Spectral analysis of Grushin type operators on the quarter plane

Abstract

We investigate spectral properties of self-adjoint extensions of the operator $$ G_{α,β}=-\Big(\frac{\partial^2}{\partial r^2}+\frac{2\a+1}{r}\frac{\partial}{\partial r} \Big) -r^2 \Big(\frac{\partial^2}{\partial s^2}+\frac{2\b+1}{s}\frac{\partial}{\partial s} \Big), $$ $\a,\b\in\R$, with domain $\D\, G_{α,β}=C^\infty(\R^2_+)\subset L^2(\R^2_+,r^{2\a+1}s^{2\b+1}drds)$, which for some specific values of $\a,\b$, is a bi-radial part of the Grushin operator. Alternatively, we investigate $G_{\alpha ,\beta }^{\circ }$, 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 ${\mathbb{G}}_{\alpha ,\beta }^{\circ }$.

32 pages, 1 figure