Feynman integral and complex classical trajectories

Let _t be an analytic solution of the Schrödinger equation with the initial condition . Let _t be the solution of the Schrödinger equation with the initial condition =, where is an analytic function. When 0, then _t (x) _t (x)¹ ( _t (x)), where _t (x) trajectory starting from x. We relate this result to Feynman's sum over trajectories and complex stochastic differential equations.     

An invariant measure for the equationu tt −u xx +u 3=0

Numerical studies of the initial boundary-value problem of the semilinear wave equationu _tt –u _xx +u ³=0 subject to periodic boundary conditionsu(t, 0)=u(t, 2),u _t (t, 0)=u _t (t, 2) and initial conditionsu(0,x)=u ₀(x),u _t(0,x)=v ₀(x), whereu ₀(x) andv ₀(x) satisfy the same periodic conditions, suggest that solutions ultimately return to a neighborhood of the initial stateu ₀(x),v ₀(x) after undergoing a possibly chaotic evolution. In this paper an appropriate abstract space is considered. In this space a finite measure is constructed. This measure is invariant under the flow generated by the Hamiltonian system which corresponds to the original equation. This enables one to verify the above returning property.     

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).     

Strong coupling in the quantum field theory with nonlocal nonpolynomial interaction

In the relativistic quantum field theory the representation for theS-matrix elements is obtained for any coupling constantsg in the case of a one component scalar field (x) with nonlocal nonpolynomial interaction _I ()=gU() when the causal function is bounded in the Euclidean region 0D _c (x _E ² D _c (0)< and the function |U(u)|1 for realu. It is proved that the two point Green function is bounded in the physical region of momenta variablep ².     

A limit theorem for stochastic acceleration

We consider the motion of a particle in a weak mean zero random force fieldF, which depends on the position,x(t), and the velocity,v(t)= (t). The equation of motion is (t)=F(x(t),v(t), ), wherex(·) andv(·) take values in ^d ,d3, and ranges over some probability space. We show, under suitable mixing and moment conditions onF, that as 0,v (t)v(t/²) converges weakly to a diffusion Markov processv(t), and ² x (t) converges weakly to , wherex=lim ² x (0).     

Minimum action solutions of some vector field equations

The system of equations studied in this paper is –u _i =g ⁱ (u) on ^d ,d2, withu: ^{d n} andg ⁱ (u)=G/u _i . Associated with this system is the action,S(u)={1/2|u|²–G(u)}. Under appropriate conditions onG (which differ ford=2 andd3) it is proved that the system has a solution,u 0, of finite action and that this solution also minimizes the action within the class {v is a solution,v has finite action,v 0}.Work partially supported by U.S. National Science Foundation Grant PHY-81-16101-A02     

On the Existence of a Maximizer for the Strichartz Inequality

It is shown that a maximizing function u _*L ² does exist for the Strichartz inequality e ^it _x ² u _L ⁶ _t (L ⁶ _x )Su _L ², with S>0 being the sharp constant.     

Long-range correlations at depinning transitions

Semi-infinite systems are considered with long-range surface fields B _z ^–(1+r) for large distancesz from the surface. The influence of such fields on the global phase diagram and on the critical singularities of depinning transitions is studied within Landau theory. For |B|0, the correlation length diverges as b ^–1/2 withb=|Bln|B^–(1+r). For finiteB, t ^v withv =(2+r)/(2+2r) wheret measures the distance from bulk coexistence. In the latter case, a Ginzburg criterion leads to the upper critical dimensiond ^*=(2+3r)/(2+r).     

Directed paths on percolation clusters

We consider a variant of the problem of directed polymers on a disordered lattice, in which the disorder is geometrical in nature. In particular, we allow a finite probability for each bond to be absent from the lattice. We show, through the use of numerical and scaling arguments on both Euclidean and hierarchical lattices, that the model has two distinct scaling behaviors, depending upon whether the concentration of bonds on the lattice is at or above the directed percolation threshold. We are particularly interested in the exponents and, defined by ft and xt , describing the free-energy and transverse fluctuations, respectively. Above the percolation threshold, the scaling behavior is governed by the standard random energy exponents (=1/3 and =2/3 in 1+1 dimensions). At the percolation threshold, we predict (and verify numerically in 1+1 dimensions) the exponents=1/2 and =v/v, where v and v are the directed percolation exponents. In addition, we predict the absence of a free phase in any dimension at the percolation threshold.     

The surfboard Schrödinger equations

We study the large time behavior of solutions of time dependent Schrödinger equationsiu/t=–(1/2)u+t V(x/t)u with bounded potentialV(x). We show that (1) if>–1, all solutions are asymptotically free ast, (2) if–1 a solution becomes asymptotically free if and only if it has the momentum support outside of suppV for large time, (3) if –1 <0 all solutions are still asymptotically modified free ast and that (4) if 0 <2, for each local minimumx ₀ ofV(x), there exist solutions which are asymptotically Gaussians centered atx=tx ₀ and spreading slowly ast.     

