{'fraction_non_alphanumeric': 0.08716136631330977, 'fraction_numerical': 0.008244994110718492, 'mean_word_length': 4.3125, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 1, '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}}
SCHRÖDINGER EQUATIONS WITH TIME-DEPENDENT STRONG MAGNETIC FIELDS

Time dependent d-dimensional Schrödinger equations i∂tu = H(t)u, H(t) = −(∂x − iA(t, x))2 +V (t, x) are considered in the Hilbert space G = L2(Rd) of square integrable functions. V (t, x) and A(t, x) are assumed to be almost critically singular with respect to the spatial variables x ∈ Rd both locally and at infinity for the operator H(t) to be essentially selfadjoint on C∞ 0 (R d). In particular, when the magnetic fields B(t, x) produced by A(t, x) are very strong at infinity, V (t, x) can explode to the negative infinity like −θ|B(t, x)| − C(|x|2 + 1) for some θ < 1 and C > 0. It is shown that such equations uniquely generate unitary propagators in G under suitable conditions on the size and singularities of the time derivatives of the potentials V̇ (t, x) and Ȧ(t, x). §