scholarly journals Investigation of the Spectral Properties of a Non-Self-Adjoint Elliptic Differential Operator

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Arezoo Ghaedrahmati ◽  
Ali Sameripour
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Spectral Properties ◽  
Differential Operators ◽  
Dimensional Space ◽  
The Other ◽  
Heat Equations ◽  
Elliptic Differential Operator ◽  
Other Hand ◽  
Adjoint Operators

Non-self-adjoint operators have many applications, including quantum and heat equations. On the other hand, the study of these types of operators is more difficult than that of self-adjoint operators. In this paper, our aim is to study the resolvent and the spectral properties of a class of non-self-adjoint differential operators. So we consider a special non-self-adjoint elliptic differential operator (Au)(x) acting on Hilbert space and first investigate the spectral properties of space H 1 = L 2 Ω 1 . Then, as the application of this new result, the resolvent of the considered operator in ℓ -dimensional space Hilbert H ℓ = L 2 Ω ℓ is obtained utilizing some analytic techniques and diagonalizable way.

Inequalities between means of positive operators

1978 ◽  
Vol 83 (3) ◽  
pp. 393-401 ◽  
Author(s):  
K. V. Bhagwat ◽  
R. Subramanian
Keyword(s):  
Hilbert Space ◽  
Dimensional Space ◽  
Adjoint Operator ◽  
Positive Definite ◽  
Inner Product ◽  
Finite Dimensional Space ◽  
Finite Dimensional ◽  
Definite Operator ◽  
Unit Operator ◽  
Adjoint Operators

One of the most fruitful – and natural – ways of introducing a partial order in the set of bounded self-adjoint operators in a Hilbert space is through the concept of a positive operator. A bounded self-adjoint operator A denned on is called positive – and one writes A ≥ 0 - if the inner product (ψ, Aψ) ≥ 0 for every ψ ∈ . If, in addition, (ψ, Aψ) = 0 only if ψ = 0, then A is called positive-definite and one writes A > 0. Further, if there exists a real number γ > 0 such that A — γI ≥ 0, I being the unit operator, then A is called strictly positive (in symbols, A ≫ 0). In a finite dimensional space, a positive-definite operator is also strictly positive.

On the study of the spectral properties of differential operators with a smooth weight function

2020 ◽  
pp. 25-47
Author(s):  
Sergey I. Mitrokhin
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Operator ◽  
Spectral Properties ◽  
Differential Operators ◽  
Asymptotic Estimates ◽  
Third Order ◽  
Indicator Diagram ◽  
Estimates Of Solutions ◽  
Smooth Coefficients

In this paper we study the spectral properties of a third-order differential operator with a summable potential with a smooth weight function. The boundary conditions are separated. The method of studying differential operators with summable potential is a development of the method of studying operators with piecewise smooth coefficients. Boundary value problems of this kind arise in the study of vibrations of rods, beams and bridges composed of materials of different densities. The differential equation defining the differential operator is reduced to the solution of the Volterra integral equation by means of the method of variation of constants. The solution of the integral equation is found by the method of successive Picard approximations. Using the study of an integral equation, we obtained asymptotic formulas and estimates for the solutions of a differential equation defining a differential operator. For large values of the spectral parameter, the asymptotics of solutions of the differential equation that defines the differential operator is derived. Asymptotic estimates of solutions of a differential equation are obtained in the same way as asymptotic estimates of solutions of a differential operator with smooth coefficients. The study of boundary conditions leads to the study of the roots of the function, presented in the form of a third-order determinant. To get the roots of this function, the indicator diagram wasstudied. The roots of this equation are in three sectors of an infinitely small size, given by the indicator diagram. The article studies the behavior of the roots of this equation in each of the sectors of the indicator diagram. The asymptotics of the eigenvalues of the differential operator under study is calculated. The formulas found for the asymptotics of eigenvalues allow us to study the spectral properties of the eigenfunctions of the differential operator under study.

scholarly journals On Dirichlet's boundary value problem for some formally hypoelliptic differntial operators

1988 ◽  
Vol 44 (3) ◽  
pp. 324-349 ◽  
Author(s):  
Niels Jacob
Keyword(s):  
Hilbert Space ◽  
Dirichlet Problem ◽  
Differential Operator ◽  
Boundary Value Problem ◽  
Differential Operators ◽  
Boundary Value ◽  
Divergence Form ◽  
Alternative Theorem ◽  
Sesquilinear Form ◽  
Homogeneous Dirichlet Problem

AbstractFor a class of formally hypoelliptic differential operators in divergence form we prove a generalized Gårding inequality. Using this inequality and further properties of the sesquilinear form generated by the differential operator a generalized homogeneous Dirichlet problem is treated in a suitable Hilbert space. In particular Fredholm's alternative theorem is proved to be valid.

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.

scholarly journals Asymptotic Variational Formulae for Eigenvalues

1963 ◽  
Vol 6 (1) ◽  
pp. 15-25 ◽  
Author(s):  
C.A. Swanson
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Component ◽  
Differential Operators ◽  
Open Interval ◽  
Variational Problems ◽  
Elliptic Differential Operator ◽  
Additional Boundary Condition ◽  
Half Open Interval ◽  
Ordinary Differential Operators

The eigenvalues of a second order self-adjoint elliptic differential operator on Riemannian n-space R will be considered. Our purpose is to obtain asymptotic variational formulae for the eigenvalues under the topological deformations of (i) removing an ɛ -cell (and adjoining an additional boundary condition on the boundary component thereby introduced); and (ii) attaching an ɛ -handle, valid on a half-open interval 0 < ɛ ≤ ɛo. In particular the formulae will exhibit the non-analytic nature of the variation. Similar variational problems for singular ordinary differential operators have been considered by the writer in [3] and [4].

Singular differential operators with spectra discrete and bounded below

1979 ◽  
Vol 84 (1-2) ◽  
pp. 117-134 ◽  
Author(s):  
Don B. Hinton ◽  
Roger T. Lewis
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Ordinary Differential Operator ◽  
Adjoint Operators ◽  
Bounded Below

SynopsisA weighted, formally self-adjoint ordinary differential operator l of order 2n is considered, and conditions are given on the coefficients of l which ensure that all self-adjoint operators associated with l have a spectrum which is discrete and bounded below. Both finite and infinite singularities are considered. The results are obtained by the establishment of certain conditions which imply that l is non-oscillatory.

scholarly journals Spectrums of Solvable Pantograph Differential-Operators for First Order

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Z. I. Ismailov ◽  
P. Ipek
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Operator Theory ◽  
Differential Operators ◽  
Finite Interval ◽  
Exact Formula ◽  
Operator Expression ◽  
Minimal Operator ◽  
First Order ◽  
Delay Differential

By using the methods of operator theory, all solvable extensions of minimal operator generated by first order pantograph-type delay differential-operator expression in the Hilbert space of vector-functions on finite interval have been considered. As a result, the exact formula for the spectrums of these extensions is presented. Applications of obtained results to the concrete models are illustrated.

Examples of degenerate symmetric differential operators with infinite deficiency indices in L2(ℝm)

1999 ◽  
Vol 129 (1) ◽  
pp. 165-179
Author(s):  
Yurii B. Orochko
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Differential Expression ◽  
Symmetric Operator ◽  
Differential Operators ◽  
Separable Hilbert Space ◽  
Adjoint Operator ◽  
Deficiency Indices ◽  
Image Position ◽  
Symmetric Differential Operators

For an unbounded self-adjoint operator A in a separable Hilbert space ℌ and scalar real-valued functions a(t), q(t), r(t), t ∊ ℝ, consider the differential expressionacting on ℌ-valued functions f(t), t ∊ ℝ, and degenerating at t = 0. Let Sp denotethe corresponding minimal symmetric operator in the Hilbert space (ℝ) of ℌ-valued functions f(t) with ℌ-norm ∥f(t)∥ square integrable on the line. The infiniteness of the deficiency indices of Sp, 1/2 < p < 3/2, is proved under natural restrictions on a(t), r(t), q(t). The conditions implying their equality to 0 for p ≥ 3/2 are given. In the case of a self-adjoint differential operator A acting in ℌ = L2(ℝn), the first of these results implies examples of symmetric degenerate differential operators with infinite deficiency indices in L2(ℝm), m = n + 1.

scholarly journals On commutativity of C*-algebras

1987 ◽  
Vol 29 (1) ◽  
pp. 93-97 ◽  
Author(s):  
C.-S. Lin
Keyword(s):  
Hilbert Space ◽  
The Other ◽  
Nilpotent Operator ◽  
C Algebra ◽  
C Algebras ◽  
Adjoint Operators ◽  
Numerical Index

Two numerical characterizations of commutativity for C*-algebra (acting on the Hilbert space H) were given in [1]; one used the norms of self-adjoint operators in (Theorem 2), and the other the numerical index of (Theorem 3). In both cases the proofs were based on the result of Kaplansky which states that if the only nilpotent operator in is 0, then is commutative ([2] 2.12.21, p. 68). Of course the converse also holds.

Spurious Structure from Approximations to the Dyson Equation

1972 ◽  
Vol 50 (19) ◽  
pp. 2286-2293 ◽  
Author(s):  
Clas Blomberg ◽  
Birger Bergersen
Keyword(s):  
Spectral Function ◽  
Spectral Properties ◽  
Perturbation Expansion ◽  
Dyson Equation ◽  
The Self ◽  
The Other ◽  
Exact Results ◽  
Electron Level ◽  
Self Energy ◽  
Other Hand

The method of calculating electron spectral properties by substituting perturbation theoretic results for the self-energy into the Dyson equation is investigated for a model of a deep electron level in which exact results are known. The method gives wrong results in the most important range and also predicts spurious structure. On the other hand, if a perturbation expansion is made directly for the spectral function after extracting the appropriate energy shifts no such difficulty arises.

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