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Resolvent and Scattering Matrix at the Maximum of the Potential

Alexandrova, Ivana, Bony, Jean-François, Ramond, Thierry (2008)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 35P25, 81U20, 35S30, 47A10, 35B38.We study the microlocal structure of the resolvent of the semiclassical Schrödinger operator with short range potential at an energy which is a unique non-degenerate global maximum of the potential. We prove that it is a semiclassical Fourier integral operator quantizing the incoming and outgoing Lagrangian submanifolds associated to the fixed hyperbolic point. We then discuss two applications of this result to describing...

Resolvent and spectrum of a nonselfadjoint differential operator in a Hilbert space

Michael Gil (2012)

Annales UMCS, Mathematica

We consider a second order regular differential operator whose coefficients are nonselfadjoint bounded operators acting in a Hilbert space. An estimate for the resolvent and a bound for the spectrum are established. An operator is said to be stable if its spectrum lies in the right half-plane. By the obtained bounds, stability and instability conditions are established.

Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. II

Colin Guillarmou, Andrew Hassell (2009)

Annales de l’institut Fourier

Let M ∘ be a complete noncompact manifold of dimension at least 3 and g an asymptotically conic metric on M ∘ , in the sense that M ∘ compactifies to a manifold with boundary M so that g becomes a scattering metric on M . We study the resolvent kernel ( P + k 2 ) - 1 and Riesz transform T of the operator P = Δ g + V , where Δ g is the positive Laplacian associated to g and V is a real potential function smooth on M and vanishing at the boundary.In our first paper we assumed that P has neither zero modes nor a zero-resonance and showed...

Resolvent conditions and powers of operators

Olavi Nevanlinna (2001)

Studia Mathematica

We discuss the relation between the growth of the resolvent near the unit circle and bounds for the powers of the operator. Resolvent conditions like those of Ritt and Kreiss are combined with growth conditions measuring the resolvent as a meromorphic function.

Resonance in Preisach systems

Pavel Krejčí (2000)

Applications of Mathematics

This paper deals with the asymptotic behavior as t → ∞ of solutions u to the forced Preisach oscillator equation w ¨ ( t ) + u ( t ) = ψ ( t ) , w = u + 𝒫 [ u ] , where 𝒫 is a Preisach hysteresis operator, ψ ∈ L ∞ ( 0 , ∞ ) is a given function and t ≥ 0 is the time variable. We establish an explicit asymptotic relation between the Preisach measure and the function ψ (or, in a more physical terminology, a balance condition between the hysteresis dissipation and the external forcing) which guarantees that every solution remains bounded for all times. Examples show...

Resonances and Spectral Shift Function near the Landau levels

Jean-François Bony, Vincent Bruneau, Georgi Raikov (2007)

Annales de l’institut Fourier

We consider the 3D Schrödinger operator H = H 0 + V where H 0 = ( - i ∇ - A ) 2 - b , A is a magnetic potential generating a constant magneticfield of strength b > 0 , and V is a short-range electric potential which decays superexponentially with respect to the variable along the magnetic field. We show that the resolvent of H admits a meromorphic extension from the upper half plane to an appropriate Riemann surface ℳ , and define the resonances of H as the poles of this meromorphic extension. We study their distribution near any fixed...

Currently displaying 281 – 300 of 346