Comparison between the exact value of the spectral zeta function, Z H (1) = 5 −6/5 [3 − 2 cos(π/5)] Γ 2 (1/5)/Γ(3/5), and the results of numeric and WKB calculations supports the conjecture by Bessis that all the eigenvalues of this PTinvariant hamiltonian are real. For one-dimensional Schrödinger operators with complex potentials having a monotonic imaginary part, the eigenfunctions (and the imaginary parts of their logarithmic derivatives) have no real zeros.

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The integrated density of states of the periodic plus random one-dimensional Schrodinger operatorί Qi(ω) > 0, has Lifschitz singularities at the edges of the gaps in Sp(H ω ). We use Dirichlet-Neumann bracketing based on a specifically one-dimensional construction of bracketing operators without eigenvalues in a given gap of the periodic ones.

Abstract. The integrated density of states has C°°-like singularities, lntk(E)-k(Ec)l = -I E -Ecl-~/2q~c(E), with q~c > 0, a milder function at the edges of the spectral gaps which appear when the distribution function of the potential d# has a sufficiently large gap. The behaviour of (Pc near E~ is determined by the local continuity properties of d# near the relevant edge: (pc(E)=(9(1) if d# has an atom and (pc=(f(InlE-E~l) if # is (absolutely) continuous and power bounded.

We present the application of the inverse scattering method to the design of semiconductor heterostructures having a preset dependence of the (conduction) electrons' reflectance on the energy. The electron dynamics are described by either the effective mass Schrödinger, or by the (variable mass) BenDaniel and Duke equations. The problem of phase (re)construction for the complex transmission and reflection coefficients is solved by a combination of Padé approximant techniques, obtaining reference solutions with simple analytic properties. Reflectance-preserving transformations allow bound state and reflection resonance management. The inverse scattering problem for the Schrödinger equation is solved using an algebraic approach due to Sabatier. This solution can be mapped unitarily onto a family of BenDaniel and Duke type equations. The boundary value problem for the nonlinear equation which determines the mapping is discussed in some detail. The chemical concentration profile of heterostructures whose self consistent potential yields the desired reflectance is solved completely in the case of Schrödinger dynamics and approximately for Ben-Daniel and Duke dynamics. The Appendix contains a brief digest of results from scattering and inverse scattering theory for the one-dimensional Schrödinger equation which are used in the paper.Contents *