Matrix Finite-Zone Dirac-Type Equations

✍ Scribed by L.A. Sakhnovich

• Book ID: 102588966
• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 173 KB
• Volume: 193
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2002.3956

✦ Synopsis

Weyl-Titchmarsh matrix functions play an essential role in the spectral theory of Dirac-type equations (Oper. Theory: Adv. Appl. 107 (1999)). In this paper, we have constructed a class of Weyl-Titchmarsh matrix-functions generating potentials of finite-zone type. It has turned out that the corresponding potentials have derivatives of an arbitrary order. Using the above-mentioned results, we deduce the matrix analogue of the trace formula for finite-zone matrix potentials. In the last part of the paper, we consider separately the scalar case of Dirac-type equations. For this case, we have constructed finite-zone potentials in explicit forms and proved that these potentials are quasiperiodical. We note that for scalar Schro¨dinger equations the corresponding results are well known (see