A new approach to inverse spectral theory. III: Short-range potentials
The half-line Schrödinger operator \(Hu=-u''+qu\) is considered for \(q\in L^1[0,\infty)\). \textit{B. Simon} [Ann. Math. (2) 150, No. 3, 1029-1057 (1999; Zbl 0945.34013] showed that there is a function \(A\in L^1_{\text{loc}}[0,\infty)\) such that \[ m(-\kappa ^2)=-\kappa -\int_0^\infty A(\alpha)e^{-2\alpha\kappa} d\alpha. \] The unique solution \(f(x,\kappa)\) to \(-f''+qf=-\kappa ^2f\) is introduced as well as \(f(\kappa)=f(0,\kappa)\). Known representations of \(A\) in case \(f(0)\neq 0\) (\(q\) has no zero energy resonances) are stated. New results on \(A\) including the case \(f(0)=0\) are formulated and proved for \(q\) belonging to certain classes \(B_n^{B,l}\), i.e., \(q\) has \(n-1\) classical derivatives, \(q^{(n)}\in L^1(0,\infty)\), and \[ \int _0^\infty (1+|x|^l)e^{-Bx}|q^{(j)}(x)|dx<0 \] for \(l=0,\dots,n\). Theorem 1.3.: Let \(q\in C_n^{0,l}\), with \(l\geq 1\) or, if \(q\) has a zero energy resonance, \(l\geq 3\). Let \(\{-\kappa _j^2\}_{j=1}^J\) be the negative eigenvalues of \(H\) with boundary condition \(u(0)=0\), \(\kappa _j >0\). Then \[ A(\alpha)=B_0+\sum_{j=0}^J B_je^{2\alpha \kappa _j}+g(\alpha), \] with \(g\in L^1(0,\infty)\). Here, \(B_0=0\) if \(q\) has no zero energy resonances. In theorem 1.4 a more explicit representation is obtained for \(q\in C_n^{B,0}\) with \(B<0\). For part II see \textit{F. Gesztesy} and \textit{B. Simon} [A new approach to inverse spectral theory. II: General real potentials and the connection to the spectral measure. Ann. Math. (2), No. 2, 593-643 (2000)].
scientific article; zbMATH DE number 1506557
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