{'fraction_non_alphanumeric': 0.03820598006644518, 'fraction_numerical': 0.018272425249169437, 'mean_word_length': 5.216494845360825, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Quantitative observability for the Schrödinger and Heisenberg equations: An optimal transport approach

We establish a quantitative observation inequality for the Schrödinger and the Heisenberg equations on Rd, uniform in the Planck constant [Formula: see text]. The proof is based on the pseudometric introduced in F. Golse and T. Paul, The Schrödinger equation in the mean-field and semiclassical regime, Arch. Ration. Mech. Anal. 223 (2017) 57–94. This inequality involves only effective constants which are computed explicitly in their dependence in [Formula: see text] and all parameters involved.