The weakly coupled fractional one-dimensional Schrödinger operator with index $\bf 1<α\leq 2$

Details The weakly coupled fractional one-dimensional Schrödinger operator with index $\bf 1<α\leq 2$ The weakly coupled fractional one-dimensional Schrödinger operator with index $\bf 1<α\leq 2$

2008-12-23

arxiv.org/abs/0812.4356v1

Physics

Mathematical Physics

16 pages, 1 figure

Scientific paper

We study fundamental properties of the fractional, one-dimensional Weyl operator $\hat{\mathcal{P}}^{\alpha}$ densely defined on the Hilbert space $\mathcal{H}=L^2({\mathbb R},dx)$ and determine the asymptotic behaviour of both the free Green's function and its variation with respect to energy for bound states. In the sequel we specify the Birman-Schwinger representation for the Schr\"{o}dinger operator $K_{\alpha}\hat{\mathcal{P}}^{\alpha}-g|\hat{V}|$ and extract the finite-rank portion which is essential for the asymptotic expansion of the ground state. Finally, we determine necessary and sufficient conditions for there to be a bound state for small coupling constant $g$.

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