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

We investigate the spectral properties of the self-adjoint extensions of the operator Gα,β=-({∂²}{∂ r²}+2α+1/r∂/∂ r) -r² ({∂²}{∂ s²}+2β+1/s∂/∂ s), α,β , with domain DomGα,β=Cc∞(R²₊)⊂ L²(R²₊,r2α+1s2β+1dr\,ds) , which for some specific values of α,β , is a bi-radial part of the Grushin operator. Alternatively, we investigate G∘α,β , the Liouville form of Gα,β , which is a symmetric and nonnegative operator on L²(R²₊, dr\,ds) . One of the main tools used is an integral transform which combines the Laguerre scaled transform and the Hankel transform. The self-adjoint extensions G∘α,β of G∘α,β are defined in terms of this transform, and the spectral decompositions of them are given. Another approach to constructing self-adjoint extensions of G∘α,β , 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 G∘α,β .