2019/05/24 by Yik‐Man Chiang, Chiang, Yik-Man, Chun‐Kong Law +3
Mathematics · Physics and Astronomy · #33E10 #33E17 (secondary) #34M35 (primary) #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #Connection (principal bundle) #Cylinder #Degenerate energy levels #Eigenvalues and eigenvectors #FOS: Mathematics #FOS: Physical sciences #Geometry #Invariant (physics) #Invariant subspace #Linear subspace #Mathematical Physics (math-ph) #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Numerical methods for differential equations #Parabolic cylinder function #Parabolic partial differential equation #Partial differential equation #Physics #Pure mathematics #Quantum mechanics #Singularity #Space (punctuation)
paper · pdf · doi:10.48550/arxiv.1905.10046
openalex publication_date 2019/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We show that there is a full correspondence between the parameters space of\nthe degenerate biconfluent Heun connection (BHC) and that of Painlev 'e IV\nthat admits special solutions. The BHC degenerates when either the Stokes' data\nfor the irregular singularity at \∞ degenerates or the regular singular\npoint at the origin becomes an apparent singularity. We show that if the BHC is\nwritten as isomonodromy family of biconfluent Heun equations (BHE), then the\nBHE degenerates precisely when it admits eigen-solutions of the biconfluent\nHeun operators, after choosing appropriate accessory parameter, of specially\nconstructed invariant subspaces of finite dimensional solution spaces spanned\nby parabolic cylinder functions. We have found all eigen-solutions over this\nparameter space apart from three exceptional cases after choosing the right\naccessory parameters. These eigen-solutions are expressed as certain finite sum\nof parabolic cylinder functions. We extend the above sum to new convergent\nseries expansion in terms of parabolic cylinder functions to the BHE. The\ninfinite sum solutions of the BHE terminates precisely when the parameters of\nthe BHE assumes the same values as those of the degenerate biconfluent Heun\nconnection except at three instances after choosing the right accessory\nparameter.\n