Spectral analysis of Grushin type operators on the quarter plane

We investigate spectral properties of self-adjoint extensions of the operator $$G_{\alpha,\beta}=-\Big(\frac{\partial^2}{\partial r^2}+\frac{2\a+1}{r}\frac{\partial}{\partial r} \Big) -r^2 \Big(\frac{\partial^2}{\partial s^2}+\frac{2\b+1}{s}\frac{\partial}{\partial s} \Big),$$ $\a,\b\in\R$, with domain $\D\, G_{\alpha,\beta}=C^\infty(\R^2_+)\subset L^2(\R^2_+,r^{2\a+1}s^{2\b+1}drds)$, which for some specific values of $\a,\b$, is a bi-radial part of the Grushin operator. Alternatively, we investigate $G^\circ_{\alpha,\beta}$, the Liouville form of $G_{\alpha,\beta}$, which is a symmetric and nonnegative operator on $L^2(\R^2_+, drds)$. One of the main tools used is an integral transform which combines the Laguerre scaled transform and the Hankel transform. Self-adjoint extensions $\mathbb{G}^\circ_{\alpha,\beta}$ of $G^\circ_{\alpha,\beta}$ are defined in terms of this transform, and the spectral decompositions of them are given. Another approach to construct self-adjoint extensions of $G^\circ_{\alpha,\beta}$, 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 $\mathbb{G}^\circ_{\alpha,\beta}$.