A multi-channel scattering theory for some time dependent Hamiltonians,charge transfer problem

Scattering theory for time dependent HamiltonianH(t)=?(1/2) Δ+ΣV _j (x?q _j (t)) is discussed. The existence, asymptotic orthogonality and the asymptotic completeness of the multi-channel wave operators are obtained under the conditions that the potentials are short range: |V _j (x)|≦C _j (1+|x|)^?2?ε, roughly spoken; and the trajectoriesq _j (t) are straight lines at remote past and far future, and |q _j (t)?q _k (t)| → ∞ ast → ± ∞ (j≠k).     

Reconstruction of singularities for solutions of Schrödinger's equation

We determine the behavior in time of singularities of solutions to some Schrödinger equations onR ⁿ . We assume the Hamiltonians are of the formH ₀+V, where \(H_0 = 1/2\Delta + 1/2 \sum\limits_{k = 1}^n { \omega _k^2 x_k^2 } \) , and whereV is bounded and smooth with decaying derivatives. When all ω _k =0, the kernelk(t,x,y) of exp (?itH) is smooth inx for every fixed (t,y). When all ω₁ are equal but non-zero, the initial singularity “reconstructs” at times \(t = \frac{{m\pi }}{{\omega _1 }}\) and positionsx=(?1) ^m y, just as ifV=0;k is otherwise regular. In the general case, the singular support is shown to be contained in the union of the hyperplanes \(\{ x|x_{js} = ( - 1)^l js_{y_{js} } \} \) , when ω _j t/π=l _j forj=j ₁,...,j _r .     

The length and the relative length of tangent vectors of finsler spaces

We consider the length of a vector in a Finsler space with the fundamental function L(x,y). The length of a vector X is usually defined as the value L(x,X) of L. On the other hand, we have an essential tensor g_ij(x,y), called the fundamental tensor, and the concept of relative length |X_y| of X may be introduced by |X|_y^y = g_ij(x,y)XⁱX^j with re spect to a supporting element y. The question arises whether is L(x,X) the minimum of |X|_y or not? If there exists a supporting element y satisfying |X|_y < L(x,X), then a curve x(t) in the Finsler space will be measured shorter than the usual length, by integrating |dx/dt|_y with the field of such supporting element y(t) along the curve.     

Computable upper bounds for the adiabatic approximation errors

For a given Hermitian Hamiltonian H(s)(s∈[0,1])with eigenvalues Ek(s)and the corresponding eigenstates|Ek(s)(1 k N),adiabatic evolution described by the dilated Hamiltonian HT(t):=H(t/T)(t∈[0,T])starting from any fixed eigenstate|En(0)is discussed in this paper.Under the gap-condition that|Ek(s)-En(s)|λ0 for all s∈[0,1]and all k n,computable upper bounds for the adiabatic approximation errors between the exact solution|ψT(t)and the adiabatic approximation solution|ψadi T(t)to the Schr¨odinger equation i|˙ψT(t)=HT(t)|ψT(t)with the initial condition|ψT(0)=|En(0)are given in terms of fidelity and distance,respectively.As an application,it is proved that when the total evolving time T goes to infinity,|ψT(t)-|ψadi T(t)converges uniformly to zero,which implies that|ψT(t)≈|ψadi T(t)for all t∈[0,T]provided that T is large enough.     

Asymptotic inverse spectral problem for anharmonic oscillators

We study perturbationsL=A+B of the harmonic oscillatorA=1/2(??²+x ²?1) on ?, when potentialB(x) has a prescribed asymptotics at ∞,B(x)～|x|^?α V(x) with a trigonometric even functionV(x)=Σa _mcosω _m x. The eigenvalues ofL are shown to be λ _k =k+μ _k with small μ _k =O(k ^?γ), γ=1/2+1/4. The main result of the paper is an asymptotic formula for spectral fluctuations {μ _k }, $$\mu _k \sim k^{ - \gamma } \tilde V(\sqrt {2k} ) + c/\sqrt {2k} ask \to \infty ,$$ whose leading term \(\tilde V\) represents the so-called “Radon transform” ofV, $$\tilde V(x) = const\sum {\frac{{a_m }}{{\sqrt {\omega _m } }}\cos (\omega _m x - \pi /4)} .$$ as a consequence we are able to solve explicitly the inverse spectral problem, i.e., recover asymptotic part |x ^?α|V(x) ofB from asymptotics of {µ _k }. ₁ ^∞      

Time Quasi-Periodic Unbounded Perturbations¶of Schrödinger Operators and KAM Methods

We eliminate by KAM methods the time dependence in a class of linear differential equations in ℓ² subject to an unbounded, quasi-periodic forcing. This entails the pure-point nature of the Floquet spectrum of the operator H ₀+εP(ωt) for ε small. Here H ₀ is the one-dimensional Schr?dinger operator p ²+V, V(x)∼|x|^α, α <2 for |x|→∞, the time quasi-periodic perturbation P may grow as |x|^β, β <(α−2)/2, and the frequency vector ω is non resonant. The proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on Kuksin's estimate of solutions of homological equations with non-constant coefficients. Received: 3 October 2000 / Accepted: 20 December 2000     

Intermediate-range potential and the 1S0neutron-proton transition matrix

The half-shell transition matrix t(p, k) for the singlet s-wave neutron-proton interaction has been studied for a class of partly non-local potentials. The potentials have been generated from the experimental phase shifts by using inverse scattering theory. The non-uniqueness of the inversion solution has been exploited to construct potentials with different short-range behaviour. It is shown that the intermediate-range potential essentially determines t(p, k) for momenta p and k both less than 2fm^?. Different short-range behaviour is reflected in t(p, k) for larger momenta. The unknown high-energy phase shifts imply, however, comparable variations in that region even if the on-shell momentum k is small. The implications for nuclear structure calculations are discussed.     

Large deviations and Lifshitz singularity of the integrated density of states of random Hamiltonians

We consider the integrated density of states (IDS) ρ_∞(λ) of random Hamiltonian H_ω=?Δ+V_ω, V_ω being a random field on ? ^d which satisfies a mixing condition. We prove that the probability of large fluctuations of the finite volume IDS |Λ|^?1ρ(λ, H_Λ(ω)), Λ ? ? ^d , around the thermodynamic limit ρ_∞(λ) is bounded from above by exp {?k|Λ|},k>0. In this case ρ_∞(λ) can be recovered from a variational principle. Furthermore we show the existence of a Lifshitztype of singularity of ρ_∞(λ) as λ → 0⁺ in the case where V_ω is non-negative. More precisely we prove the following bound: ρ_∞(λ)≦exp(?kλ^?d/2) as λ → 0⁺ k>0. This last result is then discussed in some examples.     

Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity

We consider the nonlinear Schrödinger equation: (1) $${{i\partial u} \mathord{\left/ {\vphantom {{i\partial u} {\partial t}}} \right. \kern-\nulldelimiterspace} {\partial t}} = - \Delta u - \left| u \right|^{{4 \mathord{\left/ {\vphantom {4 N}} \right. \kern-\nulldelimiterspace} N}} uandu\left( {0,.} \right) = \varphi \left( . \right),$$ whereu:[0,T)×? ^N →?. For any given pointsx ₁,x ₂,...,x _k in ? ^N , we construct a solution of Eq. (1),u(t), which blows up in a finite timeT at exactlyx ₁,x ₂,...,x _k . In addition, we describe the precise behavior of the solutionu(t) whent→T, at the blow-up points {x ₁,x ₂,...,x _k } and in ? ^N ?{x ₁,x ₂,...,x _k }.     

Absence of Negative Energy Spectrum for N-Particle Hamiltonians

We consider the spectrum of the quantum Hamiltonian H for a system of N one-dimensional particles. H is given by $H = \sum\nolimits_{i = 1}^n { - \frac{1}{{2m_i }}\frac{{\partial ^2 }}{{\partial x_i^2 }}} + \sum {_{1 \leqslant i < j \leqslant N} } V_{ij} \left( {x_i - x_j } \right)$ acting in L ²(R ^N ). We assume that each pair potential is a sum of a hard core for |x|≤a, a>0, and a function V _ij (x), |x|>a, with $\smallint _a^\infty \left| {x - a} \right|\left| {V_{ij} \left( x \right)} \right|dx < \infty $ . We give conditions on V ^? _ij (x), the negative part of V _ij (x), which imply that H has no negative energy spectrum for all N. For example, this is the case if V ^? _ij (x) has finite range 2a and $$2m_i \smallint _a^{2a} \left| {x - a} \right|\left| {V_{ij}^ - \left( x \right)} \right|dx < 1.$$ If V ^? _ij is not necessarily small we also obtain a thermodynamic stability bound inf?σ(H)≥?cN, where 0<c<∞, is an N-independent constant.     

Measurements of elastic scattering in alpha-alpha and alpha-proton collisions at the CERN intersecting storage rings

We measured the elastic scattering of $\text{αα}\text{at}\text{s}\text{= 126}\text{GeV}$ and of αp at $\text{s}\text{= 89}\text{GeV}$. For αα, the differential cross section dσ/dt has a diffractive pattern minima at |t| = 0.10 and 0.38 GeV². At small |t| = 0.05?0.07 GeV², this cross section behaves like exp[(100 ± 10) t]. Extrapolating a fit to the data to the optical point, we obtained for the total cross section α_tot(αα) = 250 ± 50 mb and an integrated elastic cross section σ_e1(αα) = 45 ± mb. Another method of estimating σ_tot(αα), based on measuring the interaction rate, yielded 295 ± 40 mb. For αp, dσ/dt has aminimum at |t| = 0.20 GeV², and for 0.05 < |t| < 0.18 GeV² behaves like exp[(41 ± 2) t]. Extrapolating this slope to |t| = 0, we found σ_tot(αp) = 130 ± 20 and σ_e1(αp) = 20 ± 4mb. Results on pp elastic scattering at $\text{s}\text{= 63}\text{GeV}$ agree with previous ISR experiments.     

Asymptotic completeness for multi-particle schroedinger Hamiltonians with weak potentials

We show that the non-relativistic quantum mechanicaln-body HamiltoniansT(k)=T+kV andT, the free particle Hamiltonian, are unitarily equivalent in the center of mass system, i.e.,T(k)=W _± (k)TW _± (k) ^–1 fork sufficiently small and real. , a sum ofn(n–1)/2 real pair potentials,V _i, depending on the relative coordinatex _i R ³ of the pairi, whereV _i is required to behave like |x_i|^{– 2 –} as |x _i | and like |x_i|^{– 2 +} as |x _i |0.T(k) is the self-adjoint operator associated with the form sumT+kV. There are no smoothness requirements imposed on theV _i . Furthermore are the wave operators of time dependent scattering theory and are unitary. This result gives a quantitative form of the intuitive argument based on the Heisenberg uncertainty principle that a certain minimum potential well depth and range is needed before a bound state can be formed. This is the best possible long range behavior in the sense that ifkV _i C _i |x _i |^–b , 0<b2 for |x _i |>R _i (0<R _i <) and allC _i are negative thenT(k) has discrete eigenvalues andW _±(k) are not unitary.     

The generalized exponential-integral V(x,y)=∫1∞ exp(−xt)ln(t+y)dtt and computer algorithms for y = 0, 1

The generalized exponential-integral function V(x, y) defined here includes as special cases the function E⁽²⁾₁(x) = V(x, 0) introduced by van de Hulst and functions M₀(x) = V(x, 1) and N₀(x) = V(x, -1) introduced by Kourganoff in connection with integrals of the form ∫ E_n)t)E_m(t±x), which play an important role in the theory of monochromatic radiative transfer. Series and asymptotic expressions are derived and, for the most important special cases, y = 0 and y = 1, Chebyshev expansions and rational approximations are obtained that permit the function to be evaluated to at least 10 sf on 0<x<∞ using 16 sf arithmetic.     

Low Temperature Properties for Correlation Functions in Classical N-Vector Spin Models

We obtain convergent multi-scale expansions for the one-and two-point correlation functions of the low temperature lattice classical N - vector spin model in d S 3 dimensions, N S 2. The Gibbs factor is taken as exp[-b(1/2 ||?f||² +l/8 || |f|² - 1 ||² + v/2||f- h||²)], \exp [-\beta (1/2 ||\partial \phi||^2 +\lambda/8 ||\, |\phi|^2 - 1 ||^2 + v/2||\phi - h||^2)], where f(x), h ? R^N\phi(x), h \in R^N, x ? Z^dx \in Z^d, |h|=1, b < ￥|h|=1, \beta < \infty, l 3 ￥\lambda \geq \infty are large and 0 < v h 1. In the thermodynamic and v ˉ 0v \downarrow 0 limits, with h = e₁, and j L ‘^½ ‘, the expansion gives áf₁(x)? = 1+0(1/b^1/2)\langle \phi_1(x)\rangle = 1+0(1/\beta^{1/2}) (spontaneous magnetization), áf₁(x)f_i(y)? = 0\langle \phi_1(x)\phi_i(y)\rangle=0, áf_i (x)f_i (y)? = c₀ D^-1(x,y)+R(x,y)\langle \phi_i (x)\phi_i (y)\rangle = c_0 \Delta^{-1}(x,y)+R(x,y) (Goldstone Bosons), i = 2, 3, ?, Ni= 2, 3,\,\ldots, N, and áf₁(x)f₁(y)?^T=R￠(x,y)\langle \phi_1(x)\phi_1(y)\rangle^T=R'(x,y), where |R(x,y)||R(x,y)|, |R￠(x,y)| < 0(1)(1+|x-y|)^d-2+r|R'(x,y)|< 0(1)(1+|x-y|)^{d-2+\rho} for some „ > 0, and c₀ is aprecisely determined constant.     

Approach to saturation of the magnetization of dilute AuFe alloys

We present a reinterpretation of our recent measurements of the magnetic properties of some dilute AuFe alloys. We find that the observed approach to saturation of the magnetization for these AuFe alloys can be understood if both single-impurity (Kondo) effects and effects due to interactions between impurities via the Rudeman-Kittel-Kasuya-Yosida (RKKY) interaction, V(r) = (V₀ cos 2k_Fr)/r³, are properly included in the analysis. The analysis yields for the strength of the RKKY interaction V₀ = (1.1 ± 0.3) × 10^-36ergcm³, for the s-d exchange parameter |J| = (1.9 ± 0.3) eV, and for the Kondo temperature T_K = (0.8 ± 0.1) K. We conclude that mean free path effects do not significantly influence the observed approach to saturation of the magnetization for the AuFe alloys studied.     

On the invariants of the complex Ginzburg-Landau equation

Using Lie group theory, both the invariants and the similarity variables of the complex Ginzburg—Landau equation V_xx = a(V_t + bV) + cV|V|^2kwherea,b,c?Iandk?I are constructed.     

Self-similar magnetic structures and magnetic flux “Giant” creep

The penetration of a magnetic flux into a type-II high-T _c superconductor occupying the half-space x > 0 is considered. At the superconductor surface, the magnetic field amplitude increases in accordance with the law b(0, t) = b ₀(1 + t)^m (in dimensionless coordinates), where m > 0. The velocity of penetration of vortices is determined in the regime of thermally activated magnetic flux flow: v = v ₀exp?ub;?(U _0/T )(1-b?b/?x)?ub;, where U ₀ is the effective pinning energy and T is the thermal energy of excited vortex filaments (or their bundles). magnetic flux “Giant” creep (for which U ₀/T? 1) is considered. The model Navier-Stokes equation is derived with nonlinear “viscosity” v ∝ U ₀/T and convection velocity v _f ∝ (1 ? U ₀/T). It is shown that motion of vortices is of the diffusion type for j → 0 (j is the current density). For finite current densities 0 < j < j _c, magnetic flux convection takes place, leading to an increase in the amplitude and depth of penetration of the magnetic field into the superconductor. It is shown that the solution to the model equation is finite at each instant (i.e., the magnetic flux penetrates to a finite depth). The penetration depth x _eff ^A (t) ∝ (1 + t)^{(1 + m/2)/2} of the magnetic field in the superconductor and the velocity of the wavefront, which increases linearly in exponent m, exponentially in temperature T, and decreases upon an increase in the effective pinning barrier, are determined. A distinguishing feature of the solutions is their self-similarity; i.e., dissipative magnetic structures emerging in the case of giant creep are invariant to transformations b(x, t) = β^m b(t/β, x/β^{(1 + m/2)/2}), where β > 0.     

The Ruderman-Kittel-Kasuya-Yosida interaction in amorphous La80Au20 alloys with dilute Gd impurities

From magnetization measurements on some amorphous dilute La_80?xGd_xAu₂₀ alloys with x ? 1 we have shown that the magnetic behavior follows the scaling laws of a spin-glass system, characteristic of the 1/r³ dependence of the pairwise interaction. We have also determined the strength of the Ruderman-Kittel-Kasuya-Yosida interaction V(r) = (V₀cos 2k_Fr)/r³, to be V₀ = 0.20 × 10^?37 ergcm³. The corresponding value of the s-f exchange integral is |J_sf| = 0.14 eV, which is compared with values determined from other experiments.     

Phase transition and phonon-drag conductivity in quasi-unidimensional systems

We investigate the properties of a quasi-unidimensional system which exhibits Peierls instability at T_c(μ), where μ is the Fermi energy as measured from the middle of the conduction band. T_c(μ) decreases as 6μ6 increases. The phonon-drag diagrams contribute to fluctuation conductivity, which is proportional to $\text{(T ? T}{}_{\mathrm{c}}\text{)}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$, if |μ| > 1.056 T_c(0). The Kohn Anomalies do not occur at 2k_F unless μ = 0 or |μ| ? T.     

The local structure of the spectrum of the one-dimensional Schrödinger operator

Let \(H_V = - \frac{{d^{\text{2}} }}{{dt^{\text{2}} }} + q(t,\omega )\) be an one-dimensional random Schrödinger operator in ?²(?V,V) with the classical boundary conditions. The random potentialq(t, ω) has a formq(t, ω)=F(x _t ), wherex _t is a Brownian motion on the compact Riemannian manifoldK andF:K→R ¹ is a smooth Morse function, \(\mathop {\min }\limits_K F = 0\) . Let \(N_V (\Delta ) = \sum\limits_{Ei(V) \in \Delta } 1 \) , where Δ∈(0, ∞),E _i (V) are the eigenvalues ofH _V . The main result (Theorem 1) of this paper is the following. IfV→∞,E ₀>0,k∈Z ₊ anda>0 (a is a fixed constant) then $$P\left\{ {N_V \left( {E_0 - \frac{a}{{2V}},E_0 + \frac{a}{{2V}}} \right) = k} \right\}\xrightarrow[{V \to \infty }]{}e^{ - an(E_0 )} (an(E_0 ))^k |k!,$$ wheren(E ₀) is a limit state density ofH _V ,V→∞. This theorem mean that there is no repulsion between energy levels of the operatorH _V ,V→∞. The second result (Theorem 2) describes the phenomen of the repulsion of the corresponding wave functions.