{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,11]],"date-time":"2024-08-11T11:07:22Z","timestamp":1723374442008},"reference-count":27,"publisher":"World Scientific Pub Co Pte Lt","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Open Syst. Inf. Dyn."],"published-print":{"date-parts":[[2006,6]]},"abstract":"<jats:p>The theory of adiabatic invariants has a long history, and very important implications and applications in many different branches of physics, classically and quantally, but is rarely founded on rigorous results. Here we treat the general time-dependent one-dimensional harmonic oscillator, whose Newton equation [Formula: see text] cannot be solved in general. We follow the time-evolution of an initial ensemble of phase points with sharply defined energy E<jats:sub>0<\/jats:sub>at time t = 0 and calculate rigorously the distribution of energy E<jats:sub>1<\/jats:sub>after time t = T, which is fully (all moments, including the variance \u03bc<jats:sup>2<\/jats:sup>) determined by the first moment [Formula: see text]. For example, [Formula: see text], and all higher even moments are powers of \u03bc<jats:sup>2<\/jats:sup>, whilst the odd ones vanish identically. This distribution function does not depend on any further details of the function \u03c9(t) and is in this sense universal. In ideal adiabaticity [Formula: see text], and the variance \u03bc<jats:sup>2<\/jats:sup>is zero, whilst for finite T we calculate [Formula: see text], and \u03bc<jats:sup>2<\/jats:sup>for the general case using exact WKB-theory to all orders. We prove that if \u03c9(t) is of class [Formula: see text] (all derivatives up to and including the order m are continuous) \u03bc \u221d T<jats:sup>-(m+1)<\/jats:sup>, whilst for the class [Formula: see text] it is known to be exponential \u03bc \u221d exp (-\u03b1T).<\/jats:p>","DOI":"10.1007\/s11080-006-8222-0","type":"journal-article","created":{"date-parts":[[2006,5,19]],"date-time":"2006-05-19T16:03:13Z","timestamp":1148054593000},"page":"197-222","source":"Crossref","is-referenced-by-count":15,"title":["Energy Evolution in Time-Dependent Harmonic Oscillator"],"prefix":"10.1142","volume":"13","author":[{"given":"Marko","family":"Robnik","sequence":"first","affiliation":[{"name":"CAMTP \u2014 Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia"},{"name":"ATR Advanced Telecommunications Research Institute International, 2-2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0288, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Valery G.","family":"Romanovski","sequence":"additional","affiliation":[{"name":"CAMTP \u2014 Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2012,4,17]]},"reference":[{"key":"rf1","first-page":"450","volume":"1","author":"Einstein A.","journal-title":"Inst. intern. phys. 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