Regularization of two-term differential equations with singular coefficients by quasiderivatives
A regularization of the formal differential expression \[ l(y) = i^m y^{(m)}(t) +q(t)y(t) \] under the assumption \(t\in \mathcal{J}:=(a,b)\), \(q = Q^{\prime}\), \(Q\in {L}(\mathcal{J},\mathbb{C})\), is suggested to determine the corresponding maximal and minimal operators in the Hilbert space \({L}_2(\mathcal{J};\mathbb{C})\). Sufficient conditions are established for the uniform resolvent approximation of extensions of the constructed minimal operator \(L_{\min}\) by a family of operators from the same class, in particular, with smooth coefficients. Under the assumption that the minimal operator is symmetric, the authors describe all its selfadjoint, maximal dissipative, and maximal accumulative extensions in terms of homogeneous boundary conditions of the canonical form. These extensions are parametrized, respectively, by the unitary operators and by the contraction operators in \(\mathbb{C}^m\). The suggested parametrization is bijective and continuous.
scientific article; zbMATH DE number 6110034
scientific article