Czechoslovak Mathematical Journal, Vol. 70, No. 2, pp. 519-537, 2020
Boundary value problems for the Schrödinger equation involving the Henstock-Kurzweil integral
Salvador Sánchez-Perales, Francisco J. Mendoza-Torres
Received August 24, 2018. Published online December 12, 2019.
Abstract: In the present paper, we investigate the existence of solutions to boundary value problems for the one-dimensional Schrödinger equation $-y"+qy=f$, where $q$ and $f$ are Henstock-Kurzweil integrable functions on $[a,b]$. Results presented in this article are generalizations of the classical results for the Lebesgue integral.
Keywords: Henstock-Kurzweil integral; Schrödinger operator; ${\rm ACG}_*$-function; bounded variation function
References: [1] R. G. Bartle: A Modern Theory of Integration. Graduate Studies in Mathematics 32, AMS, Providence (2001). DOI 10.1090/gsm/032 | MR 1817647 | Zbl 0968.26001
[2] H. Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer, New York (2011). DOI 10.1007/978-0-387-70914-7 | MR 2759829 | Zbl 1220.46002
[3] R. A. Gordon: The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Graduate Studies in Mathematics 4, AMS, Providence (1994). DOI 10.1090/gsm/004 | MR 1288751 | Zbl 0807.26004
[4] I. Peral: Primer Curso de Ecuaciones en Derivadas Parciales. Addison Wesley, Boston (1995). (In Spanish.)
[5] S. Sánchez-Perales: The initial value problem for the Schrödinger equation involving the Henstock-Kurzweil integral. Rev. Unión Mat. Argent. 58 (2017), 297-306. MR 3733209 | Zbl 1382.34096
[6] C. Swartz: Norm convergence and uniform integrability for the Henstock-Kurzweil integral. Real Anal. Exch. 24 (1998), 423-426. MR 1691761 | Zbl 0943.26022
[7] E. Talvila: Henstock-Kurzweil Fourier transforms. Ill. J. Math. 46 (2002), 1207-1226. DOI 10.1215/ijm/1258138475 | MR 1988259 | Zbl 1037.42007
Affiliations: Salvador Sánchez-Perales (corresponding author), Instituto de Fisica y Matemáticas, Universidad Tecnológica de la Mixteca, Carretera a Acatlima Km. 2.5, Acatlima, 69000 Huajuapan de León, Oaxaca, Mexico, e-mail: es21254@yahoo.com.mx; Francisco J. Mendoza-Torres, Facultad de Ciencias Fisico Matemáticas, Benemérita Universidad Autónoma de Puebla, Av San Claudio, Cd Universitaria, Jardines de San Manuel, 72572 Puebla, Mexico, e-mail: jmendoza2@fcfm.buap.mx
