2024/03/11 by Gesztesy, Fritz, Hunziker, Markus
#35A24 #35J30 #35J48 (Primary) 35G05 #35P05 #47B02 (Secondary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2403.07160
Let n∈ℕ, n≥ 2. We prove that the strongly singular differential operator (Δ2 +c|x|-4)|_C0∞(ℝn \backslash \0\), c ∈ ℝ, is essentially self-adjoint in L2(ℝn; dn x) if and only if c≥ \begincases3(n+2)(6-n)amp;for 2≤ n≤ 5;
-(n(n+4)(n-4)(n-8))/(16)amp;for n≥ 6.\endcases
Via separation of variables, our proof reduces to studying the essential self-adjointness on the space C0∞((0,∞)) of fourth-order Euler-type differential operators of the form (d4)/(dr4)+c1((1)/(r2)(d2)/(dr2)+(d2)/(dr2)(1)/(r2))+(c2)/(r4), r∈(0,∞), (c1,c2)∈ ℝ2, in L2((0,∞);dr).
Our methods generalize to differential operators related to higher-order powers of the Laplacian, however, there are some nontrivial subtleties that arise. For example, the natural expectation that for m,n∈ℕ, n ≥ 2, there exist cm,n∈ℝ such that (Δm+c|x|-2m)|_C0∞(ℝn \backslash \0\) is essentially self-adjoint in L2(ℝn; dn x) if and only if c ≥ cm,n, turns out to be false. Indeed, for n=20, we prove that the differential operator ((-Δ)5+c|x|-10)|_C0∞(ℝ20 \backslash \0\), c ∈ ℝ, is essentially self-adjoint in L2( ℝ20; d20 x) if and only if c∈ [0,β]∪ [γ,∞), where β≈ 1.0436× 1010, and γ≈ 1.8324× 1010 are the two real roots of the quartic equation amp;3125z4-83914629120000z3+429438995162964368031744 z2
amp; +1045471534388841527438982355353600z
amp; +629847004905001626921946285352115240960000=0.