Manipulation of Semiclassical Photon States

Gabriel F. Calvo and Antonio Picon defined a class of operators, for use in quantum communication, that allows arbitrary manipulations of the three lowest two-dimensional Hermite-Gaussian modes {|0,0>,|1,0>,|0,1>}. Our paper continues the study of those operators, and our results fall into two categories. For one, we show that the generators of the operators have infinite deficiency indices, and we explicitly describe all self-adjoint realizations. And secondly we investigate semiclassical approximations of the propagators. The basic method is to start from a semiclassical Fourier integral operator ansatz and then construct approximate solutions of the corresponding evolution equations. In doing so, we give a complete description of the Hamilton flow, which in most cases is given by elliptic functions. We find that the semiclassical approximation behaves well when acting on sufficiently localized initial conditions, for example, finite sums of semiclassical Hermite-Gaussian modes, since near the origin the Hamilton trajectories trace out the bounded components of elliptic curves.Comment: 30 pages, 3 figures. Small corrections, mostly in Section V. To appear in the Journal of Mathematical Physic

New global stability estimates for monochromatic inverse acoustic scattering

We give new global stability estimates for monochromatic inverse acoustic scattering. These estimates essentially improve estimates of [P. Hahner, T. Hohage, SIAM J. Math. Anal., 33(3), 2001, 670-685] and can be considered as a solution of an open problem formulated in the aforementioned work

On Inverse Scattering at a Fixed Energy for Potentials with a Regular Behaviour at Infinity

We study the inverse scattering problem for electric potentials and magnetic fields in \ere^d, d\geq 3, that are asymptotic sums of homogeneous terms at infinity. The main result is that all these terms can be uniquely reconstructed from the singularities in the forward direction of the scattering amplitude at some positive energy.Comment: This is a slightly edited version of the previous pape

Intertwining technique for a system of difference Schroedinger equations and new exactly solvable multichannel potentials

The intertwining operator technique is applied to difference Schroedinger equations with operator-valued coefficients. It is shown that these equations appear naturally when a discrete basis is used for solving a multichannel Schroedinger equation. New families of exactly solvable multichannel Hamiltonians are found

Boundary relations and generalized resolvents of symmetric operators

The Kre\u{\i}n-Naimark formula provides a parametrization of all selfadjoint exit space extensions of a, not necessarily densely defined, symmetric operator, in terms of maximal dissipative (in \dC_+) holomorphic linear relations on the parameter space (the so-called Nevanlinna families). The new notion of a boundary relation makes it possible to interpret these parameter families as Weyl families of boundary relations and to establish a simple coupling method to construct the generalized resolvents from the given parameter family. The general version of the coupling method is introduced and the role of boundary relations and their Weyl families for the Kre\u{\i}n-Naimark formula is investigated and explained.Comment: 47 page

Discrete series representations for sl(2|1), Meixner polynomials and oscillator models

We explore a model for the one-dimensional quantum oscillator based upon the Lie superalgebra sl(2|1). For this purpose, a class of discrete series representations of sl(2|1) is constructed, each representation characterized by a real number beta>0. In this model, the position and momentum operators of the oscillator are odd elements of sl(2|1) and their expressions involve an arbitrary parameter gamma. In each representation, the spectrum of the Hamiltonian is the same as that of the canonical oscillator. The spectrum of the momentum operator can be continuous or infinite discrete, depending on the value of gamma. We determine the position wavefunctions both in the continuous and discrete case, and discuss their properties. In the discrete case, these wavefunctions are given in terms of Meixner polynomials. From the embedding osp(1|2)\subset sl(2|1), it can be seen why the case gamma=1 corresponds to the paraboson oscillator. Consequently, taking the values (beta,gamma)=(1/2,1) in the sl(2|1) model yields the canonical oscillator.Comment: (some minor misprints were corrected in this version

Mathematical surprises and Dirac's formalism in quantum mechanics

By a series of simple examples, we illustrate how the lack of mathematical concern can readily lead to surprising mathematical contradictions in wave mechanics. The basic mathematical notions allowing for a precise formulation of the theory are then summarized and it is shown how they lead to an elucidation and deeper understanding of the aforementioned problems. After stressing the equivalence between wave mechanics and the other formulations of quantum mechanics, i.e. matrix mechanics and Dirac's abstract Hilbert space formulation, we devote the second part of our paper to the latter approach: we discuss the problems and shortcomings of this formalism as well as those of the bra and ket notation introduced by Dirac in this context. In conclusion, we indicate how all of these problems can be solved or at least avoided.Comment: Largely extended and reorganized version, with new title and abstract and with 2 figures added (published version), 54 page

ВИКОРИСТАННЯ ПРИНЦИПУ СПІВВІДНЕСЕННЯ ДЛЯ ВИБОРУ ПРОГРАМИ ТЕХНІЧНОЇ ЕКСПЛУАТАЦІЇ КЕРОВАНИХ АВІАЦІЙНИХ ЗАСОБІВ УРАЖЕННЯ ПРИ ПЕРЕХОДІ НА ЕКСПЛУАТАЦІЮ ЗА ТЕХНІЧНИМ СТАНОМ

The article presents the current state of the available stock of guided air weapons and the problem of transition of guided air weapons to operation according to technical condition. The transfer of guided air weapons to operation according to technical condition involves increasing the role of measurement operations and control of their parameters and characteristics. This will determine the actual technical condition and make informed decisions about further operation. In general, the effective-ness of control a technical condition of aviation missiles significantly affects the readiness for use of aviation equipment and the combat capabilities of this equipment in performance of the tasks. The exciting control model and methods for selecting parameters of technical condition is universal in relation to the variants of construction of the technical maintenance program a guided air weapons allows to conduct research into these indicators. However, this program does not indicate the plural of variant that should be selected if the undetermined influence of different nature factors to carry out a set of maintenance and repair work for guided air weapons. The possibility of using the principle of correlation, when choosing the program of technical operation a guid-ed air weapons during the transition to operation according to technical condition. With the help of the correlation principle, an approach to the choice of the program of technical operation a guided air weapons can be developed in case of unknown infor-mation about the mechanisms of influence of various internal and external factors on the properties of the guided air weapons, their effect on the technical condition and features of the units of aviation guided missile that are not amenable to instrumental con-trol. To implement the control of guided air weapons with using a principle of correlation need to use the results of practical appli-cations of aviation guided missile, including other types of fire tests, sample tests using the physico-chemical analysis of powder and explosive charges.У статті розглядається сучасний стан наявного запасу керованих авіаційних засобів ураження та проблематика переходу авіаційних керованих ракет на експлуатацію за технічним станом. Показана можливість використання принципу співвіднесення для вибору програми технічної експлуатації керованих авіаційних засобів ураження при переході на експлуатацію за технічним станом. Принцип співвіднесення дає змогу враховувати необхідні та достатні умови для вибору схем контролю технічного стану керованих авіаційних засобів ураження у разі невизначеної інформації про механізми впливу на їх технічний стан