Detecting level crossings without looking at the spectrum

In many physical systems it is important to be aware of the crossings and avoided crossings which occur when eigenvalues of a physical observable are varied using an external parameter. We have discovered a powerful algebraic method of finding such crossings via a mapping to the problem of locating the roots of a polynomial in that parameter. We demonstrate our method on atoms and molecules in a magnetic field, where it has implications in the search for Feshbach resonances. In the atomic case our method allows us to point out a new class of invariants of the Breit-Rabi Hamiltonian of magnetic resonance. In the case of molecules, it enables us to find curve crossings with practically no knowledge of the corresponding Born-Oppenheimer potentials.Comment: 4 pages, new title, no figures, accepted by Phys. Rev. Let

Компарирование шумовых сигналов в миллиметровом диапазоне длин волн

The article describes the relevant problems of in-concordance arising when a unit size of the spectral power density of noise radiation (SPDNR) is transmitted from the reference source to the verifiable source of electromagnetic noise radiation (gas-discharge noise generators, avalanche-transit-time diode noise generators).The reference source of the noise radiation is a waveguide low-temperature noise generator (LNG) and verifiable noise generators (NG) are high-temperature gas-discharge noise generators, which are commonly used for the experimental evaluation of the noise characteristics of mm-wave receivers.The error of transmitting SPDNR unit size is related to the error of the reference LNG and the calibration error, the components of which are characteristics of the comparator (sensitivity, gain stability, intermediate frequency bandwidth), tract performance specifications (modulator, attenuator, valves) as well as the VSWR of a verified generator.Developed in the microwave range methods of comparation are based on the fact that the comparator intermediate frequency bandwidth is sufficiently narrow (60 MHz). This allows us to use a variety of matching devices in transmitting the SPDNRE unit size and thus significantly reduce the influence of the in-concordance error sources and thus reduce errors due to inconcordance.To increase the fluctuation sensitivity when receiving a signal from the LNG the mm-wave radiometric comparators use low-noise wide-band amplifiers of intermediate frequency with a bandwidth ΔF of 0.5 to 3 GHz, and the reception signal is carried by the main and mirror channels.The paper, for the first time, theoretically analyses the transmission of a broadband noise signal from the generator to the receiver through the passive four-pole, which includes an attenuator, a modulator, and two valves to block the modulator input and output. In the considered structural diagram of comparation the four-pole is represented as a waveguide line of length L and characterized by the scattering matrix | S |. Decreasing in-concordance errors with increasing length of the waveguide line has been experimentally proved. We investigate the particular cases:1) Both valves are ideal;2) The first valve is imperfect;3) Modules of reflection coefficients are weakly dependent on the frequency and in the bandwidth can be replaced by average values;4) Modules of reflection coefficients depend on the frequency in the band ΔF. Integration in the frequency band is replaced by summing up.Due to ambiguity of phase relationships the in-concordance error evaluation is on the maximum scale.Analysed results of numerically calculated errors due to in-concordance for various cases are shown in tables. The in-concordance error evaluation as a function of the tract length L is the nearer to the calculated one, the less is the frequency ripple of the reflection coefficient modules. Using the waveguide lines between the comparator and the NG leads to the smoothing frequency characteristics during noise signal comparation, and to reducing the errors due to inconcordance almost by an order of magnitude. This is a qualitatively new achievement for certification of the waveguide measures of SPDNR.Рассматриваются численные характеристики погрешности из-за рассогласования, при компарировании волноводного эталонного низкотемпературного генератора шума (НГШ) и волноводного высокотемпературного газоразрядного генератора шума (ГГШ). В качестве компаратора используется модуляционный радиометрический приёмник с полосой пропускания по промежуточной частоте 1 ГГц и 3 ГГц. Погрешность передачи размера единицы СПМШ связана с погрешностью эталонного НГШ и погрешностью компарирования, составляющими которой являются характеристики компаратора (чувствительность, стабильность коэффициента усиления, ширина полосы пропускания по промежуточной частоте), технические характеристики тракта (модулятор, аттенюатор, вентили), а также КСВН поверяемого генератора. Экспериментально и теоретически было показано, что включение волноводной линии между компаратором и ГШ приводит к сглаживанию частотных характеристик при компарировании шумовых сигналов и к уменьшению погрешности из-за рассогласования почти на порядок, что является качественно новым достижением при аттестации волноводных мер СПМШ. DOI: 10.7463/rdopt.0415.080122

On a linear differential game of optimal approach of many pursuers with one evader

We consider a differential game of approach of many pursuers and one evader. The motions of all players are described by linear systems of the same type. Control functions of players are subject to integral constraints. The duration of the game is fixed. The payoff functional of the differential game is the minimum of the distances between the evader and the pursuers when the game terminates. The pursuers try to minimize the payoff functional, and the evader tries to maximize it. We obtain estimates for the payoff functional of the game, which can be guaranteed by players and explicitly describe the strategies of the players. From here we obtain that in some specific cases the value of the game exists, and optimal strategies can be constructed. An illustrative example is considered