Decay of solutions of ordinary differential equations. (Q1856995)

Let \(F= w(x)^{-1} \sum^n_{k=0} p_k(x)(d/dx)^k\) be a \(n\)th-order differential operator on \(L^2_w(J)\), \(J= (a,\infty)\), and let \(G\) be the perturbed operator \(G= -F+ gFg^{-1}\), \(g(x)\geq\alpha> 0\), \(g\in C^n(J)\). The author studies the dimension of eigenspaces associated with \(F^+\) and \(F^++ G^+\), where \(F^+\), \(G^+\) are formal adjoints of \(F\), \(G\). The result is applied to obtain estimates on the growth of eigenfunctions of \(F\), choosing appropriate functions \(g(x)\). Let \(F_1\), \(F_0\) be maximal and minimal operators associated with \(F\) and assume \(\overline\lambda\) is not in the essential spectrum of \(F_0\). Suppose that we can choose \(g(x)\) so that \(R(F_0)\) is closed, \(\| Gy\|\leq c\| y\|+ d\|(F- \overline\lambda)y\|\), \(y\in D(F_0)\), and \(c+ d\| (F_0- \overline\lambda I)^{-1}\|<\|(F_0- \overline\lambda I)^{-1}\|\). Then it is proved that (i) \(\dim N((F^+-\lambda I))= \dim N(g(F^+-\lambda I)g^{-1}))\). By (i) if \(u\) is an eigenvalue of \(F\) then \(gu\) is an eigenvalue of \(gF g^{-1}\) and \(\int^\infty_a w| gu|^2\, dx< \infty\) implying that \(| gu|\to 0\), \(x\to\infty\). Three examples are given. For \(F= -(d/dx)(p(x) d/dx)+ q(x)\), on \(L^2(J)\), \(J= (a,\infty)\), \(a> 0\), we may choose \(g(x)= \exp(\sigma \int^x_a (p(x))^{-1/2} dx)\), \(\sigma> 0\) is a small number.