One-parameter semigroups

Davies, E. B.

doi:10.1017/cbo9780511618864.007

Cited by 360 publications

(366 citation statements)

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“…This operator, A, arises from the second-order differential equation (1.2) by means of the substitution 10) which leads to a first-order differential equation of the form…”

Section: · ·] Since For X ∈ (D(v ) ∩ D(h 0 )) ⊕ D(v ) X ∈ H ⊕ Hmentioning

confidence: 99%

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Spectral Theory of the Klein–Gordon Equation in Krein Spaces

Langer¹,

Najman

Tretter³

2008

Proceedings of the Edinburgh Mathematical Society

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In this paper the spectral properties of the abstract Klein-Gordon equation are studied. The main tool is an indefinite inner product known as the charge inner product. Under certain assumptions on the potential V , two operators are associated with the Klein-Gordon equation and studied in Krein spaces generated by the charge inner product. It is shown that the operators are self-adjoint and definitizable in these Krein spaces. As a consequence, they possess spectral functions with singularities, their essential spectra are real with a gap around 0 and their non-real spectra consist of finitely many eigenvalues of finite algebraic multiplicity which are symmetric to the real axis. One of these operators generates a strongly continuous group of unitary operators in the Krein space; the other one gives rise to two bounded semi-groups. Finally, the results are applied to the Klein-Gordon equation in R n .

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“…This operator, A, arises from the second-order differential equation (1.2) by means of the substitution 10) which leads to a first-order differential equation of the form…”

Section: · ·] Since For X ∈ (D(v ) ∩ D(h 0 )) ⊕ D(v ) X ∈ H ⊕ Hmentioning

confidence: 99%

“…[10,Theorem 3.1]). The next theorem establishes the existence of solutions of the abstract differential equation (1.14).…”

Section: Semi-groups Associated With Amentioning

confidence: 99%

Spectral Theory of the Klein–Gordon Equation in Krein Spaces

Langer¹,

Najman

Tretter³

2008

Proceedings of the Edinburgh Mathematical Society

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“…While the semigroup and the inverse can be studied in the framework of functional analysis as explained in [3,4,5,8,9,16], the results and methods in this paper are based on explicit formulas in hard analysis and are related to the works in [1,2,6,7,10,12,14,15].…”

Section: Cubo 12 3 (2010)mentioning

confidence: 99%

The Semigroup and the Inverse of the Laplacian on the Heisenberg Group

Dasgupta

Wong

2010

Cubo

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By decomposing the Laplacian on the Heisenberg group into a family of parametrized partial differential operatorsL τ ,τ ∈ R \ {0}, and using parametrized Fourier-Wigner transforms, we give formulas and estimates for the strongly continuous one-parameter semigroup generated byL τ , and the inverse ofL τ . Using these formulas and estimates, we obtain Sobolev estimates for the one-parameter semigroup and the inverse of the Laplacian.

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“…It only remains to characterize the generator of TL in C x C. We denote it by A and show that A = AL. This will be accomplished by the core theorem; see [9,Theorem 1.9]. We need to show that (a) D(Al) is invariant under TL{t) for t > 0,…”

Section: Jo Where T -> W(tv(t)) Is the Function Described In Lemma 2mentioning

confidence: 99%