Some theorems on perturbation theory. IV

1951 ◽  
Vol 210 (1100) ◽  
pp. 30-47 ◽  
Keyword(s):  
Perturbation Theory ◽  
Continuous Spectrum ◽  
Discrete Spectrum ◽  
Real Axis ◽  
General Analysis ◽  
The Real ◽  
Perturbation Problem ◽  
Perturbed Equation ◽  
Unperturbed Equation

A general analysis is given of the perturbation problem in which the perturbed equation has a continuous spectrum, while the unperturbed equation has a discrete spectrum. It is shown that in a certain class of cases the function which determines the spectrum has, in the perturbed case, a pole just below the real axis, so that instead of a perturbed eigenvalue there is a point in the neighbourhood of which the perturbed spectrum is highly concentrated.

scholarly journals Plummeting and blinking eigenvalues of the Robin Laplacian in a cuspidal domain

2019 ◽  
Vol 150 (6) ◽  
pp. 2871-2893 ◽  
Author(s):  
Sergei A. Nazarov ◽  
Nicolas Popoff ◽  
Jari Taskinen
Keyword(s):  
Dimension Reduction ◽  
Discrete Spectrum ◽  
Spectral Theory ◽  
Real Axis ◽  
Complex Plane ◽  
Lipschitz Domain ◽  
High Rate ◽  
The Real ◽  
Robin Laplacian ◽  
Asymptotic Forms

We consider the Robin Laplacian in the domains Ω and Ωε, ε > 0, with sharp and blunted cusps, respectively. Assuming that the Robin coefficient a is large enough, the spectrum of the problem in Ω is known to be residual and to cover the whole complex plane, but on the contrary, the spectrum in the Lipschitz domain Ωε is discrete. However, our results reveal the strange behaviour of the discrete spectrum as the blunting parameter ε tends to 0: we construct asymptotic forms of the eigenvalues and detect families of ‘hardly movable’ and ‘plummeting’ ones. The first type of the eigenvalues do not leave a small neighbourhood of a point for any small ε > 0 while the second ones move at a high rate O(| ln ε|) downwards along the real axis ℝ to −∞. At the same time, any point λ ∈ ℝ is a ‘blinking eigenvalue’, i.e., it belongs to the spectrum of the problem in Ωε almost periodically in the | ln ε|-scale. Besides standard spectral theory, we use the techniques of dimension reduction and self-adjoint extensions to obtain these results.

On Graeffe's Method for Complex Roots of Algebraic Equations

1924 ◽  
Vol 22 (2) ◽  
pp. 83-87 ◽  
Author(s):  
S. Brodetsky ◽  
G. Smeal
Keyword(s):  
Nineteenth Century ◽  
Real Axis ◽  
Practical Method ◽  
Algebraic Equations ◽  
Argand Diagram ◽  
Considerable Difficulty ◽  
The Real ◽  
Real Roots ◽  
Bipolar Coordinates ◽  
Complex Roots

The only really useful practical method for solving numerical algebraic equations of higher orders, possessing complex roots, is that devised by C. H. Graeffe early in the nineteenth century. When an equation with real coefficients has only one or two pairs of complex roots, the Graeffe process leads to the evaluation of these roots without great labour. If, however, the equation has a number of pairs of complex roots there is considerable difficulty in completing the solution: the moduli of the roots are found easily, but the evaluation of the arguments often leads to long and wearisome calculations. The best method that has yet been suggested for overcoming this difficulty is that by C. Runge (Praxis der Gleichungen, Sammlung Schubert). It consists in making a change in the origin of the Argand diagram by shifting it to some other point on the real axis of the original Argand plane. The new moduli and the old moduli of the complex roots can then be used as bipolar coordinates for deducing the complex roots completely: this also checks the real roots.

Interaction between a nanocrack with surface elasticity and a screw dislocation

2016 ◽  
Vol 22 (2) ◽  
pp. 131-143 ◽  
Author(s):  
Xu Wang ◽  
Hui Fan
Keyword(s):  
Screw Dislocation ◽  
Real Axis ◽  
Analytical Study ◽  
Logarithmic Singularity ◽  
Surface Elasticity ◽  
Spontaneous Generation ◽  
The Real ◽  
Crack Tips ◽  
Interaction Problem ◽  
The Continuum

In the present analytical study, we consider the problem of a nanocrack with surface elasticity interacting with a screw dislocation. The surface elasticity is incorporated by using the continuum-based surface/interface model of Gurtin and Murdoch. By considering both distributed screw dislocations and line forces on the crack, we reduce the interaction problem to two decoupled first-order Cauchy singular integro-differential equations which can be numerically solved by the collocation method. The analysis indicates that if the dislocation is on the real axis where the crack is located, the stresses at the crack tips only exhibit the weak logarithmic singularity; if the dislocation is not on the real axis, however, the stresses exhibit both the weak logarithmic and the strong square-root singularities. Our result suggests that the surface effects of the crack will make the fracture more ductile. The criterion for the spontaneous generation of dislocations at the crack tip is proposed.