Parabolic problems for the Anderson model

The objective of this paper is a mathematically rigorous investigation of intermittency and related questions intensively studied in different areas of physics, in particular in hydrodynamics. On a qualitative level, intermittent random fields are distinguished by the appearance of sparsely distributed sharp peaks which give the main contribution to the formation of the statistical moments. The paper deals with the Cauchy problem (/t)u(t,x)=Hu(t, x), u(0,x)=t ₀(x) 0, (t, x) ₊ × ^d , for the Anderson HamiltonianH = + (·), (x),x d where is a (generally unbounded) spatially homogeneous random potential. This first part is devoted to some basic problems. Using percolation arguments, a complete answer to the question of existence and uniqueness for the Cauchy problem in the class of all nonnegative solutions is given in the case of i.i.d. random variables. Necessary and sufficient conditions for intermittency of the fieldsu(t,·) ast are found in spectral terms ofH. Rough asymptotic formulas for the statistical moments and the almost sure behavior ofu(t,x) ast are also derived.     

Long-Time tails in a random diffusion model

Letw = {w(x)xZ^d} be a positive random field with i.i.d. distribution. Given its realization, letX _t be the position at timet of a particle starting at the origin and performing a simple random walk with jump rate w^–1(X_t). The processX={X _t:t0} combined withw on a common probability space is an example of random walk in random environment. We consider the quantities _t =(d/dt) E (X _t ² –M ^–1 t and _t(w) = (d/dt)E_w(X _t ² – M^– 1t). Here E_w. is expectation overX at fixedw and E = E_w (dw) is the expectation over bothX andw. We prove the following long-time tail results: (1) lim_t t^d/2_t= V²M^d/2–3(d/2)^d/2 and (2) lim_t t^d/4 _st(w)= Z_s weakly in path space, with {Z_s:s>0} the Gaussian process with EZ_s=0 and EZ_rZ_s= V²M^d/2–4(d)^d/2 (r + s)^–d/2. HereM and V² are the mean and variance of w(0) under . The main surprise is that fixingw changes the power of the long-time tail fromd/2 tod/4. Since , with ₀ the stationary measure for the environment process, our result (1) exhibits a long-time tail in an equilibrium autocorrelation function.     

Dynamics of vector spin glasses: The Gabay-Toulouse line

The dynamics of ann-component vector spin glass with infinite range interactions are investigated near and above the Gabay-Toulouse (GT) line. The local transverse susceptibility _T for 0 varies along the whole GT lineT _c1 (H) as ^v , with a field and temperature independent critical exponentv=1/2. The longitudinal susceptibility _L () remains analytic for all (T, H)T _c1 (H), except for a cross-over fromv=1 tov=1/2 forH0 at the freezing temperatureT=T _f . The dynamic susceptibilities _T () and _L () are already coupled above the GT line via self-energy terms. BelowT _c1 (H), this coupling is strongly enhanced by other mechanisms.     

Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity

The asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation lawsu _t +f(u) _x =u _xx with the initial datau ₀ which tend to the constant statesu _± asx±. Stability theorems are obtained in the absence of the convexity off and in the allowance ofs (shock speed)=f(u _±). Moreover, the rate of asymptotics in time is investigated. For the casef(u₊)(u_–), if the integral of the initial disturbance over (–,x) is small and decays at the algebraic rate as |x|, then the solution approaches the traveling wave at the corresponding rate ast. This rate seems to be almost optimal compared with the rate in the casef=u ²/2 for which an explicit form of the solution exists. The rate is also obtained in the casef(u _± =s under some additional conditions. Proofs are given by applying an elementary weighted energy method to the integrated equation of the original one. The selection of the weight plays a crucial role in those procedures.     

Evidence of Hg-chain formation in Hg x TiS2: a199mHg-TDPAC study

We determined the^199mHg nuclear quadrupole interaction in the misfit or superstoichiometric compound Hg _x TiS₂ by time differential perturbed angular correlation. A unique Hg-site withv _Q =511(1) MHz and =0.410(4) was observed, irrespective of the Hg-uptake (2/3x4/3). We propose a model of Hg-Hg zig-zag chains which accounts for these observations as well as for the X-ray diffraction data.     

The Homogeneous Hamilton–Jacobi and Bernoulli Equations Revisited

The one-dimensional case of the homogeneous Hamilton–Jacobi and Bernoulli equations S_t S _x ² =0, where S(x, t) is Hamilton's principal function of a free particle and also Bernoulli's momentum potential of a perfect liquid, is considered. Non-elementary solutions are looked for in terms of odd power series in t with x-dependent coefficients and even power series in x with t-dependent coefficients. In both cases, and depending upon initial conditions, unexpected regularities are observed in the terms of these expansions and this suggests that S(^x, ^t) should be written as a product of the elementary solution x²/(2^t) and a function f=f() where =(x, |t|) owing to the symmetry property which is that S(x, –t)=–S(x, t). Requiring that this Ansatz satisfies the said equation and choosing the simplest realization of (x, |t|)=₀ |t/t₀| (x/x ₀)0 with , results in a soluble ordinary differential equation, of first order in u=ln and quadratic in f. This ODE has two fixed points: f=1, obviously, and f=0, a new fixed point which is often called trivial. The phase plane (f_u, f) consists of a family of parabolas, all of which pass through the two fixed points. Explicit solutions of the general case are given close to these fixed points. A one-parameter family of solution is found to emerge from the trivial fixed point with non-trivial initial values S(x, 0). Detailed analyses of these findings will be reported elsewhere, bearing in mind that Bernoulli's equation has to be supplemented by the continuity equation satisfied by the density of the liquid.     

Absolutely continuous invariant measures for one-parameter families of one-dimensional maps

Given a one-parameter familyf (x) of maps of the interval [0, 1], we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to Lebesgue measure. We show that this set has positive measure, for two classes of maps: i)f (x)=f(x) where 0<4 andf(x) is a functionC ³-near the quadratic mapx(1–x), and ii)f (x)=f(x) (mod 1) wheref isC ³,f(0)=f(1)=0 andf has a unique nondegenerate critical point in [0, 1].     

Fock representations of the affine Lie algebraA 1 (1)

The aim of this note is to show that the affine Lie algebraA ₁ ⁽¹⁾ has a natural family _, ,v of Fock representations on the spaceC[x _i,y _j;i andj ], parametrized by (,v) C ². By corresponding the highest weight _, of _, to each (,), the parameter spaceC ₂ forms a double cover of the weight spaceC₀C –₁ with singularities at linear forms of level –2; this number is (–1)-times the dual Coxeter number. Our results contain explicit realizations of irreducible non-integrable highest wieghtA ₁ ⁽¹⁾ -modules for generic (,v).     

Absence of highest-spin ground states in the Hubbard model

The Hubbard modelH=–tc _x c _y +U n _x n _x withN electrons and periodic boundary condition is studied onv-dimensionalL ₁ × ... ×L _v lattices. It is shown that for any value ofU there is no ground state with maximal spin (S=N/2) in the following cases: (i) ^v (v2) at low electron densities; with one hole ift>0 andL _i is odd for somei; with two holes ift<0, or ift>0 and all theL _i are even. (ii) Thebcc lattice at low densities; with two holes ift<0, or ift>0 and all theL _i are even; with 2, ..., 6 holes ifL _i =L andt<0, or ift>0 andL is even. (iii) The triangular lattice at densities near 0 and 1 ift>0; with two holes ift<0; with 2, 3, 4 holes ift<0 andL ₁=L ₂. (iv) Thefcc lattice at densities near 0 and 1 ift>0; with two holes ift<0. Some results for the one dimensional model are also presented.     

On the Davey–Stewartson and Ishimori Systems

We study the initial value problem for the two-dimensional nonlinear nonlocal Schrödinger equations i u_t + u = N(v), (t, x, y) R³, u(0, x, y) = u₀(x, y), (x, y) R² (A), where the Laplacian = ² _x + ² _y, the solution u is a complex valued function, the nonlinear term N = N₁ + N₂ consists of the local nonlinear part N₁(v) which is cubic with respect to the vector v=(u,u_x,u_y,\overline{u},\overline{u}_{x},\overline{u}_{y}) in the neighborhood of the origin, and the nonlocal nonlinear part N₂(v) =(v, ^{– 1} _x K_x(v)) + (v, ^{– 1} _y K_y(v)), where (, ) denotes the inner product, and the vectors K_x (C⁴(C⁶; C))⁶ and K_y (C⁴(C⁶; C))⁶ are quadratic with respect to the vector v in the neighborhood of the origin. We assume that the components K⁽²⁾ _x = K⁽⁴⁾ _x 0, K⁽³⁾ _y = K⁽⁶⁾ _y 0. In particular, Equation (A) includes two physical examples appearing in fluid dynamics. The elliptic–hyperbolic Davey–Stewartson system can be reduced to Equation (A) with , and all the rest components of the vectors K_x and K_y are equal to zero. The elliptic–hyperbolic Ishimori system is involved in Equation (A), when , and . Our purpose in this paper is to prove the local existence in time of small solutions to the Cauchy problem (A) in the usual Sobolev space, and the global-in-time existence of small solutions to the Cauchy problem (A) in the weighted Sobolev space under some conditions on the complex conjugate structure of the nonlinear terms, namely if N(eⁱ v) = eⁱ N(v) for all R.