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

Ruth Dillhage
2008 Electronical Notes in Theoretical Computer Science  
Therefore, it is a natural question whether 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.  ... 
doi:10.1016/j.entcs.2008.03.026 fatcat:4zbhrxbk55fq3nkqyx4b3dph7i

Eigenvalue problem meets Sierpinski triangle: computing the spectrum of a non-self-adjoint random operator

Simon N. Chandler-Wilde, Ratchanikorn Chonchaiya, Marko Lindner
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  ... 
doi:10.7153/oam-05-46 fatcat:ekcseiecijgb5g6kugyfekyitm

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

Maurice Hasson, James M. Hyman, Juan M. Restrepo
2007 Computers and Mathematics with Applications  
For a 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.  ... 
doi:10.1016/j.camwa.2005.11.040 fatcat:6n3253rofrgf7kezssrtltbbxa

Remarks on A-skew-adjoint, A-almost Similarity Equivalence and Other Operators in Hilbert Space

Isaiah Nalianya Sitati
2017 Pure and Applied Mathematics Journal  
In this paper, notions 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  ... 
doi:10.11648/j.pamj.20170603.12 fatcat:ndgbwfahynfd3g2aztcggh75se

Numerical approximation of the spectrum of self-adjoint continuously invertible operators [article]

Tomáš Gergelits, Bjørn Fredrik Nielsen, Zdeněk Strakoš
2021 arXiv   pre-print
More precisely, we consider two bounded, coercive, and 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.  ... 
arXiv:2103.00849v1 fatcat:ca7uxot3sbglrj2deodzxiw3n4

The eigenvalues of an effectively determined self-adjoint operator are computable, but the sequence of eigenvalues is not

Marian Boykan Pour-El, Ian Richards
1987 Advances in Mathematics  
This paper explores 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.  ... 
doi:10.1016/0001-8708(87)90062-4 fatcat:ycdsnf4infccbjn6ceyswdjjru

Exponential Function of a bounded Linear Operator on a Hilbert Space

Baghdad Science Journal
2014 Baghdad Science Journal  
In this paper, we introduce an exponential 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.  ... 
doi:10.21123/bsj.11.3.1267-1273 fatcat:3vtzabepkbc6zklpx6zbe7arlu

Conformal deformations of immersed discs in R^3 and elliptic boundary value problems [article]

Christoph Bohle, Ulrich Pinkall
2013 arXiv   pre-print
Boundary value problems for 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.  ... 
arXiv:1301.3752v1 fatcat:l26ksljc5jbnhchv6kc6swm4be

Spectral Properties of Random Non-self-adjoint Matrices and Operators [article]

E B Davies
2000 arXiv   pre-print
Our results imply that 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.  ... 
arXiv:math/0002159v1 fatcat:fszmx32urjg3lkmdlaxx4au7ea

Finite deficiency indices and uniform remainder in Weyl's law [article]

Luc Hillairet
2010 arXiv   pre-print
We give a proof that in settings where Von Neumann deficiency indices are finite 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.  ... 
arXiv:1001.1795v2 fatcat:kszsuxpdxjbh7jejhewxqgarqm

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

Fredrik Berntsson, Anna Orlof, Johan Thim
2017 Numerical Functional Analysis and Optimization  
It is also important to have good estimates 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.  ... 
doi:10.1080/01630563.2017.1279176 fatcat:tztkhxx7prfm3oeh5tqzg7yl5a

Adjoint zero-modes as a tool to understand the Yang-Mills vacuum [article]

Margarita Garcia Perez, Antonio Gonzalez-Arroyo, Alfonso Sastre
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.  ... 
arXiv:0710.0455v1 fatcat:h565z22qxzaytljvw67oi4yvoy

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

Guy Bonneau, Jacques Faraut, Galliano Valent
2001 American Journal of Physics  
For 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.  ... 
doi:10.1119/1.1328351 fatcat:rohgm4tgffhzdj7az2b7gw6eay

Computing the Enclosures Eigenvalues Using the Quadratic Method

Shurouq Mohammad Abusamra
2019 Applied Mathematics  
In this article, We 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.  ... 
doi:10.4236/am.2019.104016 fatcat:zm2cyzwg5ngyfbzm4gxuxq745i
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