Constants of motion of the harmonic oscillator. (English) Zbl 1451.81279
Summary: We prove that Weyl quantization preserves constant of motion of the Harmonic Oscillator. We also prove that if \(f\) is a classical constant of motion and \(\mathfrak{Op}(f)\) is the corresponding operator, then \(\mathfrak{Op}(f)\) maps the Schwartz class into itself and it defines an essentially self-adjoint operator on \(L^2(\mathbb{R}^n)\). As a consequence, we obtain detailed spectral information of \(\mathfrak{Op}(f)\). A complete characterization of the classical constants of motion of the Harmonic Oscillator is given and we also show that they form an algebra with the integral Moyal star product. We give some interesting examples of constants of motion and we analyze Weinstein-Guillemin average method within our framework.
MSC:
| 81S08 | Canonical quantization |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
| 53D55 | Deformation quantization, star products |
| 34B20 | Weyl theory and its generalizations for ordinary differential equations |
| 34C14 | Symmetries, invariants of ordinary differential equations |
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