distribution coefficients of finite order. Employing the known definition of two quasi-differential expressions with nonsmooth coefficients, in the work we propose a method for obtaining asymptotic formulae for the fundamental system of solutions to the considered equation in the case when the left hand side of this equations is represented as a product of two quasi-differential expressions.The obtained results are applied for the spectral analysis of the corresponding singular differential operators. In particular, assuming that the quasi-differential expression is symmetric, by the known scheme we define the minimal closed symmetric operator generated by this expression in the space of Lebesgue square-integrable on [1, +∞) functions (in the Hilbert space ℒ 2 [1, +∞)) and we calculate the deficiency indices for this operator.Keywords: Quasi-derivative, quasi-differential expression, the main term of asymptotic of the fundamental system of solutions, minimal closed symmetric differential operator, deficiency numbers.

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