2019
Том 71
№ 6

All Issues

Goryunov A. S.

Articles: 2
Article (Ukrainian)

Convergence and Approximation of the Sturm–Liouville Operators with Potentials-Distributions

Goryunov A. S.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 602–610

We study the operators $L_n y = −(p_n y′)′+q_n y, n ∈ ℤ_{+}$, given on a finite interval with various boundary conditions. It is assumed that the function $q_n$ is a derivative (in a sense of distributions) of $Q_n$ and $1/p_n , Q_n /p_n$, and $Q^2_n/p_n $ are integrable complex-valued functions. The sufficient conditions for the uniform convergence of Green functions $G_n$ of the operators $L_n$ on the square as $n → ∞$ to $G_0$ are established. It is proved that every $G_0$ is the limit of Green functions of the operators $L_n$ with smooth coefficients. If $p_0 > 0$ and $Q_0(t) ∈ ℝ$, then they can be chosen so that $p_n > 0$ and $q_n$ are real-valued and have compact supports.

Article (Russian)

Regularization of two-term differential equations with singular coefficients by quasiderivatives

Goryunov A. S., Mikhailets V. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1190-1205

We propose a regularization of the formal differential expression $$l(y) = i^m y^{(m)}(t) + q(t)y(t),\; t \in (a, b),$$ of order $m \geq 3$ by using quasiderivatives. It is assumed that the distribution coefficient $q$ has an antiderivative $Q \in L ([a, b]; \mathbb{C})$. In the symmetric case $(Q = \overline{Q})$, we describe self-adjoint and maximal dissipative/accumulative extensions of the minimal operator and its generalized resolvents. In the general (nonselfadjoint) case, we establish conditions for the convergence of the resolvents of the considered operators in norm. The case where $m = 2$ and $Q \in L_2 ([a, b]; \mathbb{C})$ was studied earlier.