Spectral analysis of the Grushin-type operators on the quarter plane

Abstract

We investigate the spectral properties of the self-adjoint extensions of the operator

$\alpha ,\beta \in \mathbb{R}$, with domain $\mathrm{Dom}{G}_{\alpha ,\beta }=C_c^{\infty }({\mathbb{R}}_{+}^2)\subset L^2({\mathbb{R}}_{+}^2,r^{2\alpha +1}{s}^{2\beta +1}drds)$, which for some specific values of $\alpha ,\beta$, is a bi-radial part of the Grushin operator. Alternatively, we investigate $G_{\alpha ,\beta }^{\circ }$, the Liouville form of $G_{\alpha ,\beta }$, which is a symmetric and nonnegative operator on $L^2({\mathbb{R}}_{+}^2,drds)$. One of the main tools used is an integral transform which combines the Laguerre scaled transform and the Hankel transform. The self-adjoint extensions ${\mathbb{G}}_{\alpha ,\beta }^{\circ }$ of $G_{\alpha ,\beta }^{\circ }$ are defined in terms of this transform, and the spectral decompositions of them are given. Another approach to constructing self-adjoint extensions of $G_{\alpha ,\beta }^{\circ }$, based on the technique of sesquilinear forms, is also presented and then the two approaches are compared. We also derive a closed form of the heat kernel corresponding to ${\mathbb{G}}_{\alpha ,\beta }^{\circ }$.