Spectral analysis of Grushin type operators on the quarter plane

We investigate spectral properties of self-adjoint extensions of the operator $$G_{\alpha ,\beta }=-(\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{r}^2}+\frac{2\text{a}+1}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }r})-r^2(\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{s}^2}+\frac{2\text{b}+1}{s}\frac{\mathrm{\partial }}{\mathrm{\partial }s}),$$ $\text{a},\text{b}\in \mathbb{R}$, with domain $\text{D}{G}_{\alpha ,\beta }=C^{\mathrm{\infty }}({\mathbb{R}}_{+}^2)\subset L^2({\mathbb{R}}_{+}^2,r^{2\text{a}+1}{s}^{2\text{b}+1}drds)$, which for some specific values of $\text{a},\text{b}$, 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. 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 construct 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 establish a closed form of the heat kernel corresponding to ${\mathbb{G}}_{\alpha ,\beta }^{\circ }$.