Abstract. We study the eigenvalue problem −u ′′ (z) − [(iz) m + P (iz)]u(z) = λu(z) with the boundary conditions that u(z) decays to zero as z tends to infinity along the rays arg z = − π 2 ± 2π m+2 , where P (z) = a 1 z m−1 + a 2 z m−2 + · · · + a m−1 z is a real polynomial and m ≥ 2. We prove that if for some 1 ≤ j ≤ m 2 , we have (j − k)a k ≥ 0 for all 1 ≤ k ≤ m − 1, then the eigenvalues are all positive real. We then sharpen this to a slightly larger class of polynomial potentials.In particular, this implies that the eigenvalues are all positive real for the potentials αiz 3 + βz 2 + γiz when α, β, γ ∈ R with α = 0 and α γ ≥ 0, and with the boundary conditions that u(z) decays to zero as z tends to infinity along the positive and negative real axes. This verifies a conjecture of Bessis and Zinn-Justin.Preprint.
Abstract. The spectra of the Schrödinger operators with periodic potentials are studied. When the potential is real and periodic, the spectrum consists of at most countably many line segments (energy bands) on the real line, while when the potential is complex and periodic, the spectrum consists of at most countably many analytic arcs in the complex plane.In some recent papers, such operators with complex PT -symmetric periodic potentials are studied. In particular, the authors argued that some energy bands would appear and disappear under perturbations. Here, we show that appearance and disappearance of such energy bands imply existence of nonreal spectra. This is a consequence of a more general result, describing the local shape of the spectrum.In recent papers [1,2,3,4,5], appearance and disappearance of real energy bands for some complex PT -symmetric periodic potentials under perturbations have been reported. In this paper, we show that appearance and disappearance of such real energy bands imply existence of nonreal band spectra.We begin by introducing some facts on Floquet theory and the associated Hill operators H. Consider the Schrödinger equation(1) −ψ xx (E, x) + V (x)ψ(E, x) = Eψ(E, x), x ∈ R,where E ∈ C and V ∈ L
We consider the eigenvalue problems −u ′′ (z)±(iz) m u(z) = λu(z), m ≥ 3, under every rapid decay boundary condition that is symmetric with respect to the imaginary axis in the complex z-plane. We prove that the eigenvalues λ are all positive real.
Preprint.Recently, a conjecture of Bessis and Zinn-Justin has been verified by Dorey et al. [9], (and extended by the author [18]). That is, the eigenvalues λ of 1 2 z)) decaying to zero as z → ∞ along rays in S −n−1 and S n .
Abstract. We consider the non-Hermitian Hamiltonianon the real line, where P (x) is a polynomial of degree at most n ≥ 1 with all nonnegative real coefficients (possibly P ≡ 0). It is proved that the eigenvalues λ must be in the sector, we establish a zero-free region of the eigenfunction u and its derivative u ′ and we find some other interesting properties of eigenfunctions.Preprint.
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