Article
Ann. Henri Poincaré (to appear)
[DOI: 10.1007/s00023-024-01451-0]
Essential Self-Adjointness of Even-Order, Strongly Singular, Homogeneous Half-Line Differential Operators
Fritz Gesztesy, Markus Hunziker, and Gerald Teschl
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 d2n/d x2n + c/x2n , x > 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 ∈ ℝ, such that
τ2n(c)|_C0∞((0,∞)) is essentially self-adjoint if and only if c ≥ cn.
(*)
for which values of c 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 \eqref{0.1} 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).
As one of the principal results of this paper we indeed establish the existence of cn, satisfying cn ≥ (4n-1)!!/22n, such that property (*) holds.
In sharp contrast to the analogous lower semiboundedness question,
MSC2010: Primary: 34B20, 34D15, 34M03; Secondary: 34D10, 34L40.
Keywords: Homogeneous differential operators, Euler differential operator, strongly singular coefficients, essential self-adjointness.
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