On the Equivalence of Different Lax Pairs for the Kac-van Moerbeke Hierarchy

We give a simple algebraic proof that the two different Lax pairs for the Kac-van Moerbeke hierarchy, constructed from Jacobi respectively super-symmetric Dirac-type difference operators, give rise to the same hierarchy of evolution equations. As a byproduct we obtain some new recursions for computing these equations.Comment: 8 page

Trace Formulas in Connection with Scattering Theory for Quasi-Periodic Background

We investigate trace formulas for Jacobi operators which are trace class perturbations of quasi-periodic finite-gap operators using Krein's spectral shift theory. In particular we establish the conserved quantities for the solutions of the Toda hierarchy in this class.Comment: 7 page

Reconstructing Jacobi Matrices from Three Spectra

Cut a Jacobi matrix into two pieces by removing the n-th column and n-th row. We give neccessary and sufficient conditions for the spectra of the original matrix plus the spectra of the two submatrices to uniqely determine the original matrix. Our result contains Hostadt's original result as a special case

The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy

We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy with complex-valued initial data and prove unique solvability globally in time for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Ablowitz-Ladik hierarchy, which is of independent interest as it solves the inverse algebro-geometric spectral problem for general (non-unitary) Ablowitz-Ladik Lax operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin-type equations), this yields the construction of global algebro-geometric solutions of the time-dependent Ablowitz-Ladik hierarchy. The treatment of general (non-unitary) Lax operators associated with general coefficients for the Ablowitz-Ladik hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Ablowitz-Ladik hierarchy but applies generally to (1+1)-dimensional completely integrable soliton equations of differential-difference type.Comment: 47 page

Simulation of polarimetric radar variables in rain at S-, C- and X-band wavelengths

International audiencePolarimetric radar variables of rainfall events, like differential reflectivity ZDR, or specific differential phase KDP, are better suited for estimating rain rate R than just the reflectivity factor for horizontally polarized waves, ZH. A variety of physical and empirical approaches exist to estimate the rain rate from polarimetric radar observables. The relationships vary over a wide range with the location and the weather conditions. In this study, the polarimetric radar variables were simulated for S-, C- and X-band wavelengths in order to establish radar rainfall estimators for the alpine region of the form R(KDP), R(ZH, ZDR), and R(KDP), ZDR. For the simulation drop size distributions of hundreds of 1-minute-rain episodes were obtained from 2D-Video-Distrometer measurements in the mountains of Styria, Austria. The sensitivity of the polarimetric variables to temperature is investigated, as well as the influence of different rain drop shape models ? including recently published ones ? on radar rainfall estimators. Finally it is shown how the polarimetric radar variables change with the elevation angle of the radar antenna

Scattering Theory for Jacobi Operators with Steplike Quasi-Periodic Background

We develop direct and inverse scattering theory for Jacobi operators with steplike quasi-periodic finite-gap background in the same isospectral class. We derive the corresponding Gel'fand-Levitan-Marchenko equation and find minimal scattering data which determine the perturbed operator uniquely. In addition, we show how the transmission coefficients can be reconstructed from the eigenvalues and one of the reflection coefficients.Comment: 14 page

Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies

Combining algebro-geometric methods and factorization techniques for finite difference expressions we provide a complete and self-contained treatment of all real-valued quasi-periodic finite-gap solutions of both the Toda and Kac-van Moerbeke hierarchies. In order to obtain our principal new result, the algebro-geometric finite-gap solutions of the Kac-van Moerbeke hierarchy, we employ particular commutation methods in connection with Miura-type transformations which enable us to transfer whole classes of solutions (such as finite-gap solutions) from the Toda hierarchy to its modified counterpart, the Kac-van Moerbeke hierarchy, and vice versa.Comment: LaTeX, to appear in Memoirs of the Amer. Math. So

Invisibility in non-Hermitian tight-binding lattices

Reflectionless defects in Hermitian tight-binding lattices, synthesized by the intertwining operator technique of supersymmetric quantum mechanics, are generally not invisible and time-of-flight measurements could reveal the existence of the defects. Here it is shown that, in a certain class of non-Hermitian tight-binding lattices with complex hopping amplitudes, defects in the lattice can appear fully invisible to an outside observer. The synthesized non-Hermitian lattices with invisible defects possess a real-valued energy spectrum, however they lack of parity-time (PT) symmetry, which does not play any role in the present work.Comment: to appear in Phys. Rev.

Relative Oscillation Theory, Weighted Zeros of the Wronskian, and the Spectral Shift Function

We develop an analog of classical oscillation theory for Sturm-Liouville operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing zeros of solutions of one operator by weighted zeros of Wronskians of solutions of two different operators. In particular, we show that a Sturm-type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the finiteness of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein's spectral shift function is established.Comment: 26 page

Preliminary evaluation of polarimetric parameters from a new dual-polarization C-band weather radar in an alpine region

The first operational weather radar with dual polarization capabilities was recently installed in Austria. The use of polarimetric radar variables rises several expectations: an increased accuracy of the rain rate estimation compared to standard Z-R relationships, a reliable use of attenuation correction methods, and finally hydrometeor classification. In this study the polarimetric variables of precipitation events are investigated and the operational quality of the parameters is discussed. For the new weather radar also several polarimetric rain rate estimators, which are based on the horizontal polarization radar reflectivity, <i>Z</i>_H, the differential reflectivity, <i>Z</i>_DR, and the specific differential propagation phase shift, <i>K</i>_DP, have been tested. The rain rate estimators are further combined with an attenuation correction scheme. A comparison between radar and rain gauge indicates that <i>Z</i>_DR based rain rate algorithms show an improvement over the traditional Z-R estimate. <i>K</i>_DP based estimates do not provide reliable results, mainly due to the fact, that the observed <i>K</i>_DP parameters are quite noisy. Furthermore the observed rain rates are moderate, where <i>K</i>_DP is less significant than in heavy rain