Minimal Escape Velocities for Unitary Evolution Groups

Richard, Serge

doi:10.1007/s00023-004-0185-5

Cited by 3 publications

(3 citation statements)

References 16 publications

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“…It is easy to see that for The next lemma contains a similar result with the operator ln(X) replaced by X 2 . It can be obtained by an abstract argument based on Mourre theory, see [30] for the details. We will give an alternative elementary proof.…”

Section: B Propagation Of the Generator Of Dilationsmentioning

confidence: 99%

On Schrödinger Operators with Inverse Square Potentials on the Half-Line

Dereziński

Richard

2016

Ann. Henri Poincaré

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135

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Abstract. The paper is devoted to operators given formally by the expressionThis expression is homogeneous of degree minus 2. However, when we try to realize it as a self-adjoint operator for real α, or closed operator for complex α, we find that this homogeneity can be broken. This leads to a definition of two holomorphic families of closed operators on L 2 (R+), which we denote Hm,κ and H ν 0 , with m 2 = α, −1 < Re(m) < 1, and where κ, ν ∈ C ∪{∞} specify the boundary condition at 0. We study these operators using their explicit solvability in terms of Bessel-type functions and the Gamma function. In particular, we show that their point spectrum has a curious shape: a string of eigenvalues on a piece of a spiral. Their continuous spectrum is always [0, ∞[. Restricted to their continuous spectrum, we diagonalize these operators using a generalization of the Hankel transformation. We also study their scattering theory. These operators are usually non-self-adjoint. Nevertheless, it is possible to use concepts typical for the self-adjoint case to study them. Let us also stress that −1 < Re(m) < 1 is the maximal region of parameters for which the operators Hm,κ can be defined within the framework of the Hilbert space L 2 (R+).

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Section: B Propagation Of the Generator Of Dilationsmentioning

confidence: 99%

On Schrödinger Operators with Inverse Square Potentials on the Half-Line

Dereziński

Richard

2016

Ann. Henri Poincaré

Self Cite

135

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“…In particular, J does not contain any eigenvalue of H. We look to apply [Ri,Theorem 1]. Let ψ ∈ H, and assume without loss of generality that ψ = 1.…”

Section: 2mentioning

confidence: 99%

“…of the previous inequality are each less than ε/3. The second one is controlled by [Ri,Theorem 1] and the third one by functional calculus. Then…”

Section: 2mentioning

confidence: 99%

Propagation Estimates for One Commutator Regularity

Golenia

Mandich

2018

Integr. Equ. Oper. Theory

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In the abstract framework of Mourre theory, the propagation of states is understood in terms of a conjugate operator A. A powerful estimate has long been known for Hamiltonians having a good regularity with respect to A thanks to the limiting absorption principle (LAP). We study the case where H has less regularity with respect to A, specifically in a situation where the LAP and the absence of singularly continuous spectrum have not yet been established. We show that in this case the spectral measure of H is a Rajchman measure and we derive some propagation estimates. One estimate is an application of minimal escape velocities, while the other estimate relies on an improved version of the RAGE formula. Based on several examples, including continuous and discrete Schrödinger operators, it appears that the latter propagation estimate is a new result for multi-dimensional Hamiltonians.

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Minimal velocity bound for Schroedinger-type operator with fractional powers

Ishida¹

2020

Preprint

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It is known in scattering theory that the minimal velocity bound plays a conclusive role in proving the asymptotic completeness of the wave operators. In this study, we prove the minimal velocity bound and other important estimates for the two-body relativistic Schrödinger operator with fractional powers. We can assume that the pairwise potential functions belong to the broad classes which include the long-range decay and the Coulomb-type singularities. Additionally, we propose some candidates of the conjugate operator for the Mourre estimate and give their consequences. Our estimates are expected to be applied to prove the asymptotic completeness for the fractional relativistic operators in various (not only short-range but also long-range and N -body) situations.

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Minimal Escape Velocities for Unitary Evolution Groups

Cited by 3 publications

References 16 publications

On Schrödinger Operators with Inverse Square Potentials on the Half-Line

On Schrödinger Operators with Inverse Square Potentials on the Half-Line

Propagation Estimates for One Commutator Regularity

Minimal velocity bound for Schroedinger-type operator with fractional powers

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