On equations containing derivative of the delta-function

The expression $u''+a\delta'u$, which is consisted derivative of delta-function as a coefficient, is a formal expression and doesn't define operator in $L_{2}(\mathbf{R})$, because a product $\delta'u$ is not defined. So according to these reasons the study investigated the family of operators, which are approximated by the following formal expression $(L(\varepsilon, a, \phi)u)(x)=u''(x)+a(\varepsilon)\cdot (\int\psi_{\varepsilon}(y)u(y)\mathbb{d}y\cdot \phi_{\varepsilon}(y)u(y)\mathbb{d}y\cdot \psi_{\varepsilon}(x)),$ where $\phi\in D(\mathbf{R}); \phi(x)\in \mathbf{R}; \int \phi(x)\mathbb{d}x=1; \phi_{\varepsilon}(x)=\frac{1}{\varepsilon}\phi(\frac{x}{\varepsilon});$ coefficient $a(\varepsilon)$ could be real-valued and not null. The main results of the study were finding the limit in the family in sense of resolvent convergence. As the result, the five different kinds of limits of resolvents in this family had been received which are depended on a behavior of coefficient $a(\varepsilon)$ and function $\phi$ properties. Therefore the formal expression $u''+a\delta'u$ could not put in accordance to the operator in $L_{2}(\mathbf{R})$ uniquely. This is the fundamental difference with the case $u''+a\delta u$ expression for which the limit of resolvents doesn’t depend on choosing approximated family..