AVOIDED LEVEL-CROSSINGS AND RIEMANN SHEET STRUCTURE

The Riemann sheet structure of the energy levels E(n)(lambda) of an N-dimensional symmetric matrix problem of the form H0 + lambda-H-1 is discussed. It is shown that the singularities of the energy levels in the complex lambda plane are related to avoided level crossings. It is argued that locally the sheet structure is like that of a two-, three-, or four-dimensional problem as far as two, three, or four adjacent levels are concerned. Expressions are given for adjacent levels displaying explicitly the Riemann sheet structure on a semiglobal footing.