Some research to which the attached file alludes uses functions of the form Ψ(r) ∝ rν exp( −r2 / (2η2) ). These are solutions for isotropic quantum harmonic oscillators. Here, ξ Ψ(r) = (ξ'/2) ( −η2 ∇2 + η−2 r2 ) Ψ(r). And, ∇2 = r−(D−1)(∂/∂r)(rD−1)(∂/∂r) − Ωr−2. And, D is a non-negative integer (sometimes limited to D ≥ 2). And, Ω = ν(ν + D − 2). And, ξ = (D + 2ν) (ξ'/2). And, ξ and ξ '/2 are positive numbers. And, η can be an arbitrary non-zero real number (or real-number parameter). And, r is the radial coordinate. And, the angular dependence is not shown explicitly (though it is recognized via the appearance of Ω). And, ν is sometimes limited to satisfying both 2ν is an integer and 2ν ≥ −D.
The limit 2ν ≥ −D ensures that normalization pertains. In the case of 2ν = −D, normalization occurs in the limit η2 → 0.
My applications are for physics. Usually, I restrict solutions to those that satisfy Ω = ±S(S + 1), with 2S being a non-negative integer. (In physics correlating with 3-space-like dimensions, S correlates with the spin [in units of ħ]. But, it could be appropriate to consider, for example, Ω = ±J(J + 1), with 2J being a non-negative integer "total angular momentum.")
While my already-published research may not require answers to the following questions, I am curious as to answers to such questions. Perhaps, there are physics-relevant or other uses for some such answers.
One question becomes, for the set of solutions for which Ω = ±S(S + 1), to what extent does the set comprise a basis set for expressing all 'reasonably well behaved functions?' (I think of this in terms of parallels to using plane waves as basis sets in situations for which Fourier transforms pertain.)
Similar questions may pertain for cases involving single values of D.
Also, perhaps someone will advise regarding the extent to which such sets of solutions comprise a bases sets for possible bound states regarding spherically symmetric potentials.
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