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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