scholarly journals Tunneling for the Robin Laplacian in smooth planar domains

2016 ◽  
Vol 19 (01) ◽  
pp. 1650030 ◽  
Author(s):  
Bernard Helffer ◽  
Ayman Kachmar ◽  
Nicolas Raymond
Keyword(s):  
Asymptotic Formula ◽  
Planar Domain ◽  
Independent Interest ◽  
Tunneling Effect ◽  
Rigorous Derivation ◽  
Weyl Formula ◽  
Planar Domains ◽  
Robin Laplacian ◽  
Axis Of Symmetry ◽  
Semiclassical Tunneling

We study the low-lying eigenvalues of the semiclassical Robin Laplacian in a smooth planar domain with bounded boundary which is symmetric with respect to an axis. In the case when the curvature of the boundary of the domain attains its maximum at exactly two points away from the axis of symmetry, we establish an explicit asymptotic formula for the splitting of the first two eigenvalues. This is a rigorous derivation of the semiclassical tunneling effect induced by the domain’s geometry. Our approach is close to the Born–Oppenheimer one and yields, as a byproduct, a Weyl formula of independent interest.

scholarly journals Toeplitz Operators on the Bergman Space of Planar Domains with Essentially Radial Symbols

2011 ◽  
Vol 2011 ◽  
pp. 1-26 ◽  
Author(s):  
Roberto C. Raimondo
Keyword(s):  
Bergman Space ◽  
Toeplitz Operators ◽  
Boundary Behavior ◽  
Berezin Transform ◽  
Planar Domain ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Planar Domains ◽  
Necessary And Sufficient ◽  
The Relationship

We study the problem of the boundedness and compactness of when and is a planar domain. We find a necessary and sufficient condition while imposing a condition that generalizes the notion of radial symbol on the disk. We also analyze the relationship between the boundary behavior of the Berezin transform and the compactness of

scholarly journals Rational points on linear slices of diagonal hypersurfaces

2015 ◽  
Vol 218 ◽  
pp. 51-100
Author(s):  
Jörg Brüdern ◽  
Olivier Robert
Keyword(s):  
Asymptotic Formula ◽  
Rational Points ◽  
Independent Interest ◽  
The Way ◽  
Hardy Littlewood Method

AbstractAn asymptotic formula is obtained for the number of rational points of bounded height on the class of varieties described in the title line. The formula is proved via the Hardy-Littlewood method, and along the way we establish two new results on Weyl sums that are of some independent interest.

scholarly journals On the distribution of harmonic measure on simply connected planar domains

2003 ◽  
Vol 75 (2) ◽  
pp. 145-152 ◽  
Author(s):  
Dimitrios Betsakos ◽  
Alexander Yu. Solynin
Keyword(s):  
Harmonic Measure ◽  
Planar Domain ◽  
Planar Domains ◽  
Simply Connected

AbstractFor a simply connected planar domain D with 0 ∈ D and dist(0, ∂D) = 1, let hD(r) be the harmonic measure of ∂ D ∩{|Z| ≤ r} evaluated at 0. The function hD(r) is the distribution of harmonic measure. It has been studied by B. L. Walden and L. A. Ward. We continue their study and answer some questions raised by them by constructing domains with pre-specified distribution.

From Steklov to Neumann and Beyond, via Robin: The Szegő Way

2019 ◽  
Vol 72 (4) ◽  
pp. 1024-1043 ◽  
Author(s):  
Pedro Freitas ◽  
Richard S. Laugesen
Keyword(s):  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Planar Domains ◽  
Steklov Eigenvalue ◽  
Robin Laplacian ◽  
Fixed Area ◽  
Neumann Laplacian ◽  
Simply Connected ◽  
Second Eigenvalue ◽  
Simply Connected Domains

AbstractThe second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among simply-connected planar domains of fixed area when the Robin parameter is scaled by perimeter in the form $\unicode[STIX]{x1D6FC}/L(\unicode[STIX]{x1D6FA})$, and $\unicode[STIX]{x1D6FC}$ lies between $-2\unicode[STIX]{x1D70B}$ and $2\unicode[STIX]{x1D70B}$. Corollaries include Szegő’s sharp upper bound on the second eigenvalue of the Neumann Laplacian under area normalization, and Weinstock’s inequality for the first nonzero Steklov eigenvalue for simply-connected domains of given perimeter.The first Robin eigenvalue is maximal, under the same conditions, for the degenerate rectangle. When area normalization on the domain is changed to conformal mapping normalization and the Robin parameter is positive, the maximiser of the first eigenvalue changes back to the disk.

scholarly journals Rational points on linear slices of diagonal hypersurfaces

2015 ◽  
Vol 218 ◽  
pp. 51-100 ◽  
Author(s):  
Jörg Brüdern ◽  
Olivier Robert
Keyword(s):  
Asymptotic Formula ◽  
Rational Points ◽  
Independent Interest ◽  
The Way ◽  
Hardy Littlewood Method

AbstractAn asymptotic formula is obtained for the number of rational points of bounded height on the class of varieties described in the title line. The formula is proved via the Hardy-Littlewood method, and along the way we establish two new results on Weyl sums that are of some independent interest.

Heavy Tails of Discounted Aggregate Claims in the Continuous-Time Renewal Model

2007 ◽  
Vol 44 (02) ◽  
pp. 285-294 ◽  
Author(s):  
Qihe Tang
Keyword(s):  
Asymptotic Formula ◽  
Continuous Time ◽  
Heavy Tails ◽  
Tail Behavior ◽  
Renewal Model ◽  
Discounted Aggregate Claims ◽  
Aggregate Claims ◽  
Tail Asymptotic

We study the tail behavior of discounted aggregate claims in a continuous-time renewal model. For the case of Pareto-type claims, we establish a tail asymptotic formula, which holds uniformly in time.

The Simplest Patterns of Interpenetrated Honeycomb Layers – Counter examples to the Minimal Transitivity Principle?

2018 ◽  
Author(s):  
Igor Baburin
Keyword(s):  
Crystal Engineering ◽  
Structural Chemistry ◽  
Broad Context ◽  
Rigorous Derivation ◽  
New Approach

The paper calls attention to the most symmetric interpenetration patterns of honeycomb layers. To the best of my knowledge, such patterns remained unknown so far. In my contribution a rigorous derivation of such patterns is given that makes use of a new approach to interpenetrating nets. The results are presented in a broad context of structural chemistry and crystal engineering.

A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

2020 ◽  
Vol 17 (2) ◽  
pp. 256-277
Author(s):  
Ol'ga Veselovska ◽  
Veronika Dostoina
Keyword(s):  
Complex Plane ◽  
Complex Variable ◽  
Analytic Functions ◽  
Combinatorial Identities ◽  
Independent Interest ◽  
Closed Curves ◽  
In Series ◽  
Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

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