The virtual level of the Schr�dinger equation

Yafaev, D. R.

doi:10.1007/bf01088759

Cited by 13 publications

(12 citation statements)

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“…Clearly (cf. [28]), the operator h has an eigenvalue λ ≤ ε(0), that is, Ker(h − λI) = 0, if and only if the compact operator G(λ) on C(T 3 ) has an eigenvalue −1 and there exists a function ψ ∈ Ker(G + I) such that the function f given by…”

Section: Hypothesis 31 Assume That the Dispersion Relation ε(P) Is A ...mentioning

confidence: 99%

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The Threshold Effects for the Two-Particle Hamiltonians on Lattices

Albeverio

Lakaev

Makarov

et al. 2005

Commun. Math. Phys.

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For a wide class of two-body energy operators h(k) on the three-dimensional lattice Z 3 , k being the two-particle quasi-momentum, we prove that if the following two assumptions (i) and (ii) are satisfied, then for all nontrivial values k, k = 0, the discrete spectrum of h(k) below its threshold is non-empty. The assumptions are: (i) the twoparticle Hamiltonian h(0) corresponding to the zero value of the quasi-momentum has either an eigenvalue or a virtual level at the bottom of its essential spectrum and (ii) the oneparticle free Hamiltonians in the coordinate representation generate positivity preserving semi-groups.

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Section: Hypothesis 31 Assume That the Dispersion Relation ε(P) Is A ...mentioning

confidence: 99%

“…In Appendix A, for readers convenience, we give a proof of Proposition 3.9 which is a "lattice" analogue of a result due to Yafaev [28] in the continuous case.…”

Section: Introductionmentioning

confidence: 99%

The Threshold Effects for the Two-Particle Hamiltonians on Lattices

Albeverio

Lakaev

Makarov

et al. 2005

Commun. Math. Phys.

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“…Of course this definition depends not only on H but also on a class of perturbations. Other definitions (see [15] or [18]) are adapted to differential operators. Thus, the Schrödinger operator H = −∆+v(x) has a zero-energy resonance if the kernel of its resolvent R(z) = (H −z)…”

Section: 3mentioning

confidence: 99%

Spectral and scattering theory of fourth order differential operators

Yafaev

2008

American Mathematical Society Translations: Series 2

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Abstract. An ordinary differential operator of the fourth order with coefficients converging at infinity sufficiently rapidly to constant limits is considered. Scattering theory for this operator is developed in terms of special solutions of the corresponding differential equation. In contrast to equations of second order "scattering" solutions contain exponentially decaying terms. A relation between the scattering matrix and a matrix of coefficients at exponentially decaying modes is found. In the second part of the paper the operator D 4 on the half-axis with different boundary conditions at the point zero is studied. Explicit formulas for basic objects of the scattering theory are found. In particular, a classification of different types of zero-energy resonances is given.

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“…Threshold energy expansions for the resolvent of two-particle Schrödinger operators have been studied in [4,8,14,17,18,27,28,30] and have been applied to the proof of the existence of Efimov's effect in [4,18,[27][28][29].…”

Section: Introductionmentioning

confidence: 99%

The threshold effects for a family of Friedrichs models under rank one perturbations

Albeverio¹,

Lakaev

Muminov

2007

Journal of Mathematical Analysis and Applications

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A family of Friedrichs models under rank one perturbations h μ (p), p ∈ (−π, π] 3 , μ > 0, associated to a system of two particles on the three-dimensional lattice Z 3 is considered. We prove the existence of a unique eigenvalue below the bottom of the essential spectrum of h μ (p) for all non-trivial values of p under the assumption that h μ (0) has either a threshold energy resonance (virtual level) or a threshold eigenvalue. The threshold energy expansion for the Fredholm determinant associated to a family of Friedrichs models is also obtained.

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The virtual level of the Schr�dinger equation

Cited by 13 publications

References 1 publication

The Threshold Effects for the Two-Particle Hamiltonians on Lattices

The Threshold Effects for the Two-Particle Hamiltonians on Lattices

Spectral and scattering theory of fourth order differential operators

The threshold effects for a family of Friedrichs models under rank one perturbations

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