The weakly coupled fractional one-dimensional Schrödinger operator with index $1<\alpha \le 2$

Abstract

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 Schrödinger 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 $g$.

16 pages, 1 figure

References in corpus (4)

• Fractional Quantum Mechanics
• The fundamental solution of the space-time fractional diffusion equation
• Fractional Laplacian in Bounded Domains
• Equivalence of Sobolev inequalities and Lieb-Thirring inequalities

Cited by in corpus (3)

• Energy-dependent noncommutative quantum mechanics
• Weak perturbations of the p-Laplacian
• Weak coupling asymptotics for the Pauli operator in two dimensions