scholarly journals Computing the q-Numerical Range of Differential Operators

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Ahmed Muhammad ◽  
Faiza Abdullah Shareef
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Variational Methods ◽  
Schrödinger Operator ◽  
Orthonormal Basis ◽  
Differential Operators ◽  
Numerical Range ◽  
Stokes Operator ◽  
Schrodinger Operator ◽  
Operator Matrices

A linear operator on a Hilbert space may be approximated with finite matrices by choosing an orthonormal basis of thez Hilbert space. In this paper, we establish an approximation of the q-numerical range of bounded and unbounnded operator matrices by variational methods. Application to Schrödinger operator, Stokes operator, and Hain-Lüst operator is given.

scholarly journals Essential numerical ranges for linear operator pencils

2019 ◽  
Vol 40 (4) ◽  
pp. 2256-2308
Author(s):  
Sabine Bögli ◽  
Marco Marletta
Keyword(s):  
Linear Operator ◽  
Numerical Range ◽  
Projection Methods ◽  
Operator Pencil ◽  
Operator Matrices ◽  
New Concepts ◽  
Block Operator ◽  
Block Operator Matrices ◽  
Spectral Pollution ◽  
Operator Pencils

Abstract We introduce concepts of essential numerical range for the linear operator pencil $\lambda \mapsto A-\lambda B$. In contrast to the operator essential numerical range, the pencil essential numerical ranges are, in general, neither convex nor even connected. The new concepts allow us to describe the set of spectral pollution when approximating the operator pencil by projection and truncation methods. Moreover, by transforming the operator eigenvalue problem $Tx=\lambda x$ into the pencil problem $BTx=\lambda Bx$ for suitable choices of $B$, we can obtain nonconvex spectral enclosures for $T$ and, in the study of truncation and projection methods, confine spectral pollution to smaller sets than with hitherto known concepts. We apply the results to various block operator matrices. In particular, Theorem 4.12 presents substantial improvements over previously known results for Dirac operators while Theorem 4.5 excludes spectral pollution for a class of nonselfadjoint Schrödinger operators which has not been possible to treat with existing methods.

Discrete spectra criteria for singular differential operators with middle terms

1975 ◽  
Vol 77 (2) ◽  
pp. 337-347 ◽  
Author(s):  
Don B. Hinton ◽  
Roger T. Lewis
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Linear Operator ◽  
Differential Operators ◽  
Adjoint Operator ◽  
Continuous Functions ◽  
Adjoint Extension ◽  
Image Position ◽  
Discrete Spectra ◽  
Bounded Below

Let l be the differential operator of order 2n defined bywhere the coefficients are real continuous functions and pn > 0. The formally self-adjoint operator l determines a minimal closed symmetric linear operator L0 in the Hilbert space L2 (0, ∞) with domain dense in L2 (0, ∞) ((4), § 17). The operator L0 has a self-adjoint extension L which is not unique, but all such L have the same continuous spectrum ((4), § 19·4). We are concerned here with conditions on the pi which will imply that the spectrum of such an L is bounded below and discrete.

Completeness properties in L2 of the eigenfunctions of two semi-linear differential operators

1980 ◽  
Vol 88 (3) ◽  
pp. 451-468 ◽  
Author(s):  
L. E. Fraenkel
Keyword(s):  
Hilbert Space ◽  
Product Space ◽  
Orthonormal Basis ◽  
Differential Operators ◽  
Real Hilbert Space ◽  
The Real ◽  
Linear Differential Operators ◽  
Linear Problems ◽  
Linear Differential ◽  
Image Position

This paper concerns the boundary-value problemsin which λ is a real parameter, u is to be a real-valued function in C2[0, 1], and problem (I) is that with the minus sign. (The differential operators are called semi-linear because the non-linearity is only in undifferentiated terms.) If we linearize the equations (for ‘ small’ solutions u) by neglecting , there result the eigenvalues λ = n2π2 (with n = 1,2,…) and corresponding normalized eigenfunctionsand it is well known ((2), p. 186) that the sequence {en} is complete in that it is an orthonormal basis for the real Hilbert space L2(0, 1). We shall be concerned with possible extensions of this result to the non-linear problems (I) and (II), for which non-trivial solutions (λ, u) bifurcate from the trivial solution (λ, 0) at the points {n2π2,0) in the product space × L2(0, 1). (Here denotes the real line.)

Numerical Range and Convex Sets*

1974 ◽  
Vol 17 (2) ◽  
pp. 295-296 ◽  
Author(s):  
Fredric M. Pollack
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Convex Subset ◽  
Bounded Linear Operator ◽  
Convex Sets ◽  
Numerical Range ◽  
Bounded Subset ◽  
Bounded Linear ◽  
Empty Convex ◽  
Image Position

The numerical range W(T) of a bounded linear operator T on a Hilbert space H is defined byW(T) is always a convex subset of the plane [1] and clearly W(T) is bounded since it is contained in the ball of radius ‖T‖ about the origin. Which non-empty convex bounded subsets of the plane are the numerical range of an operator? The theorem we prove below shows that every non-empty convex bounded subset of the plane is W(T) for some T.

scholarly journals A result on hermitian operators

1989 ◽  
Vol 31 (1) ◽  
pp. 71-72
Author(s):  
J. E. Jamison ◽  
Pei-Kee Lin
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Numerical Range ◽  
Complex Banach Space ◽  
Bounded Linear ◽  
Image Position

Let X be a complex Banach space. For any bounded linear operator T on X, the (spatial) numerical range of T is denned as the setIf V(T) ⊆ R, then T is called hermitian. Vidav and Palmer (see Theorem 6 of [3, p. 78] proved that if the set {H + iK:H and K are hermitian} contains all operators, then X is a Hilbert space. It is natural to ask the following question.

The pseudo-spectrum of operator pencils

2019 ◽  
Vol 13 (05) ◽  
pp. 2050100
Author(s):  
A. Khellaf ◽  
H. Guebbai ◽  
S. Lemita ◽  
Z. Aissaoui
Keyword(s):  
Hilbert Space ◽  
Schrödinger Operator ◽  
Unbounded Operator ◽  
Schrodinger Operator ◽  
Bounded Operators ◽  
Numerical Example ◽  
Generalized Spectrum ◽  
Definition Of ◽  
Operator Pencils

In this paper, we propose a new definition of the pseudo-spectrum for operator pencils, which is associated with two bounded operators [Formula: see text] and [Formula: see text] defined in a Hilbert space. Unlike other definitions available in the literature, we prove, under specific conditions on [Formula: see text] and [Formula: see text], that the pseudo-spectrum for operator pencils is equal to an [Formula: see text]-neighborhood of the generalized spectrum. Moreover, we demonstrate how to use this concept to redefine the pseudo-spectrum of an unbounded operator. We illustrate its usefulness through a numerical example dealing with the Schrödinger operator.

scholarly journals Restricted interpolation by meromorphic inner functions

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Alexei Poltoratski ◽  
Rishika Rupam
Keyword(s):  
Half Plane ◽  
Schrödinger Operator ◽  
Differential Operators ◽  
Schrodinger Operator ◽  
Inner Functions ◽  
Spectral Problems ◽  
Upper Half Plane

AbstractMeromorphic Inner Functions (MIFs) on the upper half plane play an important role in applications to spectral problems for differential operators. In this paper, we survey some recent results concerning function theoretic properties of MIFs and show their connections with spectral problems for the Schrödinger operator.

Boundary Points of the Numerical Range of an Operator

1975 ◽  
Vol 17 (5) ◽  
pp. 689-692 ◽  
Author(s):  
C.-S. Lin
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Numerical Range ◽  
Bounded Linear ◽  
Boundary Points

The purpose of this note is to investigate boundary points of the numerical range of an operator in terms of inner and outer center points. Some applications on commutators are given.Throughout this note, an operator will always mean a bounded linear operator on a Hilbert space X.

scholarly journals Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator

Entropy ◽  
2020 ◽  
Vol 22 (7) ◽  
pp. 722
Author(s):  
Stefan Klus ◽  
Feliks Nüske ◽  
Boumediene Hamzi
Keyword(s):  
Schrödinger Operator ◽  
Reproducing Kernel ◽  
Differential Operators ◽  
Eigenvalue Problems ◽  
Schrodinger Operator ◽  
Koopman Operator ◽  
Reduction Techniques ◽  
Kernel Hilbert Spaces ◽  
Quantum Mechanical Systems ◽  
Matrix Eigenvalue Problems

Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems.

Relations Between Generalized Growth Conditions and Several Classes of Convexoid Operators

1977 ◽  
Vol 29 (5) ◽  
pp. 1010-1030 ◽  
Author(s):  
Takayuki Furuta
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Numerical Range ◽  
Growth Conditions ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Image Position

In this paper we shall discuss some classes of bounded linear operators on a complex Hilbert space. If T is a bounded linear operator T acting on the complex Hilbert space H, then the following two inequalities always hold:where σ(T) indicates the spectrum of T, W(T) denotes the numerical range of T defined by W(T) = {(Tx, x) : ||x|| = 1 and x ∊ H} and means the closure of W(T) respectively.

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