Relative boundedness and compactness theory for second-order differential operators
If \(N\) is an \(n\)-th order differential operator on a compact interval with bounded coefficients and invertible leading coefficient and \(M\) is an \(m\)-th order differential operator with \(m<n\), then \(M\) is relatively \(N\)-compact. However, this well-known property is no longer true, in general, if the differential operators are considered on unbounded intervals. \(M\) does not even need to be \(N\)-bounded. Necessary and sufficient conditions are given such that the maximal operator \(B_j\) associated with \(b_jy^{(j)}\) \((j=0,1)\) is relatively bounded (compact) with respect to the maximal operator \(L\) associated with the Sturm-Liouville expression \(-(py')'+qy\) on \(I=[a,\infty)\). In case \(|p'(t)|\leq k\sqrt{p(t)}\), this is a special case of earlier results of the first author. In a second result, Theorem 3.1, \(q(t)\geq K\), \(|p'(t)|\leq A_1\sqrt{p(t)q(t)}\), and \(p(t)^{1/2}|q'(t)|\leq A_2q(t)^{3/2}\) are assumed, where \(p,q\in AC_{\text{loc}}(I)\) are real valued, \(p>0\), and \(A_2<1\). For \(j=0,1\) and \(\delta>0\) let \[ g_{j,\delta}(t)=\sqrt{q(t)/p(t)}\int_t^{t+\delta\sqrt{p(t)/q(t)}} \frac{|b_j(\tau)|^2}{p(\tau)^jq(\tau)^{2-j}} d\tau. \] Then \(B_j\) is \(L\)-bounded [\(L\)-compact] if and only if \(b_j\in L^2_{\text{loc}}(I)\) and \(\sup\{g_{j,\delta}(t):t\in [a,\infty)\}<\infty\) [\(\lim_{t\to\infty}g_{j,\delta}(t)=0\)]. A similar result is obtained in case \(p(t)=t^\alpha\), \(|q(t)|\leq Mt^{\alpha-2}\), where \(\alpha\geq 2\) and \(M>0\). This result is applied to the energy operator of the hydrogen atom. In particular, it is shown that the operator with a Coulomb type potential \(V(x)=c/x\) is a relatively compact perturbation of the energy operator with potential \(0\).
scientific article; zbMATH DE number 1086037