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The holy grail of TQFT would be a prescription for the duality that interchanges the operator quantum numbers and topological quantum numbers of an arbitrary QFT. More prosaically, there is the sub-problem of finding dualities between classes of perturbative intractable QFTs and (inherently nonperturbative) TQFTs.
What makes you think $H$ has eigenfunctions with compact support? Naively it looks like any functions with compact support are square integrable on $\mathbb{R}$, and all the square integrable eigenfunctions of $H$ are known. They are just the usual harmonic oscillator wave functions, none of which has compact support.