Diffraction from Polygonal-conical Screens, an Operator Approach

Castro, L. P.; Duduchava, Roland; Speck, F.-O.

doi:10.1007/978-3-0348-0816-3_6

Cited by 8 publications

(22 citation statements)

References 39 publications

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“…We assume that the notion and basic properties of Sobolev spaces (here synonymously for fractional Sobolev spaces or Bessel potential spaces) are known using the common notation of

H^{s} = H^{s} ({double-struckR}^{n})

and H s ( Ω ) for the restricted function(al)s,

s \in double-struckR

. Further we write

H_{Σ}^{s}

for those defined on

{double-struckR}^{n - 1}

but supported on

\overset{Σ}{false}

, and

{\overset{H}{true}}^{s} (Σ)

for those defined on Σ , which admit an extension by zero within

H^{s} ({double-struckR}^{n - 1})

, see for details.…”

Section: Introduction: Formulation Of the Problems And Main Resultsmentioning

confidence: 99%

“…Hence the interface conditions can be understood in the sense of the trace theorem or by the help of representation formulas, respectively. Namely, the weak solutions u of the HE can be written in the form (, e.g. ):

\begin{matrix} u & = K_{D, Ω} (u_{0}^{+}, u_{0}^{-}) = \{\begin{matrix} {scriptK}_{D, Ω^{+}} u_{0}^{+} in Ω^{+} \\ {scriptK}_{D, Ω^{-}} u_{0}^{-} in Ω^{-} \end{matrix} \end{matrix}

\begin{matrix} K_{D, Ω^{+}} u_{0}^{+} (x) & = F_{ξ^{'} \mapsto x^{'}}^{- 1} e^{- t (ξ^{'}) x_{n}} \hat{u_{0}^{+}} (ξ^{'}) = \frac{1}{{(2 π)}^{2}} \int_{R^{2}} e^{- i ξ^{'} x^{'} - t (ξ^{'}) x_{n}} \hat{u_{0}^{+}} (ξ^{'}) d ξ^{'} \\ K_{D, Ω^{-}} u_{0}^{-} (x) \end{matrix}

…”

Section: Introduction: Formulation Of the Problems And Main Resultsmentioning

confidence: 99%

“…A milestone was the solution of the quarter‐plane problem , which caused some irritations in the literature, see , the introduction in , and Remark 5.2 in . However, it started a new method based on the use of projectors onto

H_{Σ}^{\pm 1 / 2}

, first applied to Sommerfeld‐type problems (for n = 3) with Dirichlet and Neumann conditions on certain Lipschitz screens and later for polygonal‐conical screens . In brief, those techniques yield an explicit solution particularly for the quarter‐plane problem with Dirichlet or Neumann conditions, but (still) not for the mixed (Dirichlet–Neumann) or other boundary or transmission conditions.…”

Section: Introduction: Formulation Of the Problems And Main Resultsmentioning

confidence: 99%

“…Now, let us try to work with t ( ξ ) = ( ξ 2 − k 2 ) 1/2 instead of t i =( ξ 2 +1) 1/2 where k ≠ i , moreover

frakturRek > 0, Imk > 0

. Replacing λ + ( ξ ) by the last factor of the factorization

(ξ^{2} - k^{2}) = (ξ_{m} - i {(ξ^{' 2} - k^{2})}^{1 / 2}) (ξ_{m} + i {(ξ^{' 2} - k^{2})}^{1 / 2})

we obtain similar results but no orthogonality. Remark If one tries to solve IFPs with the present kind of interface conditions for PCDs as in by analogy, one needed projectors onto

H_{Σ}^{1 / 2} \times H_{Σ}^{- 1 / 2}

along

A_{σ} (H_{Σ^{'}}^{1 / 2} \times H_{Σ^{'}}^{- 1 / 2})

, which are orthogonal with respect to some equivalent inner product in H 1/2 × H −1/2 . This seems to be impossible.…”

Section: Basic Definitions and Conceptionsmentioning

confidence: 99%

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A class of interface problems for the Helmholtz equation in

Speck

2015

Math Methods in App Sciences

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Motivated by the Sommerfeld diffraction problem, we consider interface problems for the n‐dimensional Helmholtz equation in Ω=double-struckR+n∪double-struckR−n (due to xn>0 or xn<0, respectively, with main interest in n = 3) where the interface Γ = ∂Ω is identified with double-struckRn−1 and divided into two parts, Σ and Σ′, with different transmission conditions of first and second kind. These two parts are half‐spaces of double-struckRn−1 (half‐planes for n = 3) and more general sets in the first part of the paper. The aim of this work is to construct resolvent operators acting from the interface data into the energy space H1(Ω) and representing the solution explicitly. The approach is based upon a factorization conception for Wiener–Hopf operators (according to the interface equations), the so‐called Wiener–Hopf factorization through an intermediate space, that includes Simonenko's well‐known ‘generalized factorization of matrix functions in Lp spaces’ and avoids an interpretation of the factors as unbounded operators. Copyright © 2015 John Wiley & Sons, Ltd.

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“…We assume that the notion and basic properties of Sobolev spaces (here synonymously for fractional Sobolev spaces or Bessel potential spaces) are known using the common notation of

H^{s} = H^{s} ({double-struckR}^{n})

and H s ( Ω ) for the restricted function(al)s,

s \in double-struckR

. Further we write

H_{Σ}^{s}

for those defined on

{double-struckR}^{n - 1}

but supported on

\overset{Σ}{false}

, and

{\overset{H}{true}}^{s} (Σ)

for those defined on Σ , which admit an extension by zero within

H^{s} ({double-struckR}^{n - 1})

, see for details.…”

Section: Introduction: Formulation Of the Problems And Main Resultsmentioning

confidence: 99%

\begin{matrix} u & = K_{D, Ω} (u_{0}^{+}, u_{0}^{-}) = \{\begin{matrix} {scriptK}_{D, Ω^{+}} u_{0}^{+} in Ω^{+} \\ {scriptK}_{D, Ω^{-}} u_{0}^{-} in Ω^{-} \end{matrix} \end{matrix}

\begin{matrix} K_{D, Ω^{+}} u_{0}^{+} (x) & = F_{ξ^{'} \mapsto x^{'}}^{- 1} e^{- t (ξ^{'}) x_{n}} \hat{u_{0}^{+}} (ξ^{'}) = \frac{1}{{(2 π)}^{2}} \int_{R^{2}} e^{- i ξ^{'} x^{'} - t (ξ^{'}) x_{n}} \hat{u_{0}^{+}} (ξ^{'}) d ξ^{'} \\ K_{D, Ω^{-}} u_{0}^{-} (x) \end{matrix}

…”

Section: Introduction: Formulation Of the Problems And Main Resultsmentioning

confidence: 99%

H_{Σ}^{\pm 1 / 2}

Section: Introduction: Formulation Of the Problems And Main Resultsmentioning

confidence: 99%

“…Now, let us try to work with t ( ξ ) = ( ξ 2 − k 2 ) 1/2 instead of t i =( ξ 2 +1) 1/2 where k ≠ i , moreover

frakturRek > 0, Imk > 0

. Replacing λ + ( ξ ) by the last factor of the factorization

(ξ^{2} - k^{2}) = (ξ_{m} - i {(ξ^{' 2} - k^{2})}^{1 / 2}) (ξ_{m} + i {(ξ^{' 2} - k^{2})}^{1 / 2})

we obtain similar results but no orthogonality. Remark If one tries to solve IFPs with the present kind of interface conditions for PCDs as in by analogy, one needed projectors onto

H_{Σ}^{1 / 2} \times H_{Σ}^{- 1 / 2}

along

A_{σ} (H_{Σ^{'}}^{1 / 2} \times H_{Σ^{'}}^{- 1 / 2})

, which are orthogonal with respect to some equivalent inner product in H 1/2 × H −1/2 . This seems to be impossible.…”

Section: Basic Definitions and Conceptionsmentioning

confidence: 99%

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A class of interface problems for the Helmholtz equation in

Speck

2015

Math Methods in App Sciences

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show abstract

“…The operator relations equivalent after extension (EAE) , matricial coupling (MC) and Schur coupling (SC) for Banach space operators

U

and

V

were first used to solve certain integral equations , and have found many applications since; for some recent applications, see (on diffraction theory), (on truncated Toeplitz operators), (on unbounded operator functions) and (on Wiener–Hopf factorisation). The main feature in these applications is that the relations EAE, MC and SC coincide, and that one can transfer from one to another in a constructive way.…”

Section: Introductionmentioning

confidence: 99%

Equivalence after extension and Schur coupling do not coincide on essentially incomparable Banach spaces

Horst

Messerschmidt

Ran

et al. 2019

Bull. London Math. Soc.

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In 1994, H. Bart and V. É. Tsekanovskii posed the question whether the Banach space operator relations matricial coupling (MC), equivalence after extension (EAE) and Schur coupling (SC) coincide, leaving only the implication EAE/MC ⇒ SC open. Despite several affirmative results, in this paper we show that the answer in general is no. This follows from a complete description of EAE and SC for the case that the operators act on essentially incomparable Banach spaces, which also leads to a new characterisation of the notion of essential incomparability. Concretely, the forward shift operators U on ℓp and V on ℓq, for 1⩽p,q⩽∞, p≠q, are EAE but not SC. As a corollary, SC is not transitive. Under mild assumptions, given U and V that are Atkinson or generalised invertible and EAE, we give a concrete operator W that is SC to both U and V, even if U and V are not SC themselves. Some further affirmative results for the case where the Banach spaces are isomorphic are also obtained.

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Wiener–Hopf Factorization Through an Intermediate Space

Speck

2014

Integr. Equ. Oper. Theory

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Diffraction from Polygonal-conical Screens, an Operator Approach

Cited by 8 publications

References 39 publications

A class of interface problems for the Helmholtz equation in

A class of interface problems for the Helmholtz equation in

Equivalence after extension and Schur coupling do not coincide on essentially incomparable Banach spaces

Wiener–Hopf Factorization Through an Intermediate Space

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