Behavior of generalized eigenfunctions at infinity and the Schrödinger conjecture
The authors consider the stationary Schrödinger equation \(H\psi =E\psi\), where \(H\) is the Hamiltonian. It was postulated when the quantum mechanics theory was in its infancy that only square integrable solutions, or solutions bounded at infinity have physical meaning. Schrödinger conjectured that solutions can become unbounded only in the neighborhood of a singularity of \(H\), that is in the neighborhood of a particle. Variants of this statement can be found in several textbooks on quantum mechanics. Steklov's conjecture concerns a specific system of polynomials orthogonal on an interval \(\Delta\) with respect to the weight function \(\rho(\lambda)\), \(\lambda\in \Delta\), i.e. \(\int_\Delta [P_n(\lambda) P_m (\lambda) \rho(\lambda)] d\lambda =\delta (n-m)\). Steklov's conjecture states that \(|P_n (\lambda) |\leq C\Delta'\), with \(\lambda\in \Delta' \subset \text{int} \Delta\). The recurrence relation for these polynomials is \(\alpha_{n+1} P_{n+1} (\lambda) +\beta_n P_n(\lambda) +\alpha_{n-1} P_{n-1} (\lambda)=\lambda P_n(\lambda)\). With this relation the Steklov's conjecture becomes a discrete version of the Schrödinger conjecture. In a suitable Hilbert space, such as \(L^2 (\mathbb{R}^\nu)\) the operator \(H\) is essentially self-adjoint and allows spectral decomposition. The projector of this decomposition is an integral operator whose kernel is given in terms of solutions of the problem \(H\psi= E'\psi\), \(E'\leq E\). These solutions are known as generalized eigenfunctions. Of course the spectral decomposition of \(H\) is defined modulo sets of zero \(\sigma\)-measure. The authors outline the complicated history of Schrödingers and Steklov's conjectures. For a while Steklov's conjecture was regarded as a problem in algebra. George Szegö proved Steklov's conjecture under additional assumptions. K. Maurin and A. G. Kostiushenko published independently a theorem which implied that Schrödinger conjecture is true (in fact it implied more!) But their arguments contained errors. Yu. M. Berezanskij and G. I. Kac proved a related theorem weaker than Schrödinger's, which has found many useful applications. The problem was regarded as solved after V. P. Maslov (co-author of this article) published a counterexample, which was accepted for a number of years. While analysis offered in that one-dimensional counterexample was correct the conclusion was not. The solutions obtained were not generalized eigenfunctions. The reason for this universal acceptance of the Maslov counterexample for several years was an unusual and subtle spectral situation arising in that counterexample. But the idea introduced there of constructing space barriers which are further and further spaced is fruitful and has been used since then by the authors of this article to construct more complex counterexamples to Schrödinger's conjecture. Replacing the original idea of almost nonreflexive triangular ``teeth'' in the original counterexample by more complicated scatterers, the authors produced a number of counterexamples differing from each other in technical details and dealing with different spectral types. This construction can be applied also to the discrete version, producing counterexamples to the spectral version of Steklov's conjecture. The original version of Steklov's conjecture has been shown false in an article by \textit{E. A. Rakhmanov} [Mat. Sb., Nov. Ser. 108, 581-608 (1979; Zbl 0452.33012)]. The present article contains one of several counterexamples constructed by the authors. First they offer a discussion of products of random matrices, and prove a theorem of the type proved by \textit{S. Kotani} in 1982 [North-Holland Math. Libr. 32, 225-247 (1984; Zbl 0549.60058)] who showed that positivity of Lyapunov exponents is essential in determining the point spectrum of an operator. A gap in Kotani's theorem is dealt with in this article. Let us look at the first order (vector) system \(d/dx U_E(x,a) =\left[ \begin{smallmatrix} 0 & 1\\ V(x)- E & 0 \end{smallmatrix} \right]\) corresponding to the equation \(-y''+ v(x)y= Ey\), \(U_E (a,a) =I\) (identity matrix). For any positive function \(F(x)\) existence of the limit \(\lim_{x\to \infty} \{\ln |U_E(x,a) |/ \ln F(x)\}= \gamma_F(E)\) defines the Lyapunov \(F\)-exponent \(\gamma_F(E)\). An example of the theorems previously proved by the authors and used here is the following one by A. Ya. Gordon: Let \(q\) be an integrally bounded measurable function such that for certain pairs of sequences \(T_n\to \infty\) there exists sequence of functions \(q_n(t)= q_n(t+T_n)\), such that \(\int^{Tn}_{-2 Tn} |q(t)-q_n (t+T_n)|dt< \exp \{-\alpha_nT_n\}\), \(n=1,2, \dots\). This theorem implies the absence of eigenvalues. A spectral type discovered by R. Carmona is analyzed by the authors providing a counterexample to Steklov's conjecture applied to the entire real axis. The authors comment that Morse-type potentials considered in this article could provide counterexamples to what the authors call ``naive localization theory''. In the final parts of this article the authors discuss the rare scatterer-type potentials. Examples are offered such as a potential unbounded from below, with corresponding solutions unbounded a.e. In this specific example a solution juts out from the center of the scatterer. Yet this solution is bounded in the \(L^2\) norm. To overcome some of the difficulties caused by potentials being unbounded the authors promise to exhibit more complex scatterers in an article to be published in the near future. Since the present article appeared in early 1993, while this review is written in September 1997, that promised publication should be now in print. This is obviously an important article clarifying much confusion that has existed in analysis of spectral properties of the Schrödinger operator, and summing up results of many years of work of the authors.
scientific article; zbMATH DE number 1047127