Essential Self-Adjointness of Even-Order, Strongly Singular, Homogeneous Half-Line Differential Operators

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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)nd2ndx2n+cx2n,x>0,n∈N,c∈R, 0, \\; n \\in {{\\mathbb {N}}}, \\; c \\in {{\\mathbb {R}}}, \\end{aligned}$$\\end{document}]]>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 existscn∈R,n∈N,such thatτ2n(c)|C0∞((0,∞))is essentiallyself-adjoint(∗)if andonly ifc≥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 (*) holds. In sharp contrast to the analogous lower semiboundedness question, for whichvalues ofcisτ2n(c)|C0∞((0,∞))bounded frombelow?,which permits the sharp (and explicit) answer c≥[(2n-1)!!]2/22n, n∈N, the answer for (*) 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+71009)/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).