Заседание семинара РМЦ ЮФУ

Региональный научно-​образовательный математический центр проводит еженедельный научно-​исследовательский семинар по анализу, дифференциальным уравнениям и математической физике. В пятницу 5 июля с 11:00 – 12:00 в ауд. 211 состоится лекция «Con­struc­tion of Bergman-​type repro­duc­ing ker­nels for the sta­tion­ary Schrödinger oper­a­tor in the plane with vari­able coef­fi­cients». Докладчик — Hugo Cam­pos, School of Math­e­mat­i­cal Sci­ences and Infor­ma­tion Tech­nol­ogy, Yachay Tech, Ecuador.

In the talk we deal with bound­ary value prob­lems for the oper­a­tor L=-Δ+q in a bounded sim­ply con­nected domain in the plane. We show how the Bergman-​type repro­duc­ing ker­nels [1] for the oper­a­tor L can be con­structed with the help of trans­mu­ta­tion oper­a­tor the­ory. Under cer­tain con­di­tions on the poten­tial q and with the aid of trans­mu­ta­tion oper­a­tors a com­plete sys­tem of null solu­tions of the oper­a­tor can be obtained fol­low­ing a sim­ple algo­rithm con­sist­ing in recur­sive inte­gra­tion [2]. Then the com­plete sys­tem of solu­tions is used for con­struct­ing the cor­re­spond­ing repro­duc­ing ker­nel which in turn can be used to solve the asso­ci­ated bound­ary value prob­lem. The algo­rith­mi­cal sim­plic­ity of the pro­posed method leads to numer­i­cal imple­men­ta­tion [3].
[1] S. Bergman, M. Schif­fer, Ker­nel Func­tions and Ellip­tic Dif­fer­en­tial Equa­tions in Math­e­mat­i­cal Physics, Aca­d­e­mic Press, 1953.
[2] H. M. Cam­pos, V. V. Kravchenko, S. M. Torba 2012 Trans­mu­ta­tions, L-​bases and com­plete fam­i­lies of solu­tions of the sta­tion­ary Schrödinger equa­tion in the plane. Jour­nal of Math­e­mat­i­cal Analy­sis and Appli­ca­tions, v.389, issue 2, 1222 – 1238.

[3] H. M. Cam­pos, R. Castillo, V. V. Kravchenko 2012 Con­struc­tion and appli­ca­tion Bergman-​type repro­duc­ing ker­nels for bound­ary and eigen­value prob­lems in the plane. Com­plex Vari­ables and Ellip­tic Equa­tions, Vol. 57, Nos. 7 – 8, 787 – 824.