On weak and strong irregularity
The present article is devoted to a comparative analysis of the properties of solutions to the irregular equations \[ (1)\quad aD_tu- A(t)u= \lambda u+f, \qquad (2) \quad a(t) D_tu-Au =\lambda u+f \] considered in a bounded domain having nonempty intersection with the hyperplane \(t=0\). Here \(D_t=d/dt\); \(a(t)\) is a complex-valued function; \(A(t)\) is a complex-valued or operator-valued function: \[ a(t)= \begin{cases} a_2t^{b_2}, \quad & t>0,\\ a_1 | t|^{b_1}, \quad & t<0; \end{cases} \quad A(t)= \begin{cases} t^{c_2} A_2, \quad & t>0,\\ | t |^{c_1} A_1, \quad & t<0, \end{cases} \] \(a\), \(a_1\), and \(a_2\) are complex numbers such that \(a_2= qa_1\), \(-\infty<q< +\infty\); \(0\leq b_1\), \(b_2<1\); \(c_1, c_2>-1\); \(A\), \(A_1\), and \(A_2\) are complex numbers or model operators acting in some Banach spaces \({\mathcal B}\), i.e., unbounded closed densely defined linear operators possessing the same complete system \(\{ \varphi_s\}_{s\in S}\) of eigenfunctions \(\varphi_s\) forming a Riesz basis for \({\mathcal B}\). Consider the operator \(L\) defined as \(L(D)= a(t)\cdot D_t \otimes I+ I\otimes A(t)\). We are interested in the spectrum and the properties of the system of eigenfunctions of the operator \(L\).
scientific article; zbMATH DE number 1023949
scientific article