Sch’nol’s Theorem for Strongly Local Forms

We prove a variant of Sch’nol’s theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures. As an application, we discuss quantum graphs with δor Kirchhoff boundary conditions. Dedicated to Shmuel Agmon on the occasion of his 85th birthday Introduction The behavior of solutions to elliptic partial differential equations and its interplay with spectral properties of the associated partial differential operators is a topic of fundamental interest. Our understanding today is in many aspects based on groundbreaking work by Shmuel Agmon (cf [4, 1, 3, 5, 2, 6]) to whom this article is dedicated with great admiration and gratitude. Here we explore the well known classical fact that the spectral values of Schrödinger operators H can be characterized in terms of the existence of appropriate “generalized eigenfunctions” or “eigensolutions”. One part of this characterization is Sch’nol’s theorem stating that existence of an eigensolution of Hu = λu “with enough decay” guarantees λ ∈ σ(H). We refer to the original result [29] by Sch’nol from 1957 that was rediscovered by Simon, [30], as well as the discussion in [13]. Clearly, if u ∈ D(H) then λ is an eigenvalue. But much less restrictive growth conditions suffice to construct a Weyl sequence from u by a cut-off procedure. One of the main objectives of the present paper is to provide a proof along these lines for a great variety of operators. In our framework, the principal part H0 of H is the selfadjoint operator associated with a strongly local regular Dirichlet form E and H = H0+μ with a measure perturbation. Of course, this includes Schrödinger operators on manifolds and open subsets of Euclidean space, but much more singular coefficients are included. In our general Sch’nol’s theorem potentials in Lloc with form small negative and arbitrary positive part are included, thereby generalizing results that require some Kato class condition. The appropriate “decay assumption” on u that is necessary can roughly be called subexponential growth and is phrased in terms of conditions like ‖uχB(x0,rn+δ)‖ ‖uχB(x0,rn)‖ → 1 for some rn → ∞ and some fixed δ > 0. Here, χM is the characteristic function of M and B(p, s) denotes the closed ball in the intrinsic metric around p with radius s. A precise definition of the intrinsic metric is given below. For uniformly Date: February 1, 2008. 1 2 A. BOUTET DE MONVEL, D. LENZ, AND P. STOLLMANN bounded and strictly elliptic divergence form operators, one recovers the usual Euclidean balls. It is interesting to note that we use a form analog of Weyl sequences that enables us to treat partial differential operators with singular coefficients. Of course, the usual calculations of H(ηu) for a smooth cut-off function η fail in the present general context. That is already true for operators in divergence form with nondifferentiable coefficients and to our knowledge, there is no Sch’nol’s Theorem in that context available in the literature. They have to be replaced by calculations with the corresponding forms. The crucial object in that respect is the energy measure of a strongly local Dirichlet forms that supplies one with a calculus reminiscent of gradients. All this together leads to our version of Sch’nol’s theorem, Theorem 4.4 below which is one of the main results of the present paper. Apart from its generality it is also pretty simple conceptually. Another aim of the present paper is to advertise Dirichlet form techniques for quantum or metric graphs. As a space these consist of a countable family of edges (intervals) that are glued together in the sense that the Laplacian on the direct sum of intervals is equipped with certain boundary conditions for those edges that meet at a vertex. For certain types of boundary conditions one can apply the Dirichlet form framework. In this way we get a similar understanding (and a partial generalization) of results by P. Kuchment [25] on Sch’nol’s theorem for quantum graphs. Needless to say that on the other hand quantum graphs provide a wealth of examples of strongly local Dirichlet forms. While Sch’nol’s theorem had already been known for quantum graphs, the way to interpret them as Dirichlet forms opens a powerful arsenal of analytic and probabilistic techniques. Quite a number of results in operator and perturbation theory have been established in the Dirichlet form setting and can readily be applied to quantum graphs. This can be illustrated by the “converse” of Sch’nol’s theorem. Proving results on “expansion in generalized eigenfunctions” one gets the fact that for spectrally almost λ ∈ σ(H) there exists a solution that doesn’t increase too seriously. In the context of Dirichlet forms that has been established in [12]; see also the references in there and the discussion in [13]. Together with what we said above, the results from [12] can directly be applied to certain quantum graphs which yields a partial converse of Kuchment’s results in [25] that seems to be new. At least in terms of existing proofs the “Sch’nol part” of the characterization of the spectrum in terms of eigenfunctions appears to be the easier one. That is reflected in the fact that we needed more restrictive conditions in [12] to establish an eigenfunction expansion than what we need in the present paper. That refers to conditions on the underlying operator as well as to conditions on the measure perturbation, where a Kato type condition is needed in [12]. The conclusion from the latter paper is that for spectrally almost λ ∈ σ(H) there is a “subexponentially bounded” eigensolution. To see that this is compatible with the growth condition referred to above is the third main result of the present paper. We should, moreover, mention our version of the Caccioppoli inequality, Theorem 3.1 below. For the unperturbed operator H0 such an inequality can be found in [10]. Our version