Renormalizing rectangles and other topics in random matrix theory

We consider random Hermitian matrices made of complex or realM×N rectangular blocks, where the blocks are drawn from various ensembles. These matrices haveN pairs of opposite real nonvanishing eigenvalues, as well asM–N zero eigenvalues (forM>N). These zero eigenvalues are kinematical in the sense that they are independent of randomness. We study the eigenvalue distribution of these matrices to leading order in the large-N, M limit in which the rectangularityr=M/N is held fixed. We apply a variety of methods in our study. We study Gaussian ensembles by a simple diagrammatic method, by the Dyson gas approach, and by a generalization of the Kazakov method. These methods make use of the invariance of such ensembles under the action of symmetry groups. The more complicated Wigner ensemble, which does not enjoy such symmetry properties, is studied by large-N renormalization techniques. In addition to the kinematical -function spike in the eigenvalue density which corresponds to zero eigenvalues, we find for both types of ensembles that if |r–1| is held fixed asN, theN nonzero eigenvalues give rise to two separated lobes that are located symmetrically with respect to the origin. This separation arises because the nonzero eigenvalues are repelled macroscopically from the origin. Finally, we study the oscillatory behavior of the eigenvalue distribution near the endpoints of the lobes, a behavior governed by Airy functions. Asr1 the lobes come closer, and the Airy oscillatory behavior near the endpoints that are close to zero breaks down. We interpret this breakdown as a signal thatr1 drives a crossover to the oscillation governed by Bessel functions near the origin for matrices made of square blocks.     

Weakly coupled states on branching graphs

We consider a Schrödinger particle on a graph consisting of N links joined at a single point. Each link supports a real locally integrable potential V _j ; the self-adjointness is ensured by the type boundary condition at the vertex. If all the links are semi-infinite and ideally coupled, the potential decays as x ^–1– along each of them, is nonrepulsive in the mean and weak enough, the corresponding Schrödinger operator has a single negative eigenvalue; we find its asymptotic behavior. We also derive a bound on the number of bound states and explain how the coupling constant may be interpreted in terms of a family of squeezed potentials.     

Many-body systems and the Efimov effect

We employ a Birman-Schwinger type analysis to derive estimates on the number of bound-states of certainN-body systems with threshold-energy =inf _ess(H) supposed to be zero. For many-body systems without any substructure we show that eigenvalues of the Schrödinger operatorH absorbed at =0 are in the point-spectrum ofH. Furthermore we characterize a multiparticle equivalent of the Efimov effect.     

Proof of the Landau-Zener formula in an adiabatic limit with small eigenvalue gaps

We consider a smooth operator-valued functionH(t,) that has two isolated non-degenerate eigenvaluesE _A (t,) andE _B (t,) for >0. We assume these eigenvalues are bounded away from the rest of the spectrum ofH(t,), but have an avoided crossing with one another with a closest approach that isO() as tends to zero. Under these circumstances, we study the small limit for the adiabatic Schrödinger equation

A GaAs–InGaAs optoelectronic switch with multiple operation states

An optoelectronic switch with both n- and p-type delta-doped (-doped) quantum wells was investigated. The -doped structures formed potential wells for the carrier accumulation and potential barriers for the carrier injection. Being possessed of -doped sheets with different doping levels, the potential barriers were sequentially collapsed to produce a double negative-differential-resistance (NDR) phenomenon in the current–voltage (I–V) characteristics of the device, due to the carrier accumulation in the potential wells. The device also showed an optical function related to the barrier heights controllable by incident light.     

Irreducible mass for the Tomimatsu-Sato space-times

A global definition of irreducible mass for the odd- T-S metrics is investigated. We find that its expression in terms of the source parameters is the same for all the members of the family and reduces to the formula that holds in the Kerr case (=1). As a consequence, we show that processes withm _ir = const no longer imply zero variations of the horizon's area for >1.     

Eigenvalue branches of the Schrödinger operatorH−λW in a gap of σ(H)

The authors study the eigenvalue branches of the Schrödinger operatorH — W in a gap of (H). In particular, they consider questions of asymptotic distribution of eigenvalues and bounds on the number of branches. They also address the completeness problem.     

Slow quenching for a one-dimensional kinetic Ising model: Residual energy and domain growth

A one-dimensional kinetic Ising model with Glauber dynamics subjected to a slow continuous quench to zero temperature is studied. For a rather general class of cooling schemes, described by a time-dependent temperatureT(t), the mean domain sizeL(t) is calculated along with the residual energye _res (r) as a function of the cooling rater. If the attempt frequency =₀ exp(–/kT), entering into the transition rates, is temperature dependent (i.e., the barrier is non-zero), the asymptotic growth ofL(t) is given byL()–L(t)~exp[–/kT(t)]. For this case the residual energy exhibits a power-law behaviore _res(r) ~r ^{/2(1 + )} forr small, where =4J/ andJ is the nearest neighbor coupling constant. For =0 and for certain cooling schemes the residual energy is zero andL(t)~t1/2, independent ofr.     

Fundamental studies and device application of δ-doping in GaAs Layers and in AlxGa1−xAs/GaAs heterostructures

In addition to the realization of atomically abrupt interfaces in III–V semiconductors by molecular beam epitaxy, the confinement of donor and acceptor impurities to an atomic plane normal to the crystal growth direction, called-doping, is important for the fabrication of artifically layered semiconductor structures. The implementation of-function-like doping profiles by using Si donors and Be acceptors generates V-shaped potential wells in GaAs and Al_xGa_1–xAs with a quasi-two-dimensional (2D) electron (or hole) gas. In this review we define three areas of fundamental and device aspects associated with-doping. (i) The prototype structure of-doping formed by a single atomic plane of Si donors in GaAs allows to study the 2D electron gas by magnetotransport and tunneling experiments, to study the metal-insulator transition, and to study central-cell and multivalley effects. In addition, non-alloyed ohmic contacts to GaAs and GaAs field-effect transistors (-FETs) with a buried 2D channel of high carrier density can be fabricated from-doped material. (ii) GaAs sawtooth doping superlattices, consisting of a periodic sequence of alternating n- and p-type-doping layers equally spaced by undoped regions, emit light of high intensity at wavelengths of 0.9 < <1.2 [m], which is attractive for application in photonic devices. The observed carrier transport normal to the layers due to tunneling indicates the feasibility of this superlattice as effective-mass filter. (iii) The confinement of donors (or acceptors) to an atomic (001) plane in selectively doped Al_xGa_1–xAs/GaAs heterostructures leads to very high mobilities, to high 2D carrier densities, and to a reduction of the undesired persistent photo-conductivity. These-doped heterostructures are thus important for application in transistors with improved current driving capabilities.Extended version of a paper presented at the18th Int. Symp. GaAs Related Compounds (Heraklion, Crete, 1987)     

Direction-Dependent Free Energy Singularity of the Antiferroelectric Asymmetric Six-Vertex Model

The transition from the ordered commensurate phase to the incommensurate Gaussian phase of the antiferroelectric asymmetric six-vertex model is investigated by keeping the temperature constant below the roughening point and varying the external fields (h, v). In the (h, v) plane, the phase boundary is approached along straight lines v = k h, where (h, v) measures the displacement from the phase boundary. It is found that the free energy singularity displays the exponent 3/2 typical of the Pokrovski–Talapov transition f const(h)^3/2 for any direction other than the tangential one. In the latter case f shows a discontinuity in the third derivative.     

Spectral and scattering theory for a Schrödinger operator with a class of momentum-dependent long-range potentials

Let be the selfadjoint operator for the static electromagnetic field where W _j for 0, 1, 2, ..., n is a sum of (i) a short-range potential and (ii) a smooth long-range potential decreasing at as |x|^- with in (0, 1]. Then for >1/2, asymptotic completeness holds for the scattering system (H, H ₀).     

Derivations commuting with abelian gauge actions on lattice systems

Let be an action of a compact abelian groupG on aC*-algebraA, and assume that the fixed-point subalgebraA is an AF-algebra. We show that if is a closed *-derivation onA commuting with , and the restriction of toA generates a one-parameter group of *-automorphisms, then itself is a generator. In particular, the result applies if is an infinite product action ofG on a UHF algebra. Furthermore, if in this situation ₁ and ₂ are two derivations both satisfying the hypotheses on , and ₁ and ₂ have the same restriction toA , then there exists a one-parameter subgroup of the action with generator ₀ such thatD(₁)D(₂)D(₀) is a joint core for the three derivations, and ₂=₁+₀ on this core.     

Power counting and regularization through loop integrations for multiple Feynman integrals in Minkowski space

A regularization procedure with a regularization parameter is developed which may be applied to multiple Feynman integrals in Minkowski space. The regularization is carried out inmomentum space and provides a rigorous method for studying Feynman integrals as multiple integrals in real variable theory. The regularized integrals are defined by changing the measure of integration _i dx _i to _i (1+x _i ² )^–/2 dx _i , >0, with a corresponding change defined inMinkowski space. We then develop a power counting convergence criterion for the absolute convergence of the integrals in terms of the parameter as a function of the so-called power asymptotic coefficients of Feynman integrands. An application to quantum electrodynamics is carried out.Work supported by the Department of National Defence Award under CRAD No. 3610-637:F4122.     

Small perturbations ofC*-dynamical systems

It is shown that if is the generator of a strongly continuous oneparameter group of *-automorphisms of aC*-algebraA and is an unbounded *-derivation ofA with the same domain as , then + is also a generator for all sufficiently small real numbers .     

Note on the interfacial tension of phase-separated polymer solutions

Earlier theoretical calculations of the interfacial tension of phase-separated polymer solutions as a function of the degree of polymerizationN and the temperatureT, based partly on the mean-field approximation, had led toN ^–1/4(1–T/T _c )^3/2 for fixedN1 andT approaching the critical solution temperatureT _c It is here remarked that the scaling procedure of de Gennes then modifies this toN ^–0.37(1–T/T _c )^1.26, which is in close accord with the experimentalN ^–0.44(1–T/T _c )^1.26. The simplest mean-field picture yieldsN ^–1/2(1–T/T _c )^3/2.     

A Fitting Procedure for Magnetic Multiphase Materials Mössbauer Spectra

In this work a practical method of fitting complex multiphase Mössbauer effect spectra is proposed. The task is simplified imposing specific restrictions to the analysing functions, which are appropriate for cases where the component phases spectra do not change substantially during the process under study. The ME spectra can be analysed using the phases subspectra, by defining only a reduced number of parameters. The constraints are equivalent to assume a Doppler velocity transformation v=(v–_m)B _m0/B _m+_m0 for each phase, where _m and B _m are fitting parameters containing information on the phase mean isomer shift and hyperfine field and _m0 and B _m0 their reference values. In this manner physically meaningful results are easy to obtain. The idea was applied to partially nitrogenated R₂Fe₁₇N _x (R= Sm and Y) and partially hydrogen-decomposed Nd–Fe–B materials.     

A one-dimensionalN Fermion problem with factorizedS matrix

A one dimensionalN Fermion problem with attractive or repulsive function interaction is solved by Bethe's hypothesis. TheS matrix factorizes and is explicitly given.     

Unbounded derivations of commutativeC*-algebras

It is shown that an unbounded *-derivation of a unital commutativeC*-algebraA is quasi well-behaved if and only if there is a dense open subsetU of the spectrum ofA such that, for anyf in the domain of , (f) vanishes at any point ofU wheref attains its norm. An example is given to show that even if is closed it need not be quasi well-behaved. This answers negatively a question posed by Sakai for arbitraryC*-algebras.It is also shown that there are no-zero closed derivations onA if the spectrum ofA contains a dense open totally disconnected subset.     

Correlations between eigenvalues of a random matrix

Exact analytical expressions are found for the joint probability distribution functions ofn eigenvalues belonging to a random Hermitian matrix of orderN, wheren is any integer andN. The distribution functions, like those obtained earlier forn=2, involve only trigonometrical functions of the eigenvalue differences.