scholarly journals Infinitely Many Embedded Eigenvalues for the Neumann--Poincaré Operator in 3D

2022 ◽  
Vol 54 (1) ◽  
pp. 343-362
Author(s):  
Wei Li ◽  
Karl-Mikael Perfekt ◽  
Stephen P. Shipman
Keyword(s):  
Embedded Eigenvalues

scholarly journals Eigenvalue bounds for non-selfadjoint Dirac operators

2021 ◽  
Author(s):  
Piero D’Ancona ◽  
Luca Fanelli ◽  
Nico Michele Schiavone
Keyword(s):  
Dirac Operator ◽  
Discrete Spectrum ◽  
Complex Plane ◽  
Dirac Operators ◽  
Resolvent Estimates ◽  
Eigenvalue Bounds ◽  
Mixed Norms ◽  
Massless Case ◽  
Abstract Version ◽  
Embedded Eigenvalues

AbstractWe prove that the eigenvalues of the n-dimensional massive Dirac operator $${\mathscr {D}}_0 + V$$ D 0 + V , $$n\ge 2$$ n ≥ 2 , perturbed by a potential V, possibly non-Hermitian, are contained in the union of two disjoint disks of the complex plane, provided V is sufficiently small with respect to the mixed norms $$L^1_{x_j} L^\infty _{{\widehat{x}}_j}$$ L x j 1 L x ^ j ∞ , for $$j\in \{1,\dots ,n\}$$ j ∈ { 1 , ⋯ , n } . In the massless case, we prove instead that the discrete spectrum is empty under the same smallness assumption on V, and in particular the spectrum coincides with the spectrum of the unperturbed operator: $$\sigma ({\mathscr {D}}_0+V)=\sigma ({\mathscr {D}}_0)={\mathbb {R}}$$ σ ( D 0 + V ) = σ ( D 0 ) = R . The main tools used are an abstract version of the Birman–Schwinger principle, which allows in particular to control embedded eigenvalues, and suitable resolvent estimates for the Schrödinger operator.

scholarly journals SCATTERING THEORY FOR LATTICE OPERATORS IN DIMENSION d ≥ 3

2012 ◽  
Vol 24 (08) ◽  
pp. 1250020 ◽  
Author(s):  
JEAN BELLISSARD ◽  
HERMANN SCHULZ-BALDES
Keyword(s):  
Scattering Theory ◽  
Tight Binding ◽  
Gradient Flow ◽  
Index Theorem ◽  
Finite Range ◽  
Dilation Operator ◽  
Classical Energy ◽  
Energy Gradient ◽  
Embedded Eigenvalues ◽  
Levinson Theorem

This paper analyzes the scattering theory for periodic tight-binding Hamiltonians perturbed by a finite range impurity. The classical energy gradient flow is used to construct a conjugate (or dilation) operator to the unperturbed Hamiltonian. For dimension d ≥ 3, the wave operator is given by an explicit formula in terms of this dilation operator, the free resolvent and the perturbation. From this formula, the scattering and time delay operators can be read off. Using the index theorem approach, a Levinson theorem is proved which also holds in the presence of embedded eigenvalues and threshold singularities.

scholarly journals RESONANCES ON HEDGEHOG MANIFOLDS

2013 ◽  
Vol 53 (5) ◽  
pp. 416-426 ◽  
Author(s):  
Pavel Exner ◽  
Jiří Lipovský
Keyword(s):  
Real Axis ◽  
Quantum Particle ◽  
Single Point ◽  
Three Dimensional ◽  
High Energy ◽  
Quantum Graphs ◽  
The Real ◽  
Momentum Plane ◽  
Embedded Eigenvalues ◽  
Aharonov Bohm

We discuss resonances for a nonrelativistic and spinless quantum particle confined to a two- or three-dimensional Riemannian manifold to which a finite number of semiinfinite leads is attached. Resolvent and scattering resonances are shown to coincide in this situation. Next we consider the resonances together with embedded eigenvalues and ask about the high-energy asymptotics of such a family. For the case when all the halflines are attached at a single point we prove that all resonances are in the momentum plane confined to a strip parallel to the real axis, in contrast to the analogous asymptotics in some metric quantum graphs; we illustrate this on several simple examples. On the other hand, the resonance behaviour can be influenced by a magnetic field. We provide an example of such a ‘hedgehog’ manifold at which a suitable Aharonov-Bohm flux leads to absence of any true resonance, i.e. that corresponding to a pole outside the real axis.

scholarly journals A STRONG OPERATOR TOPOLOGY ADIABATIC THEOREM

2002 ◽  
Vol 14 (06) ◽  
pp. 569-584 ◽  
Author(s):  
ALEXANDER ELGART ◽  
JEFFREY H. SCHENKER
Keyword(s):  
Spectral Data ◽  
Time Scale ◽  
Strong Operator Topology ◽  
Continuous Functions ◽  
Adiabatic Theorem ◽  
Strong Operator ◽  
Intrinsic Time ◽  
Embedded Eigenvalues ◽  
Additive Perturbation ◽  
Spectral Projections

We prove an adiabatic theorem for the evolution of spectral data under a weak additive perturbation in the context of a system without an intrinsic time scale. For continuous functions of the unperturbed Hamiltonian the convergence is in norm while for a larger class functions, including the spectral projections associated to embedded eigenvalues, the convergence is in the strong operator topology.

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