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Sum rules and spectral measures of Schrödinger operators with \(L^2\) potentials. (English) Zbl 1185.34131

Given \(I:= [0,\infty)\) and a real-valued function \(V\in L^2(I)\), let \(H\) be the self-adjoint realization of \(-(d/dx)^2+ V\) in \(L^2(I)\) and Dirichlet boundary condition at \(0\). In case \(V= 0\), it is clear that its spectral measure \(d\rho_0\) is supported on \(I\) and is purely absolutely continuous. A theorem by C. R. Putnam [Am. J. Math. 71, 109–111 (1949; Zbl 0031.40102)]) in conjunction with a classical result of H. Weyl [Math. Ann. 68, 220–269 (1910; JFM 41.0343.01)] implies that the essential spectrum of \(H\) is \(I\). There may be an at most countable number of negative eigenvalues \(E_j\) (zero being the only possible accumulation point).
In the course of their famous study of the Korteweg-de Vries equation, C. S. Gardner, J. N. Green, M. D. Kruskal and R. M. Miura showed that \(\sum|E_j|^{3/2}\) can be estimated in terms of the \(L^2\)-norm of \(V\) [Commun. Pure Appl. Math. 27, 97–133 (1974; Zbl 0291.35012)], R. Deift and R. Killip [Commun. Math. Phys. 203, No. 2, 341–347 (1999; Zbl 0934.34075)] proved that the spectral measure of \(H\) has an absolutely continuous component which is supported on \(I\). This absolutely continuous spectrum may, however, have to coexist with a set of eigenvalues which is possibly dense in \(I\) [S. N. Naboko, Theor. Math. Phys. 68, 646-653 (1986); translation from Teor. Mat. Fiz. 68, No. 1, 18–28 (1986; Zbl 0607.34023)] or even with a singular continuous spectrum [S. A. Denisov, J. Differ. Equations 191, No. 1, 90–104 (2003; Zbl 1032.34079)].
In the paper it is shown that a measure \(d\rho\) on \(\mathbb{R}\) is the spectral measure of \(H\) if and only if the following four conditions hold: 7mm
(i)
\(\text{supp}(d\rho)\) is the union of \(I\) and an at most countable set of negative numbers \(E_j\) with the unique accumulation point zero if it is infinite.
(ii)
The short-range part \(M_s\nu\) of the Hardy-Littlewood maximal function of a certain signed measure \(\nu\) which is associated with the Titchmarsh-Weyl \(m\)-function satisfies
\[ \int^\infty_1 k^2\log[1+ (M_s\nu(k)/k)^2]\,dk< \infty. \]
(iii)
\(\sum|E_j|^{3/2}< \infty\).
(iv)
\(\int^\infty_0 E^{1/2}\log[{1\over 4} (d\rho/d\rho_0+ d\rho_0/d\rho)+{1\over 2}]\,dE< \infty\).
Trace identities are a central ingredient of the proof which in its first part replaces (ii) and (iv) by stronger conditions. The proof is inspired by ramification of Szegő’s theorem the pervasive role of which was brought to the fore by the second author’s two-volume treatise on orthogonal polynomials on the unit circle [Orthogonal polynomials on the unit circle. Part 1: Classical theory. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1082.42020) and Orthogonal polynomials on the unit circle. Part 2: Spectral theory. Providence, RI: American Mathematical Society (2005; Zbl 1082.42021)].

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L05 General spectral theory of ordinary differential operators
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References:

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