Evans Functions, Jost Functions, and Fredholm Determinants

Gesztesy, Fritz; Latushkin, Yuri; Makarov, Konstantin A.

doi:10.1007/s00205-007-0071-7

Cited by 31 publications

(100 citation statements)

References 66 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In this paper we continue the work began in [5,6,7], and investigate further the connections of the Evans function and (modified) Fredholm determinants of the Birman-Schwinger type operators. In particular, we bring into the discussion a new element, the Gohberg-Rouche Theorem [9, Theorem XI.9.1].…”

Section: Introductionmentioning

confidence: 83%

“…As shown in [5], if the first order system (3.13) has an additional perturbation structure as in (3.7), then the Evans function can be chosen to agree with the (modified) Fredholm determinant of the operator I − K(λ). Specifically, the basis vectors u j (λ) can be chosen as the columns of the generalized matrix-valued Jost solutions of the first order system (3.7).…”

Section: Definition 26 the Evans Function E Is Defined Bymentioning

confidence: 99%

“…The main tool in our investigation is the connection of the (modified) Fredholm determinant of the Birman-Schwinger type integral operator for the linearized eigenvalue problem of a given partial differential equation, and the Evans function for the equivalent to this eigenvalue problem first order system, see [5,12,13]. Our strategy can be described as follows.…”

Section: Introductionmentioning

confidence: 99%

“…Now the abstract results from Section 2 imply that the algebraic multiplicity m(λ 0 ; H) of a discrete eigenvalue λ 0 of H coincides with the multiplicity of the zero at λ 0 for the function det 2 (I − K(·)). We recall that the main result in [5] is an explicit formula relating det 2 (I − K(·)) and the Evans function E = E(λ) for the equation…”

Section: Introductionmentioning

confidence: 99%

“…where S(λ 0 ) = 0, and Θ(λ) is a function, analytic in λ, explicitly computed in [5]. In particular, (1.1) shows that the algebraic multiplicity m(λ 0 ; H) is equal to the multiplicity of the zero at λ 0 of the Evans function.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The Algebraic Multiplicity of Eigenvalues and the Evans Function Revisited

Latushkin

Sukhtayev

2010

Math. Model. Nat. Phenom.

View full text Add to dashboard Cite

Abstract. This paper is related to the spectral stability of traveling wave solutions of partial differential equations. In the first part of the paper we use the Gohberg-Rouche Theorem to prove equality of the algebraic multiplicity of an isolated eigenvalue of an abstract operator on a Hilbert space, and the algebraic multiplicity of the eigenvalue of the corresponding Birman-Schwinger type operator pencil. In the second part of the paper we apply this result to discuss three particular classes of problems: the Schrödinger operator, the operator obtained by linearizing a degenerate system of reaction diffusion equations about a pulse, and a general high order differential operator. We study relations between the algebraic multiplicity of an isolated eigenvalue for the respective operators, and the order of the eigenvalue as the zero of the Evans function for the corresponding first order system.

show abstract

Section: Introductionmentioning

confidence: 83%

Section: Definition 26 the Evans Function E Is Defined Bymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Algebraic Multiplicity of Eigenvalues and the Evans Function Revisited

Latushkin

Sukhtayev

2010

Math. Model. Nat. Phenom.

View full text Add to dashboard Cite

show abstract

Derivatives of the Evans function and (modified) Fredholm determinants for first order systems

Das

Latushkin

2011

Math. Nachr.

Self Cite

View full text Add to dashboard Cite

The Evans function is a Wronskian type determinant used to detect point spectrum of differential operators obtained by linearizing PDEs about special solutions such as traveling waves, etc. This work is a sequel to the paper "Derivatives of (modified) Fredholm determinants and stability of standing and traveling waves", published by F. Gesztesy, K. Zumbrun and the second author in J. Math. Pures Appl. 90, 160-200 (2008), where the Evans and Jost functions for the Schrödinger equations have been considered. In the current work, we study the Evans function for the general case of linear ODE systems, and choose it to agree with the modified Fredholm determinant of the respective Birman-Schwinger type integral operator. The Evans function is thus the determinant of the matrix composed of the so-called generalized Jost solutions. These are the solutions of the homogeneous perturbed differential equation which are asymptotic to some reference solutions of the unperturbed equation. One of the main results of the current paper is a formula for the derivative of the Evans function for the first order systems. Its proof uses a matrix composed of the newly introduced modified Jost solutions. These are the solutions of an inhomogeneous perturbed differential equation with the inhomogeneous term constructed by means of the above-mentioned generalized Jost solutions.

show abstract

The Evans Function for nth-Order Operators on the Real Line

Kapitula

Promislow

2013

Applied Mathematical Sciences

View full text Add to dashboard Cite

Evans Functions, Jost Functions, and Fredholm Determinants

Cited by 31 publications

References 66 publications

The Algebraic Multiplicity of Eigenvalues and the Evans Function Revisited

The Algebraic Multiplicity of Eigenvalues and the Evans Function Revisited

Derivatives of the Evans function and (modified) Fredholm determinants for first order systems

The Evans Function for nth-Order Operators on the Real Line

Contact Info

Product

Resources

About