Successive coefficients of close-to-convex functions

2020 ◽  
Vol 32 (5) ◽  
pp. 1131-1141 ◽  
Author(s):  
Paweł Zaprawa
Keyword(s):  
Real Axis ◽  
Convex Functions ◽  
Positive Direction ◽  
The Real ◽  
Coefficient Problems ◽  
The Difference

AbstractIn this paper we discuss coefficient problems for functions in the class {{\mathcal{C}}_{0}(k)}. This family is a subset of {{\mathcal{C}}}, the class of close-to-convex functions, consisting of functions which are convex in the positive direction of the real axis. Our main aim is to find some bounds of the difference of successive coefficients depending on the fixed second coefficient. Under this assumption we also estimate {|a_{n+1}|-|a_{n}|} and {|a_{n}|}. Moreover, it is proved that {\operatorname{Re}\{a_{n}\}\geq 0} for all {f\in{\mathcal{C}}_{0}(k)}.

scholarly journals Zeros of Dirichlet L-functions near the Real Axis and Chebyshev's Bias

2001 ◽  
Vol 87 (1) ◽  
pp. 54-76 ◽  
Author(s):  
Carter Bays ◽  
Kevin Ford ◽  
Richard H Hudson ◽  
Michael Rubinstein
Keyword(s):  
Real Axis ◽  
The Real

scholarly journals Pseudo-hermitian random matrix theory: a review

2021 ◽  
Vol 2038 (1) ◽  
pp. 012009
Author(s):  
Joshua Feinberg ◽  
Roman Riser
Keyword(s):  
Real Axis ◽  
Complex Plane ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Complex Conjugate ◽  
Indefinite Metric ◽  
The Real ◽  
New Type ◽  
Diagrammatic Method

Abstract We review our recent results on pseudo-hermitian random matrix theory which were hitherto presented in various conferences and talks. (Detailed accounts of our work will appear soon in separate publications.) Following an introduction of this new type of random matrices, we focus on two specific models of matrices which are pseudo-hermitian with respect to a given indefinite metric B. Eigenvalues of pseudo-hermitian matrices are either real, or come in complex-conjugate pairs. The diagrammatic method is applied to deriving explicit analytical expressions for the density of eigenvalues in the complex plane and on the real axis, in the large-N, planar limit. In one of the models we discuss, the metric B depends on a certain real parameter t. As t varies, the model exhibits various ‘phase transitions’ associated with eigenvalues flowing from the complex plane onto the real axis, causing disjoint eigenvalue support intervals to merge. Our analytical results agree well with presented numerical simulations.

scholarly journals Uniqueness theorem for Fourier transformable measures on LCA groups

2020 ◽  
Vol 54 (2) ◽  
pp. 211-219
Author(s):  
S.Yu. Favorov
Keyword(s):  
Uniqueness Theorem ◽  
Real Axis ◽  
Almost Periodic ◽  
Periodic Measure ◽  
Locally Compact ◽  
The Real ◽  
Approach Infinity ◽  
The Masses ◽  
Lca Groups

We show that if points of supports of two discrete ”not very thick” Fourier transformable measures on locally compact abelian (LCA) groups tend to one another at infinity and the same is true for the masses at these points, then these measures coincide. The result is valid for discrete almost periodic measures on LCA groups too. Also, we show that the result is false for some discrete ”thick” measures. To do this, we construct a discrete almost periodic measure on the real axis, whose masses at the points of support tend to zero as these points approach infinity.

scholarly journals RESONANCES ON HEDGEHOG MANIFOLDS

2013 ◽  
Vol 53 (5) ◽  
pp. 416-426 ◽  
Author(s):  
Pavel Exner ◽  
Jiří Lipovský
Keyword(s):  
Real Axis ◽  
Quantum Particle ◽  
Single Point ◽  
Three Dimensional ◽  
High Energy ◽  
Quantum Graphs ◽  
The Real ◽  
Momentum Plane ◽  
Embedded Eigenvalues ◽  
Aharonov Bohm

We discuss resonances for a nonrelativistic and spinless quantum particle confined to a two- or three-dimensional Riemannian manifold to which a finite number of semiinfinite leads is attached. Resolvent and scattering resonances are shown to coincide in this situation. Next we consider the resonances together with embedded eigenvalues and ask about the high-energy asymptotics of such a family. For the case when all the halflines are attached at a single point we prove that all resonances are in the momentum plane confined to a strip parallel to the real axis, in contrast to the analogous asymptotics in some metric quantum graphs; we illustrate this on several simple examples. On the other hand, the resonance behaviour can be influenced by a magnetic field. We provide an example of such a ‘hedgehog’ manifold at which a suitable Aharonov-Bohm flux leads to absence of any true resonance, i.e. that corresponding to a pole outside the real axis.

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