Homogeneous Schrödinger operators, called also Bessel operators are given by $H_m=-\partial_x^2+(-\frac14+m^2)\frac{1}{x^{2}},$ with the boundary condition $\sim x^{m+\frac12}$ near $0$. I will discuss their role in representations of the smallest semi-simple non-compact Lie group $\mathrm{SL}(2,\mathbb{R})$.
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