Geodesic
On equations containing derivative of the delta-function
Journal of the Belarusian State University. Mathematics and Informatics, Tome 3 (2017), pp. 19-26.

Voir la notice de l'article provenant de la source Math-Net.Ru

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..
Keywords: resolvent; resolvent convergence; approximation; fundamental solution. 
@article{BGUMI_2017_3_a2,
     author = {E. V. Shkadinskaia},
     title = {On equations containing derivative of the delta-function},
     journal = {Journal of the Belarusian State University. Mathematics and Informatics},
     pages = {19--26},
     publisher = {mathdoc},
     volume = {3},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/BGUMI_2017_3_a2/}
}
TY  - JOUR
AU  - E. V. Shkadinskaia
TI  - On equations containing derivative of the delta-function
JO  - Journal of the Belarusian State University. Mathematics and Informatics
PY  - 2017
SP  - 19
EP  - 26
VL  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BGUMI_2017_3_a2/
LA  - ru
ID  - BGUMI_2017_3_a2
ER  - 
%0 Journal Article
%A E. V. Shkadinskaia
%T On equations containing derivative of the delta-function
%J Journal of the Belarusian State University. Mathematics and Informatics
%D 2017
%P 19-26
%V 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BGUMI_2017_3_a2/
%G ru
%F BGUMI_2017_3_a2
E. V. Shkadinskaia. On equations containing derivative of the delta-function. Journal of the Belarusian State University. Mathematics and Informatics, Tome 3 (2017), pp. 19-26. http://geodesic.mathdoc.fr/item/BGUMI_2017_3_a2/

[1] A. B. Antonevich, Ya. V. Radyno, “Ob obschem metode postroeniya algebr obobschennykh funktsii”, Dokl. AN SSSR, 43(3) (1991), 680–684 | Zbl

[2] J. F. Colombeau, “New generalized functions and multiplication of distributions”, Amsterdam: North-Holland, 1984 | MR | Zbl

[3] F. Gesztesy, B. Simon, “Rank-one pertubations at infinite coupling”, J. Funct. Anal., 128 (1995), 245–252 | DOI | MR | Zbl

[4] A. B. Antonevich, T. A. Romanchuk, “On equations with delta-shaped coefficients: the finite-dimensional perturbations approach”, Integral transforms Special Funct, 3/4 (2009), 239–246 | DOI | MR | Zbl

[5] L. P. Nizhnik, “Operator Shredingera s d-vzaimodeistviem”, Funkts. analiz i ego pril., 37(1) (2003), 85–88 | Zbl