Spectral Simplicity and Asymptotic Separation of Variables

We describe a method for comparing the spectra of two real-analytic families, (a _t ) and (q _t ), of quadratic forms that both degenerate as a positive parameter t tends to zero. We suppose that the family (a _t ) is amenable to ‘separation of variables’ and that each eigenspace of a _t is 1-dimensional for some t. We show that if (q _t ) is asymptotic to (a _t ) at first order as t → 0, then the eigenspaces of (q _t ) are also 1-dimensional for all but countably many t. As an application, we prove that for the generic triangle (simplex) in Euclidean space (constant curvature space form) each eigenspace of the Laplacian acting on Dirichlet functions is 1-dimensional.     

Erratum to: A Highly Stable Source of Spectral Lines

Optics and Spectroscopy - The ACKNOWLEDGMENTS should read as follows:     

Separation of Variables for Symplectic Characters

We perform separation of variables for the symplectic Weyl character using Sklyanin’s scheme. Viewing the characters as eigenfunctions of a quantum integrable system, we explicitly construct the separating operator using the Q-operator method. We also construct the inverse of the separating operator, as well as the factorised Hamiltonian.     

Separation of Variables for Bi-Hamiltonian Systems

We address the problem of the separation of variables for the Hamilton–Jacobi equation within the theoretical scheme of bi-Hamiltonian geometry. We use the properties of a special class of bi-Hamiltonian manifolds, called N manifolds, to give intrisic tests of separability (and Stäckel separability) for Hamiltonian systems. The separation variables are naturally associated with the geometrical structures of the N manifold itself. We apply these results to bi-Hamiltonian systems of the Gel'fand–Zakharevich type and we give explicit procedures to find the separated coordinates and the separation relations.     

Separation of Variables for the Ruijsenaars System

We construct a separation of variables for the classical n-particle Ruijsenaars system (the relativistic analog of the elliptic Calogero-Moser system). The separated coordinates appear as the poles of the properly normalised eigenvector (Baker-Akhiezer function) of the corresponding Lax matrix. Two different normalisations of the BA functions are analysed. The canonicity of the separated variables is verified with the use of the r-matrix technique. The explicit expressions for the generating function of the separating canonical transform are given in the simplest cases n=2 and n=3. Taking the nonrelativistic limit we also construct a separation of variables for the elliptic Calogero-Moser system. Received: 10 January 1997 / Accepted: 1 April 1997     

连续投影算法及其在小麦近红外光谱波长选择中的应用

采用全谱建立多元校正模型时,通常计算量大,模型不够稳健,而且模型的预测精度往往也不能达到最优。文章介绍一种新的波长选择方法：采用连续投影算法(successive projections algorithm),并将其集成偏最小二乘(partial least squares)多变量校正技术构成SPA-PLS方法,用于谷物小麦近红外光谱波长优化选择及其与水分含量的定量分析。结果表明：在经SPA算法后,光谱波数可削减97.72%,后继的定量校正模型结构得到显著简化,模型的稳健性也大大增强;同时,被选取的波长物理意义明确,模型的解释能力增强,而模型的预测性能也与GA-PLS方法相当。     

Extended Group Foliation Method and Functional Separation of Variables to Nonlinear Wave Equations

Generalized functional separation of variables to nonlinear evolution equations is studied in terms of the extended group foliation method, which is based on the Lie point symmetry method. The approach is applied to nonlinear wave equations with variable speed and external force. A complete classification for the wave equation which admits functional separable solutions is presented. Some known results can be recovered by this approach.     

Extended Group Foliation Method and Functional Separation of Variables to Nonlinear Wave Equations

Generalized functional separation of variables to nonlinear evolution equations is studied in terms of the extended group foliation method, which is based on the Lie point symmetry method. The approach is applied to nonlinear wave equations with variable speed and external force. A complete classification for the wave equation which admits functional separable solutions is presented. Some known results can be recovered by this approach.     

Erratum to: Determination of Asymptotic Normalization Coefficients by Analytic Continuation of Differential Cross Sections

Physics of Atomic Nuclei - An Erratum to this paper has been published: https://doi.org/10.1134/S106377882233002X     

Asymptotic Spectral Measures, Quantum Mechanics, and E-Theory

We study the relationship between POV-measures in quantum theory and asymptotic morphisms in the operator algebra E-theory of Connes–Higson. This is done by introducing the theory of “asymptotic” PV-measures and their integral correspondence with positive asymptotic morphisms on locally compact spaces. Examples and applications involving various aspects of quantum physics, including quantum noise models, semiclassical limits, strong deformation quantizations, