Order operator

Abstract: We modify Hormander's well-known weak type (1,1) condition for integral operators (in a weakened version due to Duong and McIn- tosh) and present a weak type (p,p) condition for arbitrary operators. Given an operatorA on L2 with a bounded H ∞ calculus, we show as an application the Lr-boundedness of the H ∞ calculus for all r ∈ (p,q), provided the semigroup (e −tA ) satisfies suitable weighted Lp → Lq-norm estimates with 2 ∈ (p,q ). This generalizes results due to Duong, McIntosh and Robinson for the special case (p,q )=( 1, ∞) where these weighted norm estimates are equivalent to Poisson-type heat kernel bounds for the semigroup (e −tA ) . Their results fail to apply in many situations where our im- provement is still applicable, e.g. if A is a Schrodinger operator with a singular potential, an elliptic higher order operator with bounded measurable coefficients or an elliptic second order operator with sin- gular lower order terms.

Abstract: We have computed new exact traveling wave solutions, including complex solutions of fractional order Boussinesq-Like equations, occurring in physical sciences and engineering, by applying Exp-function method. The method is blended with fractional complex transformation and modified Riemann-Liouville fractional order operator. Our obtained solutions are verified by substituting back into their corresponding equations. To the best of our knowledge, no other technique has been reported to cope with the said fractional order nonlinear problems combined with variety of exact solutions. Graphically, fractional order solution curves are shown to be strongly related to each other and most importantly, tend to fixate on their integer order solution curve. Our solutions comprise high frequencies and very small amplitude of the wave responses.

Abstract: We consider a quantum many-body system on a lattice which exhibits a spontaneous symmetry breaking in its infinite-volume ground states, but in which the corresponding order operator does not commute with the Hamiltonian. Typical examples are the Heisenberg antiferromagnet with a Neel order and the Hubbard model with a (superconducting) off-diagonal long-range order. In the corresponding finite system, the symmetry breaking is usually “obscured” by “quantum fluctuation” and one gets a symmetric ground state with a long-range order. In such a situation, Horsch and von der Linden proved that the finite system has a low-lying eigenstate whose excitation energy is not more than of orderN −1, whereN denotes the number of sites in the lattice. Here we study the situation where the broken symmetry is a continuous one. For a particular set of states (which are orthogonal to the ground state and with each other), we prove bounds for their energy expectation values. The bounds establish that there exist ever-increasing numbers of low-lying eigenstates whose excitation energies are bounded by a constant timesN −1. A crucial feature of the particular low-lying states we consider is that they can be regarded as finite-volume counterparts of the infinite-volume ground states. By forming linear combinations of these low-lying states and the (finite-volume) ground state and by taking infinite-volume limits, we construct infinite-volume ground states with explicit symmetry breaking. We conjecture that these infinite-volume ground states are ergodic, i.e., physically natural. Our general theorems not only shed light on the nature of symmetry breaking in quantum many-body systems, but also provide indispensable information for numerical approaches to these systems. We also discuss applications of our general results to a variety of interesting examples. The present paper is intended to be accessible to readers without background in mathematical approaches to quantum many-body systems.

Abstract: The work is giving estimations of the discrepancy between solutions of the initial and the homogenized problems for a one-dimensional second-order elliptic operators with random coefficients satisfying strong or uniform mixing conditions. We obtain several sharp estimates in terms of the corresponding mixing coefficient. In the mathematical literature there are now many papers devoted to homogenization of random op- erators with coefficients being stationary random field (see, for instance, (3) and references therein) and of operators posed in randomly perforated domain (see (2,3)). But all these results are mainly giving the convergence of the solutions towards the solution of the limit (or homogenized) equation, without estimate of the residual. The first successful attempt to give such an estimate is the work of Yurinski (6), where the expectation of some norm of the residual for the divergence form second-order elliptic random operator is estimated by a positive power of a small parameter " that characterizes the microscopic length scale. This power of " depends only on the dimension of the space, the ellipticity constant and on some characteristics of the mixing conditions; but this power is implicit and could not be computed explicitly. Later, similar problems have been studied for symmetric elliptic systems (5); in this case the residual is estimated by some negative power ofj log"j, which could not, once more, be computed explicitly. The aim of our paper is to investigate in the one-dimensional case the probabilistic property of the residual. We assume that the coefficients of the operator is a stationary random field satisfying strong or uniform mixing condition. The first two sections deal with the case when the corresponding mixing coefficient decays like a negative power of a distance. Namely, in the first part, we suppose that >1, i.e., that the random variables a(x, )a nda(x +d,) are weakly dependent for large d. This allows to apply the central limit theorem. In the second part we study the case when the mixing properties of the random field are not so good, i.e., when 6 1. Finally, in the third part, using large deviation type estimates and assuming that the mixing coefficient decays like the exponent of some power of the distance, we get more precise bounds in probability for the fields with such "good" mixing properties. It should be noted that