Truncated Quantum Observables and their Semiclassical Limit

For quantum observables H truncated on the range of orthogonal projections $$\Pi _N$$ of rank N, we study the corresponding Weyl symbol in the phase space in the semiclassical limit of vanishing Planck constant $$\hbar \rightarrow 0$$ and large quantum number $$N\rightarrow \infty $$ , with $$\hbar N$$ fixed. Under certain assumptions, we prove the $$L^2$$ - convergence of the Weyl symbols to a symbol truncated (hence, in general discontinuous) on the classically allowed region in phase space. As an illustration of the general theorems we analyse truncated observables for the harmonic oscillator and for a free particle in a one-dimensional box. In the latter case, we also compute the microscopic pointwise limit of the symbols near the boundary of the classically allowed region.