Exponential Decay of Eigenfunctions of the Bethe–Salpeter Operator

Exponential decay of the bound states of the Bethe–Salpeter operator is proven. Explicit estimate on the decay rates are derived, and several norm estimates of operators involving the free Dirac operator and exponential weight functions are given.     

Phase-Averaged Transport¶for Quasi-Periodic Hamiltonians

For a class of discrete quasi-periodic Schr?dinger operators defined by covariant representations of the rotation algebra, a lower bound on phase-averaged transport in terms of the multifractal dimensions of the density of states is proven. This result is established under a Diophantine condition on the incommensuration parameter. The relevant class of operators is distinguished by invariance with respect to symmetry automorphisms of the rotation algebra. It includes the critical Harper (almost-Mathieu) operator. As a by-product, a new solution of the frame problem associated with Weyl–Heisenberg–Gabor lattices of coherent states is given. Received: 30 May 2001 / Accepted: 2 January 2002     

Determinant of Wave Operator as a Measure for Resonance and Bound States in One-Dimensional Potential Scattering System

The determinant of the wave operator or its transposed operator (source operator) works as a measure for resonance and bound states in one-dimensional potential scattering system. This fact is based on an identity proved here: the determinant of the wave or source operator equals to the transmission coefficient, which represents the field amplitude in the forward side of the scatterer. Utilizing this measure, resonance and bound states in one-dimensional system are properly assigned without solving wave equation. In future, local enhancement of optical near-field in three-dimensional system, i.e., local plasmon resonance will be treated, generalizing the present method.     

General solution of relativistic one-channel inverse scattering problem

We find explicitly in the p-representation the kernels of the logarithms of unitary operators transforming the free Hamiltonian into the general solution of the one-channel inverse scattering problem, when bound states are absent. Then we construct the transformation of the Hamiltonian adding to its spectrum the given set of bound states without changing the scattering operator. Other generators of the Poincaré group are constructed in a similar way. The proof of some relevant limits is given.     

La méthode de la fonction génératrice et les développements en quasibosons

An isomorphism between the oH' subspace of the physical bound states and a normalized oF functional space is established. This is done by introducing a generating function for these oH' states. By choosing for the oH-space the Bargmann's space, an isomorphism is obtained between an oH_F space of the fermion states and an oH_B space of the boson states. An expansion of any quasiboson operator of oH_B is thus feasible. This is particularly interesting for the Hamiltonian operator. In this case the expansion obtained, similar to the Beliaev-ZeIevinski and Marumori expansions, appear to be simple and more convergent.     

Observable estimation of bipartite mixed-state entanglement

An analytic lower bound is presented for bipartite mixed-state entanglement which is obtained from convex roof construction of a variational definition of pure-state negativity. It is shown that this lower bound can be directly measured by one single projection operator or a few local observables in experiments without the requirement of simultaneous multiple copies of states. In particular, we show that the lower bound can serve as an exact entanglement measure for isotropic states.     

Microlocalization of Resonant States and Estimates of the Residue of the Scattering Amplitude

We obtain some microlocal estimates of the resonant states associated to a resonance z₀ of an h-differential operator. More precisely, we show that the normalized resonant states are outside the set of trapped trajectories and are in the incoming area of the phase space. As an application, we show that the residue of the scattering amplitude of a Schrödinger operator is small in some directions under an estimate of the norm of the spectral projector. Finally we prove such a bound in some examples.     

Bound States for the Three-Dimensional Aharonov-Bohm Quantum Wire

The existence of bound states for the three-dimensional Aharonov-Bohm quantum wire, suggested by Dunne and Jaffe, is established provided the wire is thin enough. Recent results on the Aharonov-Bohm Schrödinger operator on a disk are used.     

Bound State Effects in Quantum Transport Theory

A quantum kinetical equation for the one-particle density operator for inhomogeneous multi component non-ideal gases is derived taking into account retardation effects and external fields. It is shown that the interactions give rise to additional contributions to the quantities in the transport equation. In some terms (e.g., internal energy, pressure, effective external force) the interaction effects are strong in the presence of bound states and yield contributions of the order exp (—E₁₀/kT) where E₁₀ is the ground state energy of bound pairs of particles. Other terms (e.g. friction forces) are rather insensitive with respect to the presence of bound states. An interpretation of these effects in terms of the mass action law is given using the principle of equivalence between bound states and particles.     

Eigenvalue problem for arbitrary linear combinations of a boson annihilation and creation operator

The eigenvalue problem for arbitrary linear combinations kα + μα^? of a boson annihilation operator α and a boson creation operator α^? is solved. It is shown that these operators possess nondegenerate eigenstates to arbitrary complex eigenvalues. The expansion of these eigenstates into the basic set of number states | n >, (n = 0, 1, 2, …), is found. The eigenstates are normalizable and are therefore states of a Hilbert space for | ζ | < 1 with ζ ? μ/k and represent in this case squeezed coherent states of minimal uncertainty product. They can be considered as states of a rigged Hilbert space for | ζ | ? 1. A completeness relation for these states is derived that generalizes the completeness relation for the coherent states | α 〉. Furthermore, it is shown that there exists a dual orthogonality in the entire set of these states and a connected dual completeness of the eigenstates on widely arbitrary paths over the complex plane of eigenvalues. This duality goes over into a selfduality of the eigenstates of the hermitian operators kα + k* α^? to real eigenvalues. The usually as nonexistent considered eigenstates of the boson creation operator α^? are obtained by a limiting procedure. They belong to the most singular case among the considered general class of eigenstates with ζ ? μ/k as a parameter.     

Fonctions Propres et Non-Existence Absolue D'Etats Liés dans Certains Systèmes Quantiques

One proves a condition of absence of bound states for a Schrödinger operator independent of any self-adjoint extension. The method of proof uses Feynman Kac formula.     

Metastable States in Parametrically Excited Multimode Hamiltonian Systems

 Consider a linear autonomous Hamiltonian system with m time periodic bound state solutions. In this paper we study their dynamics under time almost periodic perturbations which are small, localized and Hamiltonian. The analysis proceeds through a reduction of the original infinite dimensional dynamical system to the dynamics of two coupled subsystems: a dominant m-dimensional system of ordinary differential equations (normal form), governing the projections onto the bound states and an infinite dimensional dispersive wave equation. The present work generalizes previous work of the authors, where the case of a single bound state is considered. Here, the interaction picture is considerably more complicated and requires deeper analysis, due to a multiplicity of bound states and the very general nature of the perturbation's time dependence. Parametric forcing induces coupling of bound states to continuum radiation modes, of bound states directly to bound states, as well as coupling among bound states, which is mediated by continuum modes. Our analysis elucidates these interactions and we prove the metastability (long life time) and eventual decay of bound states for a large class of systems. The key hypotheses for the analysis are: appropriate local energy decay estimates for the unperturbed evolution operator, restricted to the continuous spectral part of the Hamiltonian, and a matrix Fermi Golden rule condition, which ensures coupling of bound states to continuum modes. Problems of the type considered arise in many areas of application including ionization physics, quantum molecular theory and the propagation of light in optical fibers in the presence of defects. Received: 13 March 2002 / Accepted: 2 January 2003 Published online: 14 April 2003 Communicated by J.L. Lebowitz     

An Equivalent Reduction of the Bethe-Salpeter Equation for Multi-Fermion Bound States

By means of the renormalized Dirac spinor wave function and through. the introduction of an effective interaction operator, the exact ~ethe-saypetere quation for multi-fermion bound states has been reduced to an equivalent Pauli-Schrodinger equation. As a result, the specific form of the latter equation in the static approximation has directly been given as well. In comparison of the effective interaction operator appearing in the Pauli-Schrodinger equation with the corresponding S-matrix, a substantial difference between both interactions acting in the bound state and the scattering state emerges which is important to determine an interaction potential in the bound state.     

Lower bound of local quantum uncertainty for high-dimensional bipartite quantum systems

Quantum correlations are of fundamental importance in quantum phenomena and studies related to quantum information processing. The measurement of quantum correlations is a central challenge. A recently proposed measure of quantum correlations,local quantum uncertainty(LQU), satisfies all the physical requirements as a measure of quantum correlations. This study derives a closed-form lower bound of the LQU for arbitrary-dimensional bipartite quantum states using operator relaxation. We also compared the lower bound with the optimized LQU for several typical sets of quantum states. The results show that the lower bound is near to the optimized LQU for three-dimensional bipartite quantum systems.     

Spectral problem for branching chain quantum graph

Branching chain of rings as a quantum graph is considered. We use the transfer matrix method to obtain the spectral equation. The existence of bound states is proved. The discrete spectrum of the Schrödinger operator for the system is described. We find the dependence of the eigenvalues positions on the branching angle.     

Sum rules for radiative quarkonium decays from short-range interactions

Sum rules previously derived for proton decay are extended and applied to treat effects of bound states spectroscopy on radiative quarkonium decays. The transition involves boson (photon or pion) emission followed by quark annihilation. Sum rules for the contributions from different intermediate bound states are derived by using closure and the assumptions (1) that the boson emission is described by a plane wave or multipole operator which satisfies a wave equation, and (2) that the annihilation depends on the bound state wave function or its derivative at the origin.     

Gauge-Independence and the Two-Body Problem in QED

I describe a gauge-independent approach to the relativistic two-body bound state and scattering problems in quantum field theory. The basic tool is an ordinary three-dimensional equation involving a potential operator V which gets contributions from both irreducible and reducible diagrams. In QED the resultant V is independent of the choice of covariant gauge used for the photon propagator, unlike the kernel in the Bethe–Salpeter equation. As an illustration, a problem concerning spin-independent level shifts in two-body bound states is analyzed.     

Donor-like excited states of exciton bound to neutral acceptor in ZnTe

Photoexcitation spectroscopy has been used to study the excited states of the neutral c-acceptor bound exciton complex A^c₁ in ZnTe. We have detected four excited states at ～ 11.2 meV above the bound exciton ground state. Zeeman effects on these excited states have also been studied. The results show that they correspond to excitations of the bound electron to donor-like 2p and 2s orbital states. This represents an unambiguous experimental evidence of the pseudo-donor model previously suggested by Rühle and Bimberg for acceptor bound exciton complexes when m_e ? m_h.     

Entanglement capacity of two-qubit unitary operator for rank two mixed states

The entanglement capacity of two-qubit unitary operator acting on rank two mixed states in concurrence is discussed. The condition of perfect entangler is the same as that acting on pure states and the entanglement capacity is the mixing parameter v1. For non-perfect entangler, the upper and lower bound of the entanglement capacity are given.