scholarly journals Eigenvalue bounds for non-selfadjoint Dirac operators

2021 ◽  
Author(s):  
Piero D’Ancona ◽  
Luca Fanelli ◽  
Nico Michele Schiavone
Keyword(s):  
Dirac Operator ◽  
Discrete Spectrum ◽  
Complex Plane ◽  
Dirac Operators ◽  
Resolvent Estimates ◽  
Eigenvalue Bounds ◽  
Mixed Norms ◽  
Massless Case ◽  
Abstract Version ◽  
Embedded Eigenvalues

AbstractWe prove that the eigenvalues of the n-dimensional massive Dirac operator $${\mathscr {D}}_0 + V$$ D 0 + V , $$n\ge 2$$ n ≥ 2 , perturbed by a potential V, possibly non-Hermitian, are contained in the union of two disjoint disks of the complex plane, provided V is sufficiently small with respect to the mixed norms $$L^1_{x_j} L^\infty _{{\widehat{x}}_j}$$ L x j 1 L x ^ j ∞ , for $$j\in \{1,\dots ,n\}$$ j ∈ { 1 , ⋯ , n } . In the massless case, we prove instead that the discrete spectrum is empty under the same smallness assumption on V, and in particular the spectrum coincides with the spectrum of the unperturbed operator: $$\sigma ({\mathscr {D}}_0+V)=\sigma ({\mathscr {D}}_0)={\mathbb {R}}$$ σ ( D 0 + V ) = σ ( D 0 ) = R . The main tools used are an abstract version of the Birman–Schwinger principle, which allows in particular to control embedded eigenvalues, and suitable resolvent estimates for the Schrödinger operator.

Variational Principles

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Variational Principle ◽  
Lower Semicontinuity ◽  
Variational Principles ◽  
Perfect Information ◽  
Perturbation Functions ◽  
Abstract Version ◽  
Recursive Constructions ◽  
Natural Description

This chapter describes smooth variational principles (of Ekeland type) as infinite two-player games. These variational principles are based on a simple but careful recursive choice of points where certain functions that change during the process have values close to their infima. Like many other recursive constructions, the choice has a natural description using the language of infinite two-player games with perfect information. The chapter first considers the perturbation game used in Theorem 7.2.1 to formulate an abstract version of the variational principle before showing how to specialize it to more standard formulations. It then examines the bimetric variant of the smooth variational principle, along with the perturbation functions that are relatively simple. It concludes with an assessment of cases when completeness and lower semicontinuity hold only in a bimetric sense.

Factorization of Affinities

1979 ◽  
Vol 31 (2) ◽  
pp. 354-362 ◽  
Author(s):  
Erich W. Ellers
Keyword(s):  
Plane Motion ◽  
Valuable Insight ◽  
Abstract Version ◽  
Insight Into ◽  
Refined Version

The decomposition of mappings into a minimal number of simple mappings is a common sight in geometry. One well-known instance is the representation of a plane motion by three reflections (see e.g. H. S M. Coxeter [3]) or the representation of equiaffinities by a minimal number of shears or reflections ([14], [5], [7], [8]). Theorems of this nature not only give valuable insight into the nature of the mapping, but they are also often used as a base for characterization theories (see e.g. F. Bachmann [2], M. Götzky [10]). A more abstract version of the same type of results is the famous Cartan-Dieudonné theorem. Its usefulness is indisputable. P. Scherk [13] gave a refined version of this theorem.

An Abstract Version of a Result of Fong and Sucheston

1978 ◽  
Vol 21 (2) ◽  
pp. 247-248
Author(s):  
P. E. Kopp
Keyword(s):  
General Result ◽  
Representation Theorem ◽  
Analytic Proof ◽  
Abstract Version ◽  
Operator Version

Nagel [3] has given a purely functional-analytic proof of Akcoglu and Sucheston's operator version [1] of the Blum-Hanson theorem. The purpose of this note is to show that the same techniques may be applied to obtain a proof, in the context of (AL)-spaces, of a more general result due to Fong and Sucheston [2]. By Kakutani's representation theorem, any (AL)-space can of course be represented as an L-1-space. Thus the present result is simply a reformulation of that of Fong and Sucheston.

Permission Schemas and the Selection Task

1993 ◽  
Vol 46 (4) ◽  
pp. 637-651 ◽  
Author(s):  
Richard A. Griggs ◽  
James R. Cox
Keyword(s):  
Factorial Design ◽  
Schema Theory ◽  
Selection Task ◽  
Error Patterns ◽  
Standard Version ◽  
Correct Performance ◽  
Abstract Version ◽  
University Undergraduates ◽  
Proportion Correct

Cheng and Holyoak's abstract permission schema version of Wason's selection task and the standard abstract version of the task were examined in two experiments, each a factorial design with type of problem (permission vs. standard), presence or absence of a checking context, explicit or implicit negatives on the not-p and not-q cards, and presence or absence of a rule clarification statement as factors. The original permission problem violation-type instruction was employed in Experiment 1, and Margolis's not-p and not-q violation instruction (Griggs & Jackson, 1990) was used in Experiment 2. Subjects were 640 university undergraduates, with each subject solving only one problem. The major findings for permission tasks were: (1) facilitation for the abstract permission version was replicated but found to be dependent upon the presence of explicit negatives on the not-p and not-q cards; and (2) this facilitation was enhanced by the Margolis not-p and not-q instruction. Per Girotto, Mazzocco, and Cherubini (1992), these findings and the observed error patterns are consistent with pragmatic schema theory. The major findings for the standard version of the task were: (1) none of the factors significantly impacted proportion correct [performance was poor, ≤10% correct in 15 of 16 conditions] and (2) the number of not-p & not-q incorrect selections was increased significantly for the not-p and not-q instruction. These results are discussed in terms of Manktelow and Over's argument that the standard abstract task and the permission schema version are actually different problems.

scholarly journals Maximal functions and transference for groups of operators

2000 ◽  
Vol 43 (1) ◽  
pp. 57-71 ◽  
Author(s):  
Gordon Blower
Keyword(s):  
Banach Spaces ◽  
Laplace Operator ◽  
Maximal Function ◽  
Maximal Functions ◽  
Abstract Version ◽  
The Laplace Operator

AbstractLet Δ be the Laplace operator on ℝd and 1 < δ < 2. Using transference methods we show that, for max {q, q/(q – 1)} < 4d/(2d + 1 – δ), the maximal function for the Schrödinger group is in Lq, for f ∈ Lq with Δδ/2f ∈Lq. We obtain a similar result for the Airy group exp it Δ3/2. An abstract version of these results is obtained for bounded C0-groups eitL on subspaces of Lp spaces. Certain results extend to maximal functions defined for functions with values in U M D Banach spaces.

A Proposed Model for the Collection and Curation of Slide Sets

2013 ◽  
Author(s):  
Alan Bilansky
Keyword(s):  
Intellectual Content ◽  
Abstract Version ◽  
Proposed Model

Slides sets, created in PowerPoint and by other means, should be collected and curated. The value of possible collections of slide sets from institutional and other contexts is discussed. This paper proposes a model: the essential intellectual content of a slide set can be identified as an abstract version of the information that survives translation of the slide set into PDF--text, formulae, important graphics. TEI offers a suitable XML vocabulary for this purpose. Alternate models and vocabularies are considered.

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