On the Asymptotic Behavior of Solutions to Two-Term Differential Equations with Singular Coefficients

Linear differential operators with distribution coefficients of various singularity orders

“…In particular, Mirzoev and Shkalikov 24 obtained the regularization matrices for the two-term differential operators 𝑦 (2n) + q𝑦, where q = 𝜎 (k) , k = 0, 1, … , n, and 𝜎 is a regular function. That research was continued in Konechnaja et al, 25,26 where the solution asymptotics as x → ∞ were investigated for the two-term differential equation…”

“…In particular, Mirzoev and Shkalikov 24 obtained the regularization matrices for the two‐term differential operators $$$ {y}^{(2n)}+ qy $$$, where $$$ q={\sigma}^{(k)},k=0,1,\dots, n $$$, and $$$ \sigma $$$ is a regular function. That research was continued in Konechnaja et al, 25,26 where the solution asymptotics as $$$ x\to \infty $$$ were investigated for the two‐term differential equation $$$$ {\left(p(x){y}^{(n)}\right)}^{(n)}+q(x)y=\lambda y,\kern0.30em x>0. $$$$ The analogous solution asymptotics for some other classes of higher order differential equations with distribution coefficients were obtained in Konechnaja and Mirzoev 18 and Konechnaya 27 …”

Solving An Inverse Problem For The Sturm-Liouville Operator With A Singular Potential By Yurko's Method

“…The basic results of inverse problem theory were obtained for operators induced by the Sturm-Liouville expression ℓy := −y ′′ +q(x)y with regular (i.e., square integrable) potential q (see [1,2,3,4]). In recent years, spectral analysis of differential operators with singular coefficients from spaces of distributions has attracted much attention of mathematicians (see [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]). Properties of spectral characteristics and solutions of differential equations with singular coefficients were studied in [5,6,7,8,9,10,11,12,13,14].…”

“…In recent years, spectral analysis of differential operators with singular coefficients from spaces of distributions has attracted much attention of mathematicians (see [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]). Properties of spectral characteristics and solutions of differential equations with singular coefficients were studied in [5,6,7,8,9,10,11,12,13,14]. Some aspects of inverse problem theory for differential operators with singular coefficients have been investigated in [15,16,17,18,19,20,21,22,23,24,…”

On the Weyl--type solutions of differential systems with a singularity. Case of discontinuous potential

“…In this paper, we consider the case when the function q(•) is allowed to be non-differential and even discontinuous. Differential systems with discontinuous coefficients are investigated intensively during last two decades (see, for instance, [16], [17]), one should also mention an important role played by such systems in theory of ordinary differential operators with distribution coefficients (see, for instance, [18], [19], [20]). Investigation of such systems required further nontrivial development of the spectral theory methods (see [21], [22], [20]).…”