2023/11/16 by ritz Gesztesy, Gesztesy, Fritz, Markus Hunziker +3 · 1 citation
Computer Science · Mathematics · #34D15 #34L40 #34M03 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Primary: 34B20 #Secondary: 34D10 #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2311.09771
openalex publication_date 2023/11/16 · openalex created_date 2023/11/18 · openalex updated_date 2026/07/28
We consider essential self-adjointness on the space C0∞((0,∞)) of even order, strongly singular, homogeneous differential operators associated with differential expressions of the type τ2n(c) = (-1)n \fracd2nd x2n + \fraccx2n, x gt; 0, n ∈ ℕ, c ∈ ℝ, in L2((0,∞);dx). While the special case n=1 is classical and it is well-known that τ2(c)|C0^∞((0,∞)) is essentially self-adjoint if and only if c ≥ 3/4, the case n ∈ ℕ, n ≥ 2, is far from obvious. In particular, it is not at all clear from the outset that there exists cn ∈ ℝ, n ∈ ℕ, such that τ2n(c)|C0^∞((0,∞)) is essentially self-adjoint if and only if c ≥ cn. As one of the principal results of this paper we indeed establish the existence of cn, satisfying cn ≥ (4n-1)!!/22n, such that property \eqref0.1 holds. In sharp contrast to the analogous lower semiboundedness question, for which values of c \it is τ2n(c)|C0∞((0,∞)) bounded from below?, which permits the sharp (and explicit) answer c ≥ [(2n -1)!!]2/22n, n ∈ ℕ, the answer for \eqref0.1 is surprisingly complex and involves various aspects of the geometry and analytical theory of polynomials. For completeness we record explicitly, c1 = 3/4, c2= 45, c3 = 2240 (214+7 √(1009) )/27, and remark that cn is the root of a polynomial of degree n-1. We demonstrate that for n=6,7, cn are algebraic numbers not expressible as radicals over ℚ (and conjecture this is in fact true for general n ≥ 6).