Geodesic
Construction of regularized asymptotics for the solution of a singularly perturbed mixed problem on the half-axis for the inhomogeneous Schr\"odinger-type equation with the potential $V(x)=x$
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 2, Tome 231 (2024), pp. 27-43.

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In this paper, we propose a method for constructing an asymptotic solution to a singularly perturbed mixed problem on the half-axis for a nonstationary inhomogeneous Schrödinger-type equation in the coordinate representation in the case of violation of the stability conditions for the spectrum of the limit operator. The chosen profile of the potential energy leads to a spectral singularity of the limit operator, which, within the framework of S. A. Lomov's regularization method, is usually called a strong turning point.
Keywords: singularly perturbed problem, asymptotic solution, turning point, regularization method, semiclassical approximation 
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A. G. Eliseev; P. V. Kirichenko. Construction of regularized asymptotics for the solution of a singularly perturbed mixed problem on the half-axis for the inhomogeneous Schr\"odinger-type equation with the potential $V(x)=x$. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 2, Tome 231 (2024), pp. 27-43. http://geodesic.mathdoc.fr/item/INTO_2024_231_a2/

[1] Arnold V. I., “O matritsakh, zavisyaschikh ot parametrov”, Usp. mat. nauk., 26:2 (158) (1971), 101–114 | MR

[2] Bobodzhanov A. A., Safonov V. F., Kurs vysshei matematiki. Singulyarno vozmuschennye uravneniya i metod regulyarizatsii, MEI, M., 2012

[3] Bobodzhanov A. A., Safonov V. F., “Regulyarizovannaya asimptotika reshenii integrodifferentsialnykh uravnenii s chastnymi proizvodnymi s bystro izmenyayuschimisya yadrami”, Ufim. mat. zh., 10:2 (2018), 3–12 | MR | Zbl

[4] Eliseev A. G., “Primer resheniya singulyarno vozmuschennoi zadachi Koshi dlya parabolicheskogo uravneniya pri nalichii «silnoi» tochki povorota”, Differ. uravn. protsessy upravl., 2022, no. 3, 46–58 | Zbl

[5] Eliseev A. G., “Regulyarizovannoe reshenie singulyarno vozmuschennoi zadachi Koshi pri nalichii irratsionalnoi «prostoi» tochki povorota”, Differ. uravn. protsessy upravl., 2020, no. 2, 15–32 | Zbl

[6] Eliseev A. G., Kirichenko P. V., “Regulyarizovannoe asimptoticheskoe reshenie zadachi Koshi dlya neodnorodnogo uravneniya Shrëdingera v kvaziklassicheskom priblizhenii v prisutstvii «silnoi» tochki povorota u predelnogo operatora”, Differ. uravn. protsessy upravl., 2023, no. 1, 110–124

[7] Eliseev A. G., Kirichenko P. V., “Reshenie singulyarno vozmuschennoi zadacha Koshi pri nalichii «slaboi» tochki povorota u predelnogo operatora”, Sib. elektron. mat. izv., 17 (2020), 51–60 | Zbl

[8] Eliseev A. G., Kirichenko P. V., “Cingulyarno vozmuschennaya zadacha Koshi pri nalichii «slaboi» tochki povorota pervogo poryadka u predelnogo operatora s kratnym spektrom”, Differ. uravn., 58:6 (2022), 733–746 | Zbl

[9] Eliseev A. G., Lomov S. A., “Teoriya singulyarnykh vozmuschenii v sluchae spektralnykh osobennostei predelnogo operatora”, Mat. sb., 131:173 (1986), 544–557 | Zbl

[10] Eliseev A. G., Ratnikova T. A., “Singulyarno vozmuschennaya zadacha Koshi pri nalichii ratsionalnoi «prostoi» tochki povorota”, Differ. uravn. protsessy upravl., 2019, no. 3, 63–73 | Zbl

[11] Kirichenko P. V., “Singulyarno vozmuschennaya zadacha Koshi dlya parabolicheskogo uravneniya pri nalichii «slaboi» tochki povorota u predelnogo operatora”, Mat. zametki SVFU., no. 3, 3–15 | MR

[12] Landau L. D., Lifshits E. M., Kurs teoreticheskoi fiziki. T. 3. Kvantovaya mekhanika (nerelyativistskaya teoriya), Nauka, M., 2008 | MR

[13] Lomov S. A., Vvedenie v obschuyu teoriyu singulyarnykh vozmuschenii, Nauka, M., 1981

[14] Lomov S. A., Lomov I. S., Osnovy matematicheskoi teorii pogranichnogo sloya, MGU, M., 2011

[15] Lomov S. A., Safonov V. F., “Regulyarizatsii i asimptoticheskie resheniya dlya singulyarno vozmuschennykh zadach s tochechnymi osobennostyami spektra predelnogo operatora”, Ukr. mat. zh., 36:2 (1984), 172–180 | MR | Zbl

[16] Tikhonov A. N., Samarskii A. A., Uravneniya matematicheskoi fiziki, Nauka, M., 1977 | MR

[17] Elsgolts L. E., Differentsialnye uravneniya i variatsionnoe ischislenie, Nauka, M., 1969

[18] Liouville J., “Second mémoire sur le développement des fonctions ou parties de functions en séries dont les divers termes sont assujétis á satisfaire á une même équation différentielle du second ordre, contenant un paramétre variable”, J. Math. Pures Appl., 1837, 16–35