[Paper Review] Gordon type Theorem for measure perturbation

Christian Seifert|arXiv (Cornell University)|Aug 8, 2016

Spectral Theory in Mathematical Physics5 references3 citations

TL;DR

This paper generalizes Gordon's theorem on the absence of eigenvalues for one-dimensional Schrödinger operators to the case of measure-valued potentials. By introducing the concept of 'Gordon measures'—locally bounded signed measures approximable by periodic measures in a specific asymptotic sense—it proves that such operators $ H = -\Delta + \mu $ have no embedded or isolated eigenvalues, extending classical results on $ L^1_{\text{loc}} $ potentials to more singular, measure-based settings.

ABSTRACT

Generalizing the concept of Gordon potentials to measures we prove a version of Gordon's theorem for measures as potentials and show absence of eigenvalues for these one-dimensional Schrödinger operators.

Motivation & Objective

• To extend Gordon's theorem on eigenvalue absence from $ L^1_{\text{loc}} $ potentials to more singular, measure-valued potentials.
• To define and characterize a class of measures—'Gordon measures'—that generalize the notion of Gordon potentials to the measure setting.
• To establish the self-adjointness and spectral properties of Schrödinger operators $ H = -\Delta + \mu $ with such measures using form methods and approximation techniques.
• To demonstrate that the absence of eigenvalues persists under this generalization, even for singular, signed, locally finite Borel measures.

Proposed method

• Introduce the notion of uniformly locally bounded signed Borel measures, ensuring form boundedness with respect to the Laplacian.
• Define a Gordon measure as a uniformly locally bounded measure that can be approximated in total variation on expanding intervals by periodic measures with diverging periods.
• Use Sobolev embedding and $ L^2 $-norm estimates to prove form boundedness of the negative part of the measure, enabling the definition of the Schrödinger operator via quadratic forms.
• Establish a direct operator definition via the auxiliary operator $ T $, showing that the form-based operator $ H $ is a restriction of $ T $.
• Apply a limiting argument based on the approximation of $ \mu $ by periodic measures $ \mu^m $, using uniform estimates on solution norms over intervals $ [-p_m, 2p_m] $.
• Use a contradiction argument: if an $ L^2 $ eigenfunction existed, its derivative would decay at infinity, but the approximation forces non-decaying oscillations, contradicting $ L^2 $-integrability.

Experimental results

Research questions

• RQ1 Can Gordon's theorem on the absence of eigenvalues be extended from $ L^1_{\text{loc}} $ potentials to more singular, measure-valued potentials?
• RQ2 What conditions on a signed Borel measure $ \mu $ ensure that the Schrödinger operator $ H = -\Delta + \mu $ has no eigenvalues?
• RQ3 How can one define and characterize a class of measures that generalize the classical Gordon potentials in the context of measure perturbations?
• RQ4 What role does the approximation of a measure by periodic measures—on intervals growing with the period—play in spectral non-embeddedness?

Key findings

• The class of Gordon measures includes all generalized Gordon potentials and all locally bounded periodic measures, such as $ \mu = \sum_{n \in \mathbb{Z}} \delta_{n+1/2} $.
• For any Gordon measure $ \mu $, the Schrödinger operator $ H = -\Delta + \mu $ defined via form methods has no eigenvalues.
• The proof relies on the fact that any $ L^2 $ eigenfunction would have to satisfy a uniform lower bound on $ |u(x)|^2 + |u'(x)|^2 $ at infinity, contradicting $ L^2 $-integrability.
• The approximation condition $ \lim_{m \to \infty} e^{Cp_m} |\mu - \mu^m|([-p_m, 2p_m]) = 0 $ ensures that the solution behavior of $ H $ and $ H^m $ (with periodic $ \mu^m $) become indistinguishable on large intervals.
• The result holds even for signed, singular measures, provided they are uniformly locally bounded and satisfy the approximation condition with diverging periods.