East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
권호 표지
DOI QR코드

DOI QR Code

REGULARITY OF THE SCHRÖDINGER EQUATION FOR A CAUCHY-EULER TYPE OPERATOR

  • CHO, HONG RAE (DEPARTMENT OF MATHEMATICS, PUSAN NATIONAL UNIVERSITY) ;
  • LEE, HAN-WOOL (DEPARTMENT OF MATHEMATICS, PUSAN NATIONAL UNIVERSITY) ;
  • CHO, EUNSUNG (DEPARTMENT OF MATHEMATICS, PUSAN NATIONAL UNIVERSITY)
  • Received : 2018.08.23
  • Accepted : 2018.10.20
  • Published : 2019.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

We consider the initial value problem of the Schrodinger equation for an interesting Cauchy-Euler type operator ${\mathfrak{R}}$ on ${\mathbb{C}}^n$ that is an analogue of the harmonic oscillator in ${\mathbb{R}}^n$. We get an appropriate $L^1-L^{\infty}$ dispersive estimate for the solution of the initial value problem.

Keywords

References

  1. V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform I, Comm. Pure Appl. Math. 14 (1961), 187-214. https://doi.org/10.1002/cpa.3160140303
  2. H. R. Cho, H. Choi, and H.-W. Lee, Boundedness of the Segal-Bargmann Transform on Fractional Hermite-Sobolev Spaces, Journal of Function Spaces, Volume 2017 (2017), Article ID 9176914, 6 pages.
  3. K.-J. Engel and R. Nagel. One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics 194, Springer-Verlag, Berlin, New York, 2000.
  4. R. P. Feynman and A. R. Hibbs, Quantum mechanics and path integrals, McGraw-Hill, Maidenhead, 1965.
  5. A. K. Nandakumaran and P. K. Ratnakumar, Schrodinger equation and the oscillatory semigroup for the Hermite operator, J. Funct. Anal. 224 (2005), 371-385. https://doi.org/10.1016/j.jfa.2004.12.011
  6. E. Stein and G. Weiss, Interpolation of operators with change of measures, Trans. Amer. Math. Soc. 87 (1958), 159-172. https://doi.org/10.1090/S0002-9947-1958-0092943-6
  7. K. Zhu, Analysis on Fock spaces, Graduate Texts in Mathematics, 263. Springer, New York, 2012.