Imaginary-time path integral for a relativistic spinless particle in an electromagnetic field

A rigorous path integral representation of the solution of the Cauchy problem for the pure-imaginary-time Schrödinger equation _t (t, x)=–[H–mc ²](t,x) is established.H is the quantum Hamiltonian associated, via the Weyl correspondence, with the classical Hamiltonian [(cp–eA(x))²+m ² c ⁴]^1/2+e(x) of a relativistic spinless particle in an electromagnetic field. The problem is connected with a time homogeneous Lévy process.     

The Biot-Savart-Ampère law and the vacuum fieldB (3)

It is shown that the longitudinal vacuum fieldB ⁽³⁾ emerges from the Biot-Savart-Ampère law governing the motion of an electron with intrinsic spin moving at the speed of light, in which case the expression forB ⁽³⁾ is identical with that obtained from the Dirac equation of one electron accelerated to the speed of light by an electromagnetic field. Use of an O(3), non-Abelian, gauge geometry forB ⁽³⁾ identifies the quantized photon momentum appearing in the Dirac equation witheA ⁽⁰⁾, wheree is the charge on the electron andA ⁽⁰⁾ the amplitude of the vector potential. The condition =eA ⁽⁰⁾ can be obtained in turn from the relativistic Hamilton-Jacobi equation of an electron accelerated to the speed of light by an electromagnetic field.     

Exponential decay in the stark effect

LetH=–+V+Fx ₁ withV(x ₁,x ) analytic in the first variable andV(x ₁+ia, x ) bounded and decreasing to zero asx for eacha . Let be an eigenvector of –+V with negative eigenvalue. Among our results we show that forF0, (,e ^–H ) decays exponentially at a rate governed by the positions of the resonances ofH. This exponential decay is in marked contrast to conventional atomic resonances for which power law decay is the rule.Research supported by NSF Grant No. MCS 78-00101.     

Invariant curves for area-preserving twist maps far from integrable

One-parameter families of area-preserving twist maps of the formF (x, y)=(x +y +f(x),y +f(x)) are considered. Various invariant curves, for the maps corresponding tof(x)=sin andf(x)=sinx+(1/50) sin(5x), are rigorously constructed forlarge values of the nonlinearity parameter . For larger values of , close to critical, some numerical experiments are briefly discussed.     

Construction of analytic KAM surfaces and effective stability bounds

A class of analytic (possibly) time-dependent Hamiltonian systems withd degrees of freedom and the corresponding class of area-preserving, twist diffeomorphisms of the plane are considered. Implementing a recent scheme due to Moser, Salamon and Zehnder, we provide a method that allows us to construct explicitly KAM surfaces and, hence, to give lower bounds on their breakdown thresholds. We, then, apply this method to the HamiltonianHy ²/2+(cosx+cos(x–t)) and to the map (y,x)(y+ sinx,x+y+ sinx) obtaining, with the aid of computer-assisted estimations, explicit approximations (within an error of 10^–5) of the golden-mean KAM surfaces for complex values of with || less or equal than, respectively, 0.015 and 0.65. (The experimental numerical values at which such surfaces are expected to disappear are about, respectively, 0.027 and 0.97.) A possible connection between break-down thresholds and singularities in the complex -plane is pointed out.To our friend and colleague Paola CalderoniSupported by Consiglio Nazionale delle Ricerche, Italy     

On the asymptotic behavior of the solutions of the Caldirola-Kanai equation

We study in this Letter the asymptotic behavior, as t+, of the solutions of the one-dimensional Caldirola-Kanai equation for a large class of potentials satisfying the condition V(x)+ as |x|. We show, first of all, that if I is a closed interval containing no critical points of V, then the probability P _t (t) of finding the particle inside I tends to zero as t+. On the other hand, when I contains critical points of V in its interior, we prove that P _t (t) does not oscillate indefinitely, but tends to a limit as t+. In particular, when the potential has only isolated critical points x ₁, ..., x _N our results imply that the probability density of the particle tends to in the sense of distributions.Supported by Fulbright-MEC grant 85-07391.     

Boson fields under a general class of cut-off interactions

We consider a boson field (x) under an interaction of the form V( _k (x))dx, where _k (x) is the momentum cut-off field, andV() is a continuous bounded function. Under a weak regularity condition onV(), we prove that the total energy operator is self adjoint, that the asymptotic fields exist and that the scattering operator exists.     

Vector two-point functions in maximally symmetric spaces

We obtain massive and massless vector two-point functions in maximally symmetric spaces (and vacua) of any number of dimensions. These include de Sitter space and anti-de Sitter space, and their Euclidean analogsS ⁿ andH ⁿ. Our method is based on a simple way of constructing every possible maximally symmetric bitensorT _a...bc...d(x, x) which carries tangent-space indicesa...b atx andc...d atx.     

Energy spectrum and angular distribution of muons from the decay of heavy leptons produced in colliding electron-positron beams

On the basis of an arbitrary (V, A) structure of the neutral weak ¯ee and LL currents (L=^–, M⁰) a study is made of the processes of production in colliding electron-positron beams of pairs of heavy leptons with subsequent decays in accordance with the schemes e⁺e^––(µ^–v_µv) + ⁺( anything) and e⁺e^–M⁰(µ^–e⁺v_e) + M⁰( anything). The energy spectrum and asymmetry of the distribution of the produced muons are investigated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 66–70, February, 1981.     

Existence of solutions for Schrödinger evolution equations

We study the existence, uniqueness and regularity of the solution of the initial value problem for the time dependent Schrödinger equationiu/t=(–1/2)u+V(t,x)u,u(0)=u ₀. We provide sufficient conditions onV(t,x) such that the equation generates a unique unitary propagatorU(t,s) and such thatU(t,s)u ₀C ¹(,L ²) C ⁰(H ²( ⁿ )) foru ₀H ²( ⁿ ). The conditions are general enough to accommodate moving singularities of type x^–2+(n4) or x^–n/2+(n3).     

Scaling solutions of Smoluchowski's coagulation equation

We investigate the structure of scaling solutions of Smoluchowski's coagulation equation, of the formc _k (t)s(t)^– (k/s(t)), wherec _k (t) is the concentration of clusters of sizek at timet,s(t) is the average cluster size, and(x) is a scaling function. For the rate constantK(i, j) in Smoluchowski's equation, we make the very general assumption thatK(i, j) is a homogeneous function of the cluster sizesi andj:K(i,j)=a ^– K(ai,aj) for alla>0, but we restrict ourselves to kernels satisfyingK(i, j)/j0 asj. We show that gelation occurs if>1, and does not occur if1. For all gelling and nongelling models, we calculate the time dependence ofs(t), and we derive an equation for(x). We present a detailed analysis of the behavior of(x) at large and small values ofx. For all models, we find exponential large-x behavior: (x)A x ^– e ^–x asx and, for different kernelsK(i, j), algebraic or exponential small-x behavior: (x)Bx ^– or (x)=exp(–Cx ^–|| + ...) asx0.     

A supersymmetric transfer matrix and differentiability of the density of states in the one-dimensional Anderson model

LetH=–+V onl ²(), whereV(x),x, are i.i.d.r.v.'s with common probability distributionv. Leth(t)=e ^–itv dv(v) and letk(E) be the integrated density of states. It is proven: (i) Ifh isn-times differentiable withh ^(j)(t)=O((1+|t|)^–) for some >0,j=0, 1, ...,n, thenk(E) is aC ⁿ function. In particular, ifv has compact support andh(t)=O((1+|t|)^–) with >0, thenk(E) isC . This allowsv to be singular continuous. (ii) Ifh(t)=O(e ^–|t|) for some >0 thenk(E) is analytic in a strip about the real axis.The proof uses the supersymmetric replica trick to rewrite the averaged Green's function as a two-point function of a one-dimensional supersymmetric field theory which is studied by the transfer matrix method.Research partially supported by the NSF under grant MC-8301889     

The ground state energy of Sdhrödinger operators

We studye()=inf spec(-+V) and examine whene()<0 for all 0. We prove thatc ²e()–d ² for suitableV and all small ||.Research partially funded under NSF grant number DMS-9101716.     

Superstability estimates for anharmonic systems

We consider an anharmonic crystal described by variablesS _x ,x ^d ,S _x , with one-body interaction ¦S _x ¦ and nearest neighbor (n.n.) two body interaction ¦S _x –S _y ¦. We prove that, for ^d bounded, , where is the correlation function for the free boundary condition Gibbs state in ,>0 and are suitable constants independent of and . This generalizes previous results obtained in the case.Research partially supported by Consiglio Nazionale delle Ricerche.     

Existence,uniqueness, and nondegeneracy of positive solutions of semilinear elliptic equations

We study positive solutions of the Dirichlet problem: u(x)+f(u(x))=0,xD ⁿ ,u(x)=0,xD ⁿ , whereD ⁿ is ann-ball. We find necessary and sufficient conditions for solutions to be nondegenerate. We also give some new existence and uniqueness theorems.Research supported in part by NSF Contract Number MCS 80-02337     

The spectrum of a Schrödinger operator inL p ℝ v isp-independent

LetH _p =–1/2+V denote a Schrödinger operator, acting inL _p ^v , 1p. We show that (H _p )=(H ₂) for allp[1, ], for rather general potentialsV.     

Radiation effects in the production and decay of ψ mesons by leptons

A new expression _e is obtained for resonance in the reaction e⁺e^– ar ^+– with allowance for the radiative corrections, which also contain the emission of hard photons by the final leptons and the final value of the energy resolution E. An expression is obtained for the total probability of radiative lepton decay. A numerical analysis of the experimental data is made for the reaction ^+–, and the parameters e⁺e^– (3, 1) e⁺e^–, and _e ² /, _e, and are determined for (3, 1) resonance-with allowance for the radiative corrections.This paper was read (November 17–21, 1975) at a session on high-energy physics of the Department of Nuclear Physics of the Academy of Sciences of the USSR.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 29–34, March, 1977.     

Decay of the Bethe–Salpeter Kernel and Bound States for Lattice Classical Ferromagnetic Spin Systems at High Temperature

We consider lattice classical ferromagnetic spin systems at high temperature (1) with nearest neighbor interactions and even single-spin distributions (ssd). Associated with each system is an imaginary time lattice quantum field theory. It is known that there is a particle of mass m–ln in the energy-momentum spectrum. If s ⁴–3s ²²<0, where s ^k is the kth moment of the ssd, and is sufficiently small, we show that in the two-particle subspace there is no mass spectrum up to 2m. For >0 we show that the only mass spectrum in (m, 2m) is a bound state of mass m _b=2m+ln(1–)+O(), where =(+2s ²²)^–1. A bound on the decay of the kernel of a Bethe–Salpeter equation is obtained and used to prove these results.     

Poincaré Algebra and Space-Time Critical Dimensions for Paraspinnings Strings

In this paper, we paraquantize the spinning string theory in the Neveu-Shwarz model. Unlike the Ardalan and Mansouri work [Phys. Rev. D, Vol. 9, (1974) 3341], the paraquantum system is such that both the center of mass variables and the excitation modes of the string verify paracommutation relations. The commutators of the Poincaré algebra are satisfied, except the [p ,p ] one, since one can only write [p ,p ]= 0, for Q1. Because of the relation [x ,x ] =,0 and with the sole use of the trilinear relations, we find existence possibilities of spinning strings defined in a noncommutative space-time at space-time dimensions other than D=10.     

An Inverse Problem for an Harmonic Oscillator Perturbed by Potential: Uniqueness

Consider the perturbed harmonic oscillator Ty = -y" + x²y + q(x)y on L²(R) where the real potential q satisfy some assumption on infinity (the case q L²(R), (t+1)^-rdt), r < 1 is covered).