symmetries and (related) recursion operators of linear evolution equations对称性和(相关)递归线性演化方程的运营商.pdfVIP

Symmetry 2010, 2, 98-111; doi:10.3390/sym2010098 OPEN ACCESS symmetry ISSN 2073-8994 /journal/symmetry Article Symmetries and (Related) Recursion Operators of Linear Evolution Equations Giampaolo Cicogna ` Dipartimento di Fisica “E. Fermi” dell’Universita di Pisa and INFN, sez. di Pisa, Largo B. Pontecorvo 3, Ed. B-C, I-56127–Pisa, Italy; E-Mail: cicogna@df.unipi.it Received: 28 December 2009; in revised form: 26 January 2010 / Accepted: 29 January 2010 / Published: 5 February 2010 ¨ Abstract: Significant cases of time-evolution equations, the linear Schrodinger and the Fokker–Planck equation are considered. It is known that equations of this type can be trans- formed, in some cases, into a highly simplified form. The properties of these equations in their initial and their simplified form are compared, showing in particular that this trans- formation partially prevents a clear understanding and a full application of the (physically relevant) notion of the so-called step up/down operators. These operators are shown to be recursion operators, related to the Lie point symmetries of the equations, which are also carefully discussed. ¨ Keywords: recursion operators; step up/down operators; Lie point symmetries; Schrodinger equation; Fokker–Planck