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 $\mathbb{ G}^\circ_{ \alpha,\beta}$.
Categories: math.FA