AIMS Mathematics, 2018, 3(1): 21-34. doi: 10.3934/Math.2018.1.21.

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Lp-analysis of one-dimensional repulsive Hamiltonian with a class of perturbations

1 Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda-shi, Chiba-ken 278-8510, Japan
2 Department of Mathematics, Faculty of Science Division I, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, 162-8601, Tokyo, Japan

The spectrum of one-dimensional repulsive Hamiltonian with a class of perturbations $H_p=-\frac{d^2}{dx^2}-x^2+V(x)$ in $L^p(\R)$ ($1< p<\infty$) is explicitly given. It is also proved that the domain of $H_p$ is embedded into weighted $L^q$-spaces for some $q>p$. Additionally, non-existence of related Schr\"odinger ($C_0$-)semigroup in $L^p(\R)$ is shown when $V(x)\equiv 0$.
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Keywords repulsive Hamiltonian; WKB methods

Citation: Motohiro Sobajima, Kentarou Yoshii. Lp-analysis of one-dimensional repulsive Hamiltonian with a class of perturbations. AIMS Mathematics, 2018, 3(1): 21-34. doi: 10.3934/Math.2018.1.21

References

  • 1. R. Beals, R. Wong, Special functions, Cambridge Studies in Advanced Mathematics, 126, Cambridge University Press, Cambridge, 2010.
  • 2. J.-F. Bony, R. Carles, D. Hafner, et al. Scattering theory for the Schrödinger equation with repulsive potential, J. Math. Pures Appl., 84 (2005), 509–579.
  • 3. T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10, Amer. Mathematical Society, 2003.
  • 4. J. D. Dollard, C. N. Friedman, Asymptotic behavior of solutions of linear ordinary differential equations, J. Math. Anal. Appl., 66 (1978), 394–398.
  • 5. K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math., 194, Springer-Verlag, 2000.
  • 6. J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs, Oxford Univ. Press, New York, 1985.
  • 7. T. Ikebe, T. Kato, Uniqueness of the self-adjoint extension of singular elliptic differential operators, Arch. Ration. Mech. An., 9 (1962), 77–92.
  • 8. A. Ishida, On inverse scattering problem for the Schrödinger equation with repulsive potentials, J. Math. Phys., 55 (2014), 082101.
  • 9. T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin-New York, 1966.
  • 10. F. Nicoleau, Inverse scattering for a Schrodinger operator with a repulsive potential, Acta Math. Sin., 22 (2006), 1485–1492.
  • 11. G. Metafune, M. Sobajima, An elementary proof of asymptotic behavior of solutions of u'' = Vu, preprint (arXiv:1405.5659). Available from: http://arxiv.org/abs/1405.5659.
  • 12. N. Okazawa, On the perturbation of linear operators in Banach and Hilbert spaces, J. Math. Soc. Jpn, 34 (1982), 677–701.
  • 13. F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press, New York-London, 1974.
  • 14. H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Pure and Applied Mathematics, 204, Marcel Dekker, New York, 1997.

 

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