2002/01/28 by K. C. Shin · 2 citations
Mathematics · Physics and Astronomy · #Boundary (topology) #Boundary value problem #Class (philosophy) #Computer science #Conjecture #Eigenvalues and eigenvectors #Infinity #Mathematical analysis #Mathematics #Nonlinear Photonic Systems #Physics #Polynomial #Pure mathematics #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Quantum mechanics #Zero (linguistics) #hep-th #math-ph #math.MP
paper · pdf · doi:10.1007/s00220-002-0706-3
published as Commun.Math.Phys. 229 (2002) 543-564 · 22 pages, one figure. Reason for update--in order to more clearly explain which parts of the proof follow the earlier work of Dorey, Dunning and Tateo. Main change on page 3, minor changes on page 12 and 18
arxiv created 2002/01/28 · openalex publication_date 2002/09/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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=-\fracπ2± (2π)/(m+2), where P(z)=a1 zm-1+a2 zm-2+...+am-1 z is a real polynomial and m≥ 2. We prove that if for some 1≤ j≤(m)/(2), we have (j-k)ak≥ 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 αiz3+βz2+γiz when α,βand γare all real with α\not=0 and αγ≥ 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.