scholarly journals Global Persistence of the Unit Eigenvectors of Perturbed Eigenvalue Problems in Hilbert Spaces: The Odd Multiplicity Case

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 561
Author(s):  
Pierluigi Benevieri ◽  
Alessandro Calamai ◽  
Massimo Furi ◽  
Maria Patrizia Pera
Keyword(s):  
Hilbert Spaces ◽  
Eigenvalue Problems ◽  
Eigenvalues And Eigenvectors ◽  
Index Zero ◽  
Global Continuation ◽  
Banach Manifolds ◽  
Unperturbed Problem ◽  
Fredholm Maps

We study the persistence of eigenvalues and eigenvectors of perturbed eigenvalue problems in Hilbert spaces. We assume that the unperturbed problem has a nontrivial kernel of odd dimension and we prove a Rabinowitz-type global continuation result. The approach is topological, based on a notion of degree for oriented Fredholm maps of index zero between real differentiable Banach manifolds.

On the product formula for the oriented degree for Fredholm maps of index zero between Banach manifolds

2002 ◽  
Vol 48 (6) ◽  
pp. 853-867 ◽  
Author(s):  
Pierluigi Benevieri ◽  
Massimo Furi ◽  
Maria Patrizia Pera
Keyword(s):  
Product Formula ◽  
Index Zero ◽  
Banach Manifolds ◽  
Fredholm Maps

scholarly journals Perturbation results involving the 1-Laplace operator

2019 ◽  
Vol 12 (3) ◽  
pp. 277-302 ◽  
Author(s):  
Samuel Littig ◽  
Friedemann Schuricht
Keyword(s):  
Critical Point ◽  
Laplace Operator ◽  
Critical Point Theory ◽  
Eigenvalue Problems ◽  
Point Theory ◽  
Nonsmooth Critical Point Theory ◽  
Unperturbed Problem ◽  
Nonsmooth Critical Point

AbstractWe consider perturbed eigenvalue problems of the 1-Laplace operator and verify the existence of a sequence of solutions. It is shown that the eigenvalues of the perturbed problem converge to the corresponding eigenvalue of the unperturbed problem as the perturbation becomes small. The results rely on nonsmooth critical point theory based on the weak slope.

scholarly journals GLOBAL BRANCH OF SOLUTIONS FOR NON-LINEAR SCHRÖDINGER EQUATIONS WITH DEEPENING POTENTIAL WELL

2006 ◽  
Vol 92 (3) ◽  
pp. 655-681 ◽  
Author(s):  
C. A. STUART ◽  
HUAN-SONG ZHOU
Keyword(s):  
Potential Well ◽  
Index Zero ◽  
Topological Methods ◽  
Open Set ◽  
Connected Sets ◽  
Non Linear ◽  
Asymptotic Linearization ◽  
Positive Eigenfunction ◽  
Fredholm Maps ◽  
Global Branch

We consider the stationary non-linear Schrödinger equation\begin{equation*}\Delta u + \{1 + \lambda g(x)\} u = f(u)\mbox{with}u \in H^{1} (\mathbb{R}^{N}), u \not\equiv 0,\end{equation*} where $\lambda >0$ and the functions $f$ and $g$ are such that\begin{equation*} \lim_{s \rightarrow 0}\frac{f(s)}{s} = 0 \mbox{and} 1 < \alpha + 1 = \lim _{|s| \rightarrow \infty}\frac{f(s)}{s} < \infty\end{equation*} and \begin{equation*} g(x)\equiv 0 \mbox{on} \bar{\Omega}, g(x)\in (0, 1] \mbox{on} {\mathbb{R}^{N}} \setminus {\overline{\Omega}} \mbox{and} \lim_{|x| \rightarrow + \infty} g(x) = 1 \end{equation*} for some bounded open set $\Omega \in \mathbb{R}^{N}$. We use topological methods to establish the existence of two connected sets $\mathcal{D}^{\pm}$ of positive/negative solutions in $\mathbb{R} \times W^{2, p} (\mathbb{R}^{N})$ where $p \in [2, \infty) \cap (\frac{N}{2},\infty)$ that cover the interval $(\alpha,\Lambda(\alpha))$ in the sense that \begin{align*} P \mathcal{D}^{\pm} & = (\alpha, \Lambda(\alpha)) \text{where}P(\lambda, u) = \lambda \text{and furthermore,} \\ \lim_{\lambda \rightarrow \Lambda(\alpha)-}\left\Vert u_{\lambda} \right\Vert _{L^{\infty} (\mathbb{R}^{N})} & = \lim_{\lambda \rightarrow \Lambda (\alpha )-} \left\Vert u_{\lambda} \right\Vert _{W^{2, p}(\mathbb{R}^{N})} = \infty \text{ for }(\lambda, u_{\lambda}) \in \mathcal{D}^{\pm}. \end{align*} The number $\Lambda(\alpha)$ is characterized as the unique value of $\lambda$ in the interval $(\alpha, \infty)$ for which the asymptotic linearization has a positive eigenfunction. Our work uses a degree for Fredholm maps of index zero.

New scheme for large-scale eigenvalue problems of the RPA-type matrix equation

1992 ◽  
Vol 70 (2) ◽  
pp. 296-300 ◽  
Author(s):  
Susumu Narita ◽  
Tai-ichi Shibuya
Keyword(s):  
Eigenvalue Problem ◽  
Convergence Rate ◽  
Matrix Equation ◽  
Large Scale ◽  
Eigenvalue Problems ◽  
Type Equation ◽  
Fast Convergence ◽  
New Method ◽  
Eigenvalues And Eigenvectors ◽  
Numerical Tests

A new method is proposed for obtaining a few eigenvalues and eigenvectors of a large-scale RPA-type equation. Some numerical tests are carried out to study the convergence behaviors of this method. It is found that the convergence rate is very fast and quite satisfactory. It depends strongly on the way of estimating the deviation vectors. Our proposed scheme gives a better estimation for the deviation vectors than Davidson's scheme. This scheme is applicable to the eigenvalue problems of nondiagonally dominant matrices as well. Keywords: large-scale eigenvalue problem, RPA-type equation, fast convergence.

scholarly journals Global Persistence of the Unit Eigenvectors of Perturbed Eigenvalue Problems in Hilbert Spaces

2020 ◽  
Vol 39 (4) ◽  
pp. 475-497 ◽  
Author(s):  
Pierluigi Benevieri ◽  
Alessandro Calamai ◽  
Massimo Furi ◽  
Maria Patrizia Pera
Keyword(s):  
Hilbert Spaces ◽  
Eigenvalue Problems

On the eigenvalue problem Tu -λ Su =0 with unbounded and nonsymetric operators T and S

1968 ◽  
Vol 262 (1130) ◽  
pp. 413-458 ◽  
Keyword(s):  
Approximate Solution ◽  
Iterative Method ◽  
Eigenvalue Problems ◽  
Generalized Method Of Moments ◽  
Elastic Stability ◽  
Theoretical Problem ◽  
Bounded Operators ◽  
Eigenvalues And Eigenvectors ◽  
Complete Section ◽  
Symmetric Operators

In this paper we study both the theoretical problem of the existence and the practical problem of the approximate calculation of eigenvalues and eigenvectors of (i) = 0, where T and S are some linear (in general unbounded and nonhermitian) operators in a Hilbert space. After a short discussion of a class of K -symmetric operators, in section 2 the author proves the existence of eigenvalues and eigenvectors of (i) under various conditions on T and S and investigates conditions under which the set of eigenvectors of (i) is complete. Section 3 indicates briefly the applicability and the unifying property of the generalized method of moments to the approximate solution of (i). Section 4 presents and thoroughly studies a very general new iterative method for the approximate solution of (i). The advantage of this method is that it does not require the practically inconvenient preliminary reduction of (i) to an equivalent problem with bounded operators and that under certain rather general conditions the convergence is mono tonic. Furthermore, by specializing operators and parameters, our iterative method contains as a special case, almost every known iterative method for the calculation of eigenvalues (mostly proved previously only for symmetric matrices and bounded operators). Finally the applicability and the numerical effectiveness of the iterative method is illustrated by calculating the smallest eigenvalue for a selfadjoint and nonselfadjoint eigenvalue problems arising in the problems of elastic stability.

Sign in / Sign up

Export Citation Format

Share Document