2007/02/26 by Edwin Langmann · 7 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Coupling constant #Covariant Hamiltonian field theory #Eigenfunction #Eigenvalues and eigenvectors #Good quantum number #Hamiltonian (control theory) #Hamiltonian system #Integrable system #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Waves and Solitons #Operator (biology) #Physics #Pure mathematics #Quantum #Quantum Mechanics and Non-Hermitian Physics #Quantum mechanics #Superintegrable Hamiltonian system #Type (biology) #math-ph #math.MP #nlin.SI
paper · pdf · doi:10.3842/sigma.2007.031
published in Symmetry Integrability and Geometry Methods and Applications (National Academy of Sciences of Ukraine) · This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/
arxiv created 2007/02/26 · openalex publication_date 2007/02/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08
There exists a large class of quantum many-body systems of Calogero-Sutherland type where all particles can have different masses and coupling constants and which nevertheless are such that one can construct a complete (in a certain sense) set of exact eigenfunctions and corresponding eigenvalues, explicitly. Of course there is a catch to this result: if one insists on these eigenfunctions to be square integrable then the corresponding Hamiltonian is necessarily non-hermitean (and thus provides an example of an exactly solvable PT -symmetric quantum-many body system), and if one insists on the Hamiltonian to be hermitean then the eigenfunctions are singular and thus not acceptable as quantum mechanical eigenfunctions. The standard Calogero-Sutherland Hamiltonian is special due to the existence of an integral operator which allows to transform these singular eigenfunctions into regular ones.