Operator (computer programming)

About: Operator (computer programming) is a research topic. Over the lifetime, 40896 publications have been published within this topic receiving 671452 citations. The topic is also known as: operator symbol & operator name.

Large spin limits of AdS/CFT and generalized Landau-Lifshitz equations

Abstract: We consider AdS_5 x S^5 string states with several large angular momenta along AdS_5 and S^5 directions which are dual to single-trace Super-Yang-Mills (SYM) operators built out of chiral combinations of scalars and covariant derivatives. In particular, we focus on the SU(3) sector (with three spins in S^5) and the SL(2) sector (with one spin in AdS_5 and one in S^5), generalizing recent work hep-th/0311203 and hep-th/0403120 on the SU(2) sector with two spins in S^5. We show that, in the large spin limit and at the leading order in the effective coupling expansion, the string sigma model equations of motion reduce to matrix Landau-Lifshitz equations. We then demonstrate that the coherent-state expectation value of the one-loop SYM dilatation operator restricted to the corresponding sector of single trace operators is also effectively described by the same equations. This implies a universal leading order equivalence between string energies and SYM anomalous dimensions, as well as a matching of integrable structures. We also discuss the more general 5-spin sector and comment on SO(6) states dual to non-chiral scalar operators.

Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators

Abstract: We completely determine necessary and sufficient conditions for the normalizability of the wave functions giving the algebraic part of the spectrum of a quasi-exactly solvable Schrodinger operator on the line. Methods from classical invariant theory are employed to provide a complete list of canonical forms for normalizable quasi-exactly solvable Hamiltonians and explicit normalizability conditions in general coordinate systems.

Dispersive estimates for principally normal pseudodifferential operators

Abstract: The aim of these notes is to describe some recent re- sults concerning dispersive estimates for principally normal pseu- dodifferential operators. The main motivation for this comes from unique continuation problems. Such estimates can be used to prove L q Carleman inequalities, which in turn yield unique continuation results for various partial differential operators with rough poten- tials. Dispersive estimates are L q estimates for nonelliptic partial differ- ential operators which are a consequence of the decay properties of their fundamental solutions. These decay properties follow from spa- tial spreading of the singularities of the solutions. Since solutions prop- agate in directions conormal to the characteristic set of the operator, this spreading can be related to nonzero curvatures of the characteristic set. Dispersive estimates for constant coefficient operators are closely related to the restriction theorem in harmonic analysis. Various types of dispersive estimates are known to be true for op- erators such as the wave operator, the Schrodinger operator and the linear KdV, see Ginibre-Velo (4), Keel-Tao (11). They have proved to be useful in the study of nonlinear problems, as well as of problems with unbounded potentials. More recently, similar estimates have been obtained for wave op- erators with variable coefficients, beginning with the smooth case in Kapitanskii (10), Mockenhaupt, Seeger and Sogge (14), up to operators with C 2 coefficients in Smith (15) and Tataru (21), (23). Similar results were obtained for the Schrodinger equation in Staffilani-Tataru (19) (C 2 coefficients) and in Burq-Gerard-Tzvetkov (1) (smooth coeffic ients). In the variable coefficient elliptic case one should also mention Sogge's L q

A proof of boundedness of the Carleson operator

Abstract: We give a simplified proof that the Carleson operator is of weak type (2, 2). This estimate is the main ingredient in the proof of Carleson’s theorem on almost everywhere convergence of Fourier series of functions in L2([0, 1]).