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

We consider essential self-adjointness on the space C₀∞((0,∞ )) C 0 ∞ ( ( 0 , ∞ ) ) of even-order, strongly singular, homogeneous differential operators associated with differential expressions of the type aligned τ ₂ₙ(c) = (-1)ⁿ d²ⁿd x²ⁿ + cx²ⁿ, x > 0, \; n ∈ N, \; c ∈ R, aligned τ 2 n ( c ) = ( - 1 ) n d 2 n d x 2 n + c x 2 n , x > 0 , n ∈ N , c ∈ R , in L²((0,∞ );dx) L 2 ( ( 0 , ∞ ) ; d x ) . While the special case $$n=1$$ n = 1 is classical and it is well known that τ ₂(c) |_C₀∞((0,∞ )) τ 2 ( c ) | C 0 ∞ ( ( 0 , ∞ ) ) is essentially self-adjoint if and only if c ≥ 3/4 c ≥ 3 / 4 , the case n ∈ N n ∈ N , n ≥ 2 n ≥ 2 , is far from obvious. In particular, it is not at all clear from the outset that $${aligned} {aligned}&there exists c_n ∈ {{R}}, n ∈ {{N}}, such that \& τ ₂ₙ(c) |_{C_0∞((0,∞ ))} \, is essentially self-adjoint (*)\\ {}& if and only if c ≥ c_n. {aligned} {aligned}$$ there exists c n ∈ R , n ∈ N , such that τ 2 n ( c ) | C 0 ∞ ( ( 0 , ∞ ) ) is essentially self - adjoint ( ∗ ) if and only if c ≥ c n . As one of the principal results of this paper we indeed establish the existence of cₙ c n , satisfying cₙ ≥ (4n-1)!! /2²ⁿ c n ≥ ( 4 n - 1 ) ! ! / 2 2 n , such that property (*) holds. In sharp contrast to the analogous lower semiboundedness question, aligned for which values of c is τ ₂ₙ(c) |_C₀∞((0,∞ )) bounded from below?, aligned for which values of c is τ 2 n ( c ) | C 0 ∞ ( ( 0 , ∞ ) ) bounded from below ? , which permits the sharp (and explicit) answer c ≥ [(2n -1)!!]² /2²ⁿ c ≥ [ ( 2 n - 1 ) ! ! ] 2 / 2 2 n , n ∈ N 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, aligned c₁&= 3/4, c₂= 45, c₃ = 2240 (214+7 √1009\, ) /27, aligned c 1 = 3 / 4 , c 2 = 45 , c 3 = 2240 ( 214 + 7 1009 ) / 27 , and remark that cₙ c n is the root of a polynomial of degree $$n-1$$ n - 1 . We demonstrate that for $$n=6,7$$ n = 6 , 7 , cₙ c n are algebraic numbers not expressible as radicals over Q Q (and conjecture this is in fact true for general n ≥ 6 n ≥ 6 ).