Shnol-type theorem for the Agmon ground state

Beckus, Siegfried; Pinchover, Yehuda

doi:10.4171/jst/296

Cited by 8 publications

(5 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, a criticality theory for Schrödinger operators on graphs has also been established by Keller, Pinchover, and Pogorzelski in [19]. The theory has witnessed its applications in the works of Murata and Pinchover and their collaborators (see recent examples in [6,18,29]). For the case of generalized Schrödinger forms, we refer to [49,50].…”

Section: Introductionmentioning

confidence: 94%

Positive solutions of the $\mathcal{A}$-Laplace equation with a potential

Hou¹,

Pinchover²,

Rasila³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we study positive solutions of the quasilinear elliptic equation, where n ≥ 2, 1 < p < ∞, the divergence of A is the well known A-Laplace operator considered in the influential book of Heinonen, Kilpeläinen, and Martio, and the potential V belongs to a certain local Morrey space. The main aim of the paper is to extend criticality theory to the operator Q ′ p,A,V . In particular, we prove an Agmon-Allegretto-Piepenbrink (AAP) type theorem, establish the uniqueness and simplicity of the principal eigenvalue of Q ′ p,A,V in a domain ω ⋐ Ω and various characterizations of criticality. Furthermore, we also study positive solutions of the equation Q ′ p,A,V [u] = 0 of minimal growth at infinity in Ω, the existence of minimal positive Green function, and the minimal decay at infinity of Hardy-weights.

show abstract

Section: Introductionmentioning

confidence: 94%

Positive solutions of the $\mathcal{A}$-Laplace equation with a potential

Hou¹,

Pinchover²,

Rasila³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…where σ dis (L V ) is the discrete spectrum. On the other hand, [11,Theorem 1.1] shows that λ * (V ) ∈ σ(L V ). Therefore, λ * (V ) is an isolated eigenvalue in σ(L V ).…”

Section: Therefore (28) Holdsmentioning

confidence: 99%

Certain Liouville properties of eigenfunctions of elliptic operators

Arapostathis

Biswas

Ganguly

2018

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

We present certain Liouville properties of eigenfunctions of secondorder elliptic operators with real coefficients, via an approach that is based on stochastic representations of positive solutions, and criticality theory of secondorder elliptic operators. These extend results of Y. Pinchover to the case of nonsymmetric operators of Schrödinger type. In particular, we provide an answer to an open problem posed by Pinchover in [Comm. Math. Phys. 272 (2007), no. 1, 75-84, Problem 5]. In addition, we prove a lower bound on the decay of positive supersolutions of general second-order elliptic operators in any dimension, and discuss its implications to the Landis conjecture.

show abstract

“…In this paper we present a criticality theory for positive Schrödinger operators on general weighted graphs. First applications of this theory were already obtained in [20] and [2].…”

Section: Introductionmentioning

confidence: 98%