Geometry of differential operators of second order,the algebra of densities,and groupoids

In our previous works, we introduced, for each (super)manifold, a commutative algebra of densities. It is endowed with a natural invariant scalar product. In this paper, we study geometry of differential operators of second order on this algebra. In the more conventional language they correspond to certain operator pencils. We consider the self-adjoint operators and analyze the operator pencils that pass through a given operator acting on densities of a particular weight. There are ‘singular values’ for pencil parameters. They are related with interesting geometric picture. In particular, we obtain operators that depend on certain equivalence classes of connections (instead of connections as such). We study the corresponding groupoids. From this point of view we analyze two examples: the canonical Laplacian on an odd symplectic supermanifold appearing in Batalin–Vilkovisky geometry and the Sturm–Liouville operator on the line, related with classical constructions of projective geometry. We also consider the canonical second order semi-density arising on odd symplectic supermanifolds, which has some similarity with mean curvature of surfaces in Riemannian geometry.     

Locally Homogeneous Four-Dimensional Manifolds of Signature (2,2)

In this paper we consider 4-dimensional neutral-signature curvature models and obtain the complete classification of the Ricci operator. We then consider the property of curvature homogeneity for the above manifolds and prove that every complete, connected and simply connected 1-curvature homogeneous 4-dimensional manifold of signature (2,2) with a non-degenerate Ricci operator is isometric to a four-dimensional Lie group equipped with a left invariant neutral metric. We also classify Ricci-parallel curvature homogeneous 4-dimensional manifolds of signature (2,2).     

The spacetime of double field theory: Review,remarks, and outlook

We review double field theory (DFT) with emphasis on the doubled spacetime and its generalized coordinate transformations, which unify diffeomorphisms and b‐field gauge transformations. We illustrate how the composition of generalized coordinate transformations fails to associate. Moreover, in dimensional reduction, the O(d,d) T‐duality transformations of fields can be obtained as generalized diffeomorphisms. Restricted to a half‐dimensional subspace, DFT includes ‘generalized geometry’, but is more general in that local patches of the doubled space may be glued together with generalized coordinate transformations. Indeed, we show that for certain T‐fold backgrounds with non‐geometric fluxes, there are generalized coordinate transformations that induce, as gauge symmetries of DFT, the requisite O(d,d;ℤ) monodromy transformations. Finally we review recent results on the α^′ extension of DFT which, reduced to the half‐dimensional subspace, yields intriguing modifications of the basic structures of generalized geometry.     

On the electronic structure of liquid metals

We consider the electronic structure of liquid metals as a problem of multiple scattering. The total scattering operator is developed in terms of the reaction operators of a single scattering event, having a ‘standing wave’ boundary condition. This method has several advantages with respect to the approach where the total scattering operator is developed in terms of the scattering operators of a single scattering. Since the reaction operators are hermitian, the occurring perturbation expansions have a real perturbation parameter, which is more easily handled. Furthermore, if in the expansion of the total scattering operator, only terms up to the second order are maintained, this approximate expression satisfies already the ‘optical theorem’, just as the exact total scattering operator. Finally, spurious damping terms are eliminated at the outset. For very dense systems we obtain practically non-attenuated waves and this can be understood as if the damping due to a single scattering event is compensated by the reaction of the surrounding scatterers.     

Remarkable properties of the broken-pair model: (I). The differential structure

The differential structure of the broken-pair model is exhibited. As a consequence, matrix elements of any operator acting in the k-broken-pair subspace (or between states with different broken-pair numbers) can be evaluated by making use of effective operators defined in the 0-broken-pair subspace only.     

General triplectic quantization

The general structure of the Sp(2)-covariant version of the field-antifield quantization of general constrained systems in the Lagrangian formalism, the so-called triplectic quantization, as presented in our previous paper with A.M. Semikhatov is further generalized and clarified. We present new unified expressions for the generating operators which are more invariant and which yield a natural realization of the operator V^a and provide for a geometrical explanation for its presence. ThisV^a operator provides then for an invariant definition of a degenerate Poisson bracket on the triplectic manifold being non-degenerate on a naturally defined submanifold. We also define inverses to non-degenerate antitriplectic metrics and give a natural generalization of the conventional calculus of exterior differential forms which, e.g., explains the properties of these inverses. Finally we define and give a consistent treatment of second class hyperconstraints.     

Algebras of Operators on Holomorphic Functions and Applications

We develop the theory of operators defined on a space of holomorphic functions. First, we characterize such operators by a suitable space of holomorphic functions. Next, we show that every operator is a limit of a sequence of convolution and multiplication operators. Finally, we define the exponential of an operator which permits us to solve some quantum stochastic differential equations.     

The model for self-dual chiral bosons as a Hodge theory

We consider (1+1) dimensional theory for a single self-dual chiral boson as a classical model for gauge theory. Using the Batalin–Fradkin–Vilkovisky (BFV) technique, the nilpotent BRST and anti-BRST symmetry transformations for this theory have been studied. In this model other forms of nilpotent symmetry transformations like co-BRST and anti-co-BRST, which leave the gauge-fixing part of the action invariant, are also explored. We show that the nilpotent charges for these symmetry transformations satisfy the algebra of the de Rham cohomological operators in differential geometry. The Hodge decomposition theorem on compact manifold is also studied in the context of conserved charges.     

Darboux transformations in integral and differential forms for generalized Schrödinger equations

The Darboux transformation operator technique is applied to the generalized Schrödinger equation with a position-dependent effective mass and with linearly energy-dependent potentials. Intertwining operators are obtained in an explicit form and used for constructing generalized Darboux transformations. An interrelation is established between the differential and integral transformation operators. It is shown how to construct the quantum well potentials in nanoelectronic with a given spectrum.     

A local approach to dimensional reduction: II. Conformal invariance in Minkowski space

We consider the problem of obtaining conformally invariant differential operators in Minkowski space. We show that the conformal electrodynamics equations and the gauge transformations for them can be obtained in the frame of the method of dimensional reduction developed in the first part of the paper. We describe a method for obtaining a large set of conformally invariant differential operators in Minkowski space.     

The exact response of a two-dimensional flat-layered medium to a probing pulse: An application to a layer stripping reconstruction procedure

Abstract

We consider a simple model problem that can be found in many fields of application such as, for example, reflection seismology. That is we consider an initial boundary value problem on a half-plane for a class of two-dimensional wave equations with a piecewise-constant coefficient. This coefficient describes the flat layered medium under consideration. An initial pulse located on the boundary of the half-plane is used to probe the medium. An integral representation of the solution of this problem is obtained by studying the spectral measures of some differential operators in one variable. This integral representation is exploited to obtain an ‘explicit’ formula for the solution of the problem considered evaluated at the location of the probing pulse. This ‘explicit’ formula is exploited to reconstruct the structure of the medium from its response to a probing pulse via a layer stripping procedure. Some numerical results obtained with this procedure on test problems are shown. The ‘explicit’ formula obtained can be used in several other contexts such as, for example, the study of perturbed flat layered media or the study of random flat layered media.     

Asymptotic Behaviour of the Spectrum of a Waveguide with Distant Perturbations

We consider a quantum waveguide modelled by an infinite straight tube with arbitrary cross-section in n-dimensional space. The operator we study is the Dirichlet Laplacian perturbed by two distant perturbations. The perturbations are described by arbitrary abstract operators “localized” in a certain sense. We study the asymptotic behaviour of the discrete spectrum of such system as the distance between the “supports” of localized perturbations tends to infinity. The main results are a convergence theorem and the asymptotics expansions for the eigenvalues. The asymptotic behaviour of the associated eigenfunctions is described as well. We provide a list of the operators, which can be chosen as distant perturbations. In particular, the distant perturbations may be a potential, a second order differential operator, a magnetic Schrödinger operator, an arbitrary geometric deformation of the straight waveguide, a delta interaction, and an integral operator.     

Perturbative quantum error correction

We derive simple necessary and sufficient conditions under which a quantum channel obtained from an arbitrary perturbation from the identity can be reversed on a given code to the lowest order in fidelity. We find the usual Knill-Laflamme conditions applied to a certain operator subspace which, for a generic perturbation, is generated by the Lindblad operators. For a weak interaction with an environment, the error space to be corrected is a subspace of that spanned by the interaction operators, selected by the environment's initial state.     

Darboux related quantum integrable systems on a constant curvature surface

We consider integrable deformations of the Laplace–Beltrami operator on a constant curvature surface, obtained through the action of first-order Darboux transformations. Darboux transformations are related to the symmetries of the underlying geometric space and lead to separable potentials which are related to the KdV equation. Eigenfunctions of the corresponding operators are related to highest weight representations of the symmetry algebra of the underlying space.     

On Bogoliubov transformations. II. The general case

A rigorous treatment of Bogoliubov transformations is presented along the same lines as in a previous paper, which dealt with a special case. As in the previous paper a formulation in terms of unitary resp. pseudo-unitary operators is used, corresponding to the CAR resp. the CCR. This leads to simple proofs of well-known necessary and sufficient conditions for the transformation to be unitarily implementable in Fock space. The normal form of the implementing operator $\text{U}$ is studied. It is proved that on the subspace of algebraic tensors $\text{U}$ equals a strongly convergent infinite series of Wick monomials that sums up to a simple exponential expression. A connection between the fermion and boson transformations studied in the previous paper is established. The analogous correspondence in the general case only holds true if the (pseudo) unitary operator equals its own inverse.     

压缩算符在自由标量场理论中的推广

引入了一种在量子场论中构造压缩算符的办法:考虑两个具有不同质量的同一标量场的自由哈密顿量,通过博戈留波夫变换,导出广义压缩算符,该算符把一个基态映射到另一个。该算符作用的有效性分别在量子场论的狄拉克表象和薛定谔泛函表象中得到了验证。我们相信,在任意实标量场理论中,只要存在两组以线性变换联系起来的生成湮灭算符,压缩算符就被类似的方法找到。     

Projection operators in nuclear structure calculations

Projection operators are an important tool in nuclear structure theory, because in many circumstances it is useful to construct wave-functions ψ which are not eigenfunctions of some operator Λ, although it is apparent that the physical states must be eigenstates of that operator. Thus one first constructs ψ and then projects from it onto eigenfunctions of Λ. We discuss the cases of angular momentum, isospin, centre of mass energy, particle number and antisymmetry. We describe the integral projection operator, an expansion in shift operators, the product operator of Löwdin and another product operator (the cosine product). Certain methods which appear in the literature are seen to be equivalent to one or the other of these. We consider factors that influence the choice of an appropriate method. Projection occurs frequently in the context of a variational method (such as Hartree-Fock or BCS). We consider the question of projection before or after variation.