JournalsrlmVol. 19, No. 2pp. 163–173

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
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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.

Cite this article

Sandro Graffi, Emanuela Caliceti, A criterion for the reality of the spectrum of <em>PT</em>-symmetric Schrödinger operators with complex-valued periodic potentials. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 19 (2008), no. 2, pp. 163–173

DOI 10.4171/RLM/515