2007/05/19 by Julius Borcea, Borcea, Julius, Rikard Bøgvad +3
Computer Science · Mathematics · #30C15 #31A35 #34E05 #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.0705.2822
openalex publication_date 2007/05/19 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
Consider a homogenized spectral pencil of exactly solvable linear\ndifferential operators T la=\∑i=0k Qi(z) lak-i frac\ndidzi, where each Qi(z) is a polynomial of degree at most i and\n la is the spectral parameter. We show that under mild nondegeneracy\nassumptions for all sufficiently large positive integers n there exist\nexactly k distinct values lan,j, 1\≤ j\≤ k, of the spectral\nparameter la such that the operator T la has a polynomial eigenfunction\npn,j(z) of degree n. These eigenfunctions split into k different\nfamilies according to the asymptotic behavior of their eigenvalues. We\nconjecture and prove sequential versions of three fundamental properties: the\nlimits \Ψj(z)=\limn\→\∞ fracpn,j'(z) lan,jpn,j(z)\nexist, are analytic and satisfy the algebraic equation \∑i=0k Qi(z)\n\Ψji(z)=0 almost everywhere in bCP. As a consequence we obtain a\nclass of algebraic functions possessing a branch near \∞\∈ bCP which is\nrepresentable as the Cauchy transform of a compactly supported probability\nmeasure.\n