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Computability of the Spectrum of Self-Adjoint Operators and the Computable Operational Calculus

2008
*
Electronical Notes in Theoretical Computer Science
*

Therefore, it is a natural question whether

doi:10.1016/j.entcs.2008.03.026
fatcat:4zbhrxbk55fq3nkqyx4b3dph7i
*the**spectrum**of*a*self*-*adjoint**operator*and its eigenvalues can be*computed*from a description*of**the**operator*. ... Additionally, we show that*the*eigenvalues*of**self*-*adjoint**operators*can be*computed*in*the*sense that we can*compute*a list*of*indices such that those elements*of**the*already*computed*dense subset*of*... ¿From*the*perspective*of**computable*analysis a natural question is whether*the**spectrum*and*the*eigenvalues*of*a*self*-*adjoint**operator*can be*computed*in some natural sense. ...##
###
Eigenvalue problem meets Sierpinski triangle: computing the spectrum of a non-self-adjoint random operator

2011
*
Operators and Matrices
*

*The*purpose

*of*this paper is to prove that

*the*

*spectrum*

*of*

*the*non-

*self*-

*adjoint*one-particle Hamiltonian proposed by J. Feinberg and A. Zee (Phys. Rev. E 59 (1999), 6433--6443) has interior points. ... We do this by first recalling that

*the*

*spectrum*

*of*this random

*operator*is

*the*union

*of*

*the*set

*of*ℓ^∞ eigenvalues

*of*all infinite matrices with

*the*same structure. ... We are grateful to Estelle Basor from

*the*American Institute

*of*Mathematics for drawing our attention to this beautiful

*operator*class. Moreover ...

##
###
A strategy for detecting extreme eigenvalues bounding gaps in the discrete spectrum of self-adjoint operators

2007
*
Computers and Mathematics with Applications
*

For a

doi:10.1016/j.camwa.2005.11.040
fatcat:6n3253rofrgf7kezssrtltbbxa
*self*-*adjoint*linear*operator*with a discrete*spectrum*or a Hermitian matrix,*the*"extreme" eigenvalues define*the*boundaries*of*clusters in*the**spectrum**of*real eigenvalues. ...*The*overall*computational*cost is quadratic in*the*size*of*a dense matrix; linear in*the*size*of*a sparse matrix. ...*The*strategy applies in a straightforward way to*the*more general problem*of*finding*the*extreme eigenvalues*of**self*-*adjoint*linear*operators*. ...##
###
Remarks on A-skew-adjoint, A-almost Similarity Equivalence and Other Operators in Hilbert Space

2017
*
Pure and Applied Mathematics Journal
*

In this paper, notions

doi:10.11648/j.pamj.20170603.12
fatcat:ndgbwfahynfd3g2aztcggh75se
*of*A-almost similarity and*the*Lie algebra*of*A-skew-*adjoint**operators*in Hilbert space are introduced. In this context, A is a*self*-*adjoint*and an invertible*operator*. ... Finally A-skew*adjoint**operators*are characterized and*the*relationship between A-*self*-*adjoint*and A-skew*adjoint**operators*is given. ... In addition*the*intersection*of**the*class*of**self*-*adjoint*and unitary*operators*yields a symmetry, i.e. { Just like -*self*-*adjoint**operators*,*the**spectrum**of*anskew-*adjoint**operator*− and*the**adjoint**operator*...##
###
Numerical approximation of the spectrum of self-adjoint continuously invertible operators
[article]

2021
*
arXiv
*
pre-print

More precisely, we consider two bounded, coercive, and

arXiv:2103.00849v1
fatcat:ca7uxot3sbglrj2deodzxiw3n4
*self*-*adjoint**operators*A, B: V↦ V^#, where V^# denotes*the*dual*of*V, and investigate*the*conditions under which*the*whole*spectrum**of*B^-1A:V↦ V ... This paper deals with*the*generalized*spectrum**of*continuously invertible linear*operators*defined on infinite dimensional Hilbert spaces. ...*The*authors thank David Krejčiřík, Josef Málek and Ivana Pultarová for stimulating discussions during*the*work on this paper. ...##
###
The eigenvalues of an effectively determined self-adjoint operator are computable, but the sequence of eigenvalues is not

1987
*
Advances in Mathematics
*

This paper explores

doi:10.1016/0001-8708(87)90062-4
fatcat:ycdsnf4infccbjn6ceyswdjjru
*the**computability*structure*of**the*eigenvalues and*spectrum*for bounded and unbounded linear*operators*on a Hilbert space. ... Let T: H -+ H be a (bounded or unbounded)*self*-*adjoint**operator*on a Hilbert space H, and let T be effectively determined. ... /*spectrum*)*of*an effectively determined bounded*self*-*adjoint**operator*. ...##
###
Exponential Function of a bounded Linear Operator on a Hilbert Space

2014
*
Baghdad Science Journal
*

In this paper, we introduce an exponential

doi:10.21123/bsj.11.3.1267-1273
fatcat:3vtzabepkbc6zklpx6zbe7arlu
*of*an*operator*defined on a Hilbert space H, and we study its properties and find some*of*properties*of*T inherited to exponential*operator*, so we study*the**spectrum*...*of*exponential*operator*e^T according to*the**operator*T. ... Hence if H is a finite dimensional Hilbert space then ( ) if and only if is eigenvalue ofIn*the*following example we are going to*compute**the**spectrum**of**the*some linear*operators*. ...##
###
Conformal deformations of immersed discs in R^3 and elliptic boundary value problems
[article]

2013
*
arXiv
*
pre-print

Boundary value problems for

arXiv:1301.3752v1
fatcat:l26ksljc5jbnhchv6kc6swm4be
*operators**of*Dirac type arise naturally in connection with*the*conformal geometry*of*surfaces immersed in Euclidean 3--space. ... Here we investigate under which conditions these boundary value problems are elliptic and*self*--*adjoint*. ... If in addition*the**operator*is formally*self*-*adjoint*, this extension is*self*-*adjoint*. As a consequence,*the**spectrum*is then real and discrete and there exists a basis*of*eigenvectors. ...##
###
Spectral Properties of Random Non-self-adjoint Matrices and Operators
[article]

2000
*
arXiv
*
pre-print

Our results imply that

arXiv:math/0002159v1
fatcat:fszmx32urjg3lkmdlaxx4au7ea
*the**spectrum**of**the*non-*self*-*adjoint*Anderson model changes suddenly as one passes to*the*infinite volume limit. ... We also describe a stochastic family*of*bounded*operators*in infinite dimensions for almost all*of*which*the*eigenvectors generate a dense linear subspace, but*the*eigenvalues do not determine*the**spectrum*... Classification*of**the**Spectrum**The*classification*of**the**spectrum**of*non-*self*-*adjoint**operators*is in a primitive state by comparison with that*of**self*-*adjoint**operators*. ...##
###
Finite deficiency indices and uniform remainder in Weyl's law
[article]

2010
*
arXiv
*
pre-print

We give a proof that in settings where Von Neumann deficiency indices are finite

arXiv:1001.1795v2
fatcat:kszsuxpdxjbh7jejhewxqgarqm
*the*spectral counting functions*of*two different*self*-*adjoint*extensions*of**the*same symmetric*operator*differ by a uniformly ... bounded term (see also Birman-Solomjak's 'Spectral Theory*of**Self*-*adjoint**operators*in Hilbert Space') >. ...*The**spectrum**of*a*self*-*adjoint**operator*with compact resolvent consists in eigenvalues*of*finite multiplicities, that form a discrete set in R. ...##
###
Page 7 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 67, Issue 1
[page]

1961
*
American Mathematical Society. Bulletin of the American Mathematical Society
*

*The*problems are then classically

*self*-

*adjoint*.

*The*

*spectrum*

*of*

*the*unperturbed equa- tion is continuous over 0<\< © and there is one isolated negative eigenvalue. ... 1961] NON-

*SELF*-

*ADJOINT*PROBLEMS IN MATHEMATICAL PHYSICS 7 problems involve essentially non-

*self*-

*adjoint*differential equations since often complex indexes

*of*refraction occur. ...

##
###
Error Estimation for Eigenvalues of Unbounded Linear Operators and an Application to Energy Levels in Graphene Quantum Dots

2017
*
Numerical Functional Analysis and Optimization
*

It is also important to have good estimates

doi:10.1080/01630563.2017.1279176
fatcat:tztkhxx7prfm3oeh5tqzg7yl5a
*of**the*error in*the**computed*eigenvalues. ... In this work we use spline interpolation to construct approximate eigenfunctions*of*a linear*operator*by using*the*corresponding eigenvectors*of*a discretized approximation*of**the**operator*. ... Note that*the**self*-*adjoint*extension for an essentially*self*-*adjoint**operator*is unique. ...##
###
Adjoint zero-modes as a tool to understand the Yang-Mills vacuum
[article]

2007
*
arXiv
*
pre-print

*The*use

*of*

*adjoint*(quasi) zero-modes

*of*

*the*Dirac

*operator*to probe

*the*Yangs-Mills vacuum has been recently advocated by Gonzalez-Arroyo and Kirchner. ... In

*the*lattice implementation

*of*this idea, we show how

*the*results improve considerably if

*the*overlap

*operator*is used instead

*of*

*the*Wilson-Dirac one. ... Also acknowledged is

*the*use

*of*

*the*MareNostrum supercomputer at

*the*BSC-CNS and

*the*IFT-UAM/CSIC

*computation*cluster. ...

##
###
Self-adjoint extensions of operators and the teaching of quantum mechanics

2001
*
American Journal of Physics
*

For

doi:10.1119/1.1328351
fatcat:rohgm4tgffhzdj7az2b7gw6eay
*the*example*of**the*infinitely deep well potential, we point out some paradoxes which are solved by a careful analysis*of*what is a truly*self*-*adjoint**operator*. ... We then describe*the**self*-*adjoint*extensions and their spectra for*the*momentum and*the*Hamiltonian*operators*in different physical situations. ...*Compute*its*adjoint*(P † , D(P † )) and determine*the*deficiency indices*of*P † . 3. When they do exist, describe*the*domains*of*all*the**self*-*adjoint*extensions. ...##
###
Computing the Enclosures Eigenvalues Using the Quadratic Method

2019
*
Applied Mathematics
*

In this article, We

doi:10.4236/am.2019.104016
fatcat:zm2cyzwg5ngyfbzm4gxuxq745i
*compute**the*enclosures eigenvalues (upper and lower bounds) using*the*quadratic method. ...*The*Schrodinger*operator*(A) (harmonic and anharmonic oscillator model) has used as an example. We study a new technique to get more accurate bounds. ... This paper shows how to*compute*enclosures*of**the*eigenvalues*of**self*-*adjoint**operators*by*the*Quadratic method. ...
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