Geometric theory of Stark resonances in multielectron systems

In this paper we consider a class of many-body systems in a weak homogeneous electric field. This class includes atoms and molecules with infinitely heavy nuclei. It follows from one of the results of this paper and a result of [S 3] that the bound states of such systems in the absence of electric field turn into resonances (which we call the Stark resonances) as soon as the electric field is switched on. (The stability part of this result was earlier proven in [HeSi] (see also [Hu 2]) under an assumption of dilation analyticity.) The main result of this paper is exponential bounds on the width (and therefore the lower exponential bounds on the life-time) of the Stark resonances. These bounds are given in terms of the Stark instanton action. In contrast to the usual (one body) action the latter is not entirely classical but incorporates certain quantum data (like ionization energies). The bounds give a partial generalization to the many electron case of the well-known Oppenheimer formula for the hydrogen.Research support by NSERC under grant NA 7901     

Existence of Stark ladder resonances

We show that resonances, in the translation analytic sense of Herbst and Howland, exist for the one dimensional Stark Hamiltonian, –d ²/dx ²+q(x)+x, withq(x) a trigonometric polynomial, provided is sufficiently large.Work partially supported by NSF grant DMS-8409630     

Dielectric resonances in disordered media

Binary disordered systems are usually obtained by mixing two ingredients in variable proportions: conductor and insulator, or conductor and super-conductor. They present very specific properties, in particular the second-order percolation phase transition, with its fractal geometry and the multi-fractal properties of the current moments. These systems are naturally modeled by regular bi-dimensional or tri-dimensional lattices, on which sites or bonds are chosen randomly with given probabilities. The two significant parameters are the ratio h = σ ₁/σ of the complex conductances, σ and σ ₁, of the two components, and their relative abundances p (or, respectively, 1 - p). In this article, we calculate the impedance of the composite by two independent methods: the so-called spectral method, which diagonalises Kirchhoff's Laws via a Green function formalism, and the Exact Numerical Renormalization method (ENR). These methods are applied to mixtures of resistors and capacitors (R-C systems), simulating e.g. ionic conductor-insulator systems, and to composites constituted of resistive inductances and capacitors (LR-C systems), representing metal inclusions in a dielectric bulk. The frequency dependent impedances of the latter composites present very intricate structures in the vicinity of the percolation threshold. In this paper, we analyse the LR-C behavior of compounds formed by the inclusion of small conducting clusters (“n-legged animals”) in a dielectric medium. We investigate in particular their absorption spectra who present a pattern of sharp lines at very specific frequencies of the incident electromagnetic field, the goal being to identify the signature of each animal. This enables us to make suggestions of how to build compounds with specific absorption or transmission properties in a given frequency domain. Received 16 August 2002 Published online 14 February 2003 RID="a" ID="a"e-mail: laurent.raymond@l2mp.fr RID="b" ID="b"e-mail: steffen.schaefer@l2mp.fr RID="c" ID="c"UMR CNRS 6137     

Nonlinear delocalization on disordered Stark ladder

We study effects of weak nonlinearity on localization of waves in disordered Stark ladder corresponding to propagation in presence of disorder and a static field. Our numerical results show that nonlinearity leads to delocalization with subdiffusive spreading along the ladder. The exponent of spreading remains close to its value in absence of the static field. The delocalization implies the existence of statistical entanglement between far away parts of the spreading wave packet indicating importance of long-range effects.     

The universality of phase resonances for disordered systems in a strong magnetic field

The density of states of a three-dimensional disordered system in a strong magnetic field with a periodic back-ground potential is examined. The calculation of the density of states is reduced to the investigation of an effective one-dimensional problem. In the limit of weak disorder the density of states shows anomalies whenever the ratio of the wavelength and the lattice constant is a rational number. These phase resonances are investigated in detail for the band edges and for the middle of the band.     

Stark ladder resonances for small electric fields

We prove the existence of resonances in the semi-classical regime of smallh for Stark ladder Hamiltonians in one-dimension. The potentialv is a real periodic function with period which is the restriction to of a function analytic in a strip about . The electric field strengthF satisfies the bounds |v|>F>0. In general, the imaginary part of the resonances are bounded above by, for some 0<1, where _T h ^-1 is the single barrier tunneling distance in the Agmon metric forv+Fx. In the regime where the distance between resonant wells is , we prove that there is at least one resonance whose width is bounded above byce ^–/F , for some ,c>0 independent ofh andF forh sufficiently small. This is an extension of the Oppenheimer formula for the Stark effect to the case of periodic potentials.Partially supported by NSF Grant DMS-8911242     

Dielectric resonances in three-dimensional binary disordered media

Powdered solids often present very specific properties due to their granular nature. Such powders are often obtained by mixing two ingredients in variable proportions: conductor and insulator, or conductor and super-conductor. In a very natural way, these systems are modeled by regular lattices, whose sites or bonds are randomly chosen with given probabilities. It is known that the electrical and optical properties of random bi-dimensional (2D) networks are well described by their conductance's poles (resonances) and residues (amplitudes). The numerical implementation of a spectral method gave the spectral density, the AC conductivity, the multi-fractal properties of the moments for the local electric field (or currents), and spectrum of resonances characteristic of some small clusters (animals). This work extends the spectral method to the three-dimensional (3D) case where the problem is more complicated because the duality property and the corresponding symmetries are broken. As in the 2D-case, the two significant parameters are the ratio of the complex conductances and of both phases, and the probability p (resp. 1-p) of (resp. ). All the resonances lie on the negative real h-axis, i.e. for pure non resistive networks in the AC case. For a static (DC) system, only the value h=0 (corresponding to a binary system with finite and , or and finite) can give a resonance. Some applications are proposed, in particular the ability for small clusters (animals with one, two or three bonds) to present a singular response for well identified frequencies of the incident electromagnetic field. Received 24 March 1999     

Stark resonances: Asymptotics and distributional Borel Sum

We prove that the Stark effect perturbation theory of a class of bound states uniquely determines the position and the width of the resonances by Distributional Borel Sum. In particular the small field asymptotics of the width is uniquely related to the large order asymptotics of the perturbation coefficients. Similar results apply to all the resonances of the anharmonic and double well oscillators.Partially supported by Ministero della Università e della Ricerca Scientifica     

Asymptotics of resonances for 1D Stark operators

We consider the Stark operator perturbed by a compactly supported potential on the real line. We determine the forbidden domain for resonances, asymptotics of resonances at high energy and asymptotics of the resonance counting function for large radius.     

Lower bounds on width of Stark resonances in one dimension

We prove a lower bound for the width of Stark resonances in one dimension.     

Phase resonances in one-dimensional disordered systems-analogy to classical mechanics

According to an analogy to classical mechanics phase resonances in disordered systems occur because of the sensitivity of closed orbits of an integrable system to an additional noise. This analogy permits a unified treatment of the phase resonances, which occur in disordered systems with a periodic background potential. Starting from a general model, we confirm exact results of Derrida and Gardner for the tight-binding model, discuss the influence of a finite correlation length in the nearly free electron case and generalize the work of Gorkov and Dorochov.     

Stark ladders of resonances: Wannier ladders and perturbation theory

LetH _B be any fixed one-dimensional Bloch Hamiltonian with only the firstm gaps open andH _F=H_B+F_x be the corresponding Stark Hamiltonian. For any positiveF small enoughH _F has onlym ladders of sharp resonances given by the analytic translation method, the decoupled band approximation and the regular perturbation theroy. This way, the Wannier conjecture becomes a definite regular perturbation theory for the Stark ladders as eigenvalues of the translated Hamiltonian.     

A note on the modulation of Stark field induced resonances

Clusters of resonances have been observed to modulate in a regular manner in the Stark spectra of the potassium atom. This phenomenon is due to the manifold structure of the atom in an electric field. The energy spacings of the regular features are related to the modulation caused by the shortest precessing trajectories but the correspondence is not exactly one-to-one.     

Discontinuities in disordered systems

The entropyS _T (j) of a two-dimensional Ising spin glass with an independent distribution of the random couplingp(J)=x·δ(J+1)+(1-x)δ(J-j) is discontinuous for temperatureT=0 and rationalj>0 and continuous elsewhere. The integrated density of frequenciesk _M (ω ²) of an one-dimensional chain of coupled oscillators with an independent distribution of the random massesp(m)=x·δ(m-1)+(1-x)δ(m-M) has the same behaviour, whereω ² corresponds toj andM to 1/T. The discontinuity points for infiniteM are, for sufficiently large but finiteM, special, frequencies, wherek _M (ω ²) has a Lifshitz singularity.     

Wannier–Stark resonances in Zener tunneling diodes with GaAs/AlAs superlattices

We present DC transport measurements of the valence to conduction band (Zener) tunneling current in ap–i–ndiode with an ultrathin intrinsic layer containing a (GaAs)₅/(AlAs)₂multi-quantum well structure. According to recent theoretical predictions, the DC current should show maxima as a function of the reverse bias voltage that reflect the formation of Wannier–Stark resonances. So far, Wannier–Stark resonances have only been observed optically and never in a regime of strong Zener tunneling. Experimentally, we find the second derivative of the current-voltage characteristics to show a weak oscillatory structure indeed, indicating the existence of Wannier–Stark resonances in Zener tunneling.     

Off-perturbative states in disordered systems

The systematic approach for the off-perturbative calculations in disordered systems is developed. The proposed scheme is applied for the random temperature and the random field ferromagnetic Ising models. It is shown that away from the critical point, in the paramagnetic phase of the random temperature model, and in the ferromagnetic phase of the random field one, the free energy contains non-analytic contributions which have the form of essential singularities. It is demonstrated that these contributions appear due to localized in space instanton-like excitations.     

Instanton in disordered Peierls systems

We study disordered Peierls systems described by the fluctuating gap model. We show that the typical electron states with energies lying deep inside the pseudogap are localized near large disorder fluctuations (instantons), which have the form of a soliton-antisoliton pair. Using the “saddle-point” method we obtain the average density of states and the average optical absorption coefficient at small energy.     

Wave functions in disordered systems

Particular solutions of the stationary Schrödinger equation for ad-dimensional disordered tight binding model are found. The particular solution is defined by boundary conditions on one face of the system. The determination of the rate of growth of the mean square wave function leads to an exactly soluble eigenvalue problem ind – 1 dimensions. Ford 2 there are three types of particular wave functions in which the mean square amplitude (a) grows exponentially (b) decays exponentially (c) does not grow or decay but oscillates.Supported in part by the National Science Foundation under grant No. DMR 78-10276.