Spectral Theory and Differential Operators

Davies, E. B.

doi:10.1017/cbo9780511623721

Cited by 506 publications

(482 citation statements)

References 0 publications

Supporting

Mentioning

475

Contrasting

Unclassified

Order By: Relevance

“…(1.8) 9) are the zeroes of the polynomial det τ E − A(λ * , l) taken counting multiplicity and ordered as follows…”

Section: Statement Of the Problem And Formulation Of The Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Asymptotic Behaviour of the Spectrum of a Waveguide with Distant Perturbations

Борисов

2007

Math Phys Anal Geom

View full text Add to dashboard Cite

We consider the waveguide modelled by a n-dimensional infinite tube. The operator we study is the Dirichlet Laplacian perturbed by two distant perturbations. The perturbations are described by arbitrary abstract operators "localized" in a certain sense, and the distance between their "supports" tends to infinity. We study the asymptotic behaviour of the discrete spectrum of such system. The main results are a convergence theorem and the asymptotics expansions for the eigenvalues. The asymptotic behaviour of the associated eigenfunctions is described as well. We also provide some particular examples of the distant perturbations. The examples are the potential, second order differential operator, magnetic Schrödinger operator, curved and deformed waveguide, delta interaction, and integral operator.

show abstract

“…(1.8) 9) are the zeroes of the polynomial det τ E − A(λ * , l) taken counting multiplicity and ordered as follows…”

Section: Statement Of the Problem And Formulation Of The Resultsmentioning

confidence: 99%

“…Quite similar problems deal with the case when the small parameter at higher derivative is replaced by a large distance between wells. We mention papers [11], [13], [15] as well as the book [9,Sec. 8.6] devoted to such problems (see also bibliography of these works).…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Behaviour of the Spectrum of a Waveguide with Distant Perturbations

Борисов

2007

Math Phys Anal Geom

View full text Add to dashboard Cite

show abstract

“…It follows that Q(u, v) uniquely determines a nonnegative, densely defined, self-adjoint operator on H 2 such that Dom( 1/2 ) = Dom(Q) and Q(u, v) = ( u, v) for all u ∈ Dom( ) and v ∈ Dom(Q). (We refer the reader to [D95,K76,RS] for detail on sesquilinear forms and self-adjoint operators.) The spectrum spec ( ) of is a non-empty closed subset of [0, ∞) and the infimum of the spectrum is given by inf spec ( ) = inf{Q(u, u); u ∈ Dom(Q), u = 1}.…”

Section: Preliminariesmentioning

confidence: 99%

Hearing pseudoconvexity with the Kohn Laplacian

2004

Math. Ann.

View full text Add to dashboard Cite

show abstract

“…[14], Chapter 4). Thus in particular we can choose Φ continuous inside Ω, but not necessarily smooth.…”

Section: Proofmentioning

confidence: 99%

Bound states in straight quantum waveguides with combined boundary conditions

Dittrich

Křı́ž

2002

Journal of Mathematical Physics

View full text Add to dashboard Cite

We investigate the discrete spectrum of the Hamiltonian describing a quantum particle living in the two-dimensional straight strip. We impose the combined Dirichlet and Neumann boundary conditions on different parts of the boundary. Several statements on the existence or the absence of the discrete spectrum are proven for two models with combined boundary conditions. Examples of eigenfunctions and eigenvalues are computed numerically.

show abstract

Spectral Theory and Differential Operators

Cited by 506 publications

References 0 publications

Asymptotic Behaviour of the Spectrum of a Waveguide with Distant Perturbations

Asymptotic Behaviour of the Spectrum of a Waveguide with Distant Perturbations

Hearing pseudoconvexity with the Kohn Laplacian

Bound states in straight quantum waveguides with combined boundary conditions

Contact Info

Product

Resources

About