JournalsrlmVol. 19 , No. 2DOI 10.4171/rlm/515

A criterion for the reality of the spectrum of <em>PT</em>-symmetric Schrödinger operators with complex-valued periodic potentials

  • Sandro Graffi

    Università di Bologna, Italy
  • Emanuela Caliceti

    Università di Bologna, Italy
A criterion for the reality of the spectrum of <em>PT</em>-symmetric Schrödinger operators with complex-valued periodic potentials cover

Abstract

Consider in L2(R)L^2(\R) the \Sc\ operator family H(g):=dx2+Vg(x)H(g):=-d^2_x+V_g(x) depending on the real parameter gg, where Vg(x)V_g(x) is a complex-valued but PTPT symmetric periodic potential. An explicit condition on VV is obtained which ensures that the spectrum of H(g)H(g) is purely real and band shaped; furthermore, a further condition is obtained which ensures that the spectrum contains at least a pair of complex analytic arcs.