Asymptotic formulae with remainder estimates for eigenvalue branches of the Schrödinger operator $H - \lambda W$ in a gap of $H$

Abstract

Abstract. The Floquet theory provides a decomposition of a periodic Schrödinger operator into a direct integral, over a torus, of operators on a basic period cell. In this paper, it is proved that the same transform establishes a unitary equivalence between a multiplier by a decaying potential and a pseudo-differential operator on the torus, with an operator-valued symbol. A formula for the symbol is given.As applications, precise remainder estimates and two-term asymptotic formulas for spectral problems for a perturbed periodic Schrödinger operator are obtained.0.1. There are a number of articles on the discrete spectrum in a gap of the essential spectrum of the Schrödinger operator H = −∆ + V (see [1,3,4,7,8,9,12,13,17,20] and the bibliography in [1,3,4]). The majority of them [1,3,4,7,8,9,21] deal with the existence of eigenvalue branches and with estimates or asymptotics for the number N ± (t; H − E; W ) of branches of H ∓ µW with 0 < µ < t which cross the energy level E ∈ R − σ(H) (here W is a potential decaying at infinity). In [12,13,17], the principal term of the asymptotics of a series of eigenvalues accumulating to a boundary point E of the essential spectrum was computed, without a remainder estimate.For motivation from solid state physics (impurities in crystals), see [1] and the bibliography there.In this paper, we obtain the principal term of the asymptotics with remainder estimate for N ± (t; H − E; W ); W is smooth and non-negative, and V is periodic.The main idea of the paper is a reduction to a pseudodifferential operator (pdo) on a torus, with operator-valued symbol. For N − (t; H − E; W ), further reductions allow one to obtain bounds in terms of the counting function of the spectrum of a pdo acting in sections of a finite-dimensional fibering over a torus. These bounds give the principal terms of the asymptotics as t → ∞, with remainder estimates which are as sharp as those for the corresponding pdo; if for the latter an asymptotic formula with two terms of the asymptotics is known, then the same formula is valid for N − (t; H − E; W ).In [1], only the principal term of the asymptotics of N − (t; H − E; W ) was found, without a remainder estimate, and W was assumed to stabilize to a positive spherically homogeneous potential; on the other hand, V was not necessarily periodic.Received by the editors May 15, 1995 and, in revised form, December 9, 1995 Mathematics Subject Classification. Primary 35P20. The author was supported in part by ISF grant RNH 000. Note that the reduction to an operator with operator-valued symbol explains why the classical Weyl formula with the ordinary symbol predicts false asymptotics for N − (t; H; W ) (as was observed in [1]): for the ordinary symbol under consideration, the uncertainty principle fails since derivatives of this symbol do not admit good estimates (so that the function h, which plays a crucial role in calculi of pseudodifferential operators [10, Chapter 18], is not small outside some compact), whereas the operator-valued symbol enjoys more favourable...

On absence of embedded eigenvalues for schrÖdinger operators with perturbed periodic potentials

“…In [11] a one-dimensional example is provided where a not very fast decaying perturbation of a periodic potential creates embedded eigenvalues. There are many papers devoted to studying the behavior of the point spectrum in the gaps of the continuous spectrum (see, for instance, [1] - [6], [9], [14], [17], [23]- [25], [31], and [33] - [35]). However, apparently the only known result on the absence of embedded eigenvalues relates to the one-dimensional case of the Hill's operator…”