Fuzzy circle and new fuzzy sphere through confining potentials and energy cutoffs

Guided by ordinary quantum mechanics we introduce new fuzzy spheres SΛd of dimensions d=1,2: we consider an ordinary quantum particle in D=d+1 dimensions subject to a rotation invariant potential well V(r) with a very sharp minimum on a sphere of unit radius. Imposing a sufficiently low energy cutoff to 'freeze' the radial excitations makes only a finite-dimensional Hilbert subspace accessible and on it the coordinates noncommutative à la Snyder; in fact, on it they generate the whole algebra of observables. The construction is equivariant not only under rotations - as Madore's fuzzy sphere - , but under the full orthogonal group O(D). Making the cutoff and the depth of the well dependent on (and diverging with) a natural number Λ, and keeping the leading terms in 1∕Λ, we obtain a sequence SΛd of fuzzy spheres converging to the sphere Sd in the limit Λ→∞ (whereby we recover ordinary quantum mechanics on Sd). These models may be useful in condensed matter problems where particles are confined on a sphere by an (at least approximately) rotation-invariant potential, beside being suggestive of analogous mechanisms in quantum field theory or quantum geometry.