Spectral Projections of the Complex Cubic Oscillator

2013/10/17 by Raphaël Henry · 1 citation
Chemistry · Mathematics · Physics and Astronomy · #Algorithm #Alpha (finance) #Chemistry #Combinatorics #Eigenvalues and eigenvectors #Exponential function #Exponential growth #Geometry #Instability #Mathematical analysis #Mathematical physics #Mathematics #Operator (biology) #Physics #Pi #Projection (relational algebra) #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Quantum mechanics #Spectral Theory in Mathematical Physics #Statistics #math.SP

paper · pdf · doi:10.1007/s00023-013-0292-2

arxiv created 2013/10/17 · openalex publication_date 2013/12/19 · arxiv updated 2015/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We prove the spectral instability of the complex cubic oscillator -(d²)/(dx²)+ix³+iαx for non-negative values of the parameter α, by getting the exponential growth rate of ‖Π_n(α)‖, where Π_n(α) is the spectral projection associated with the n-th eigenvalue of the operator. More precisely, we show that for all non-negative α lim_n→+∞(1)/(n)log‖Π_n(α)‖ = \fracπ√(3).

Citations

• Making sense of non-Hermitian Hamiltonians

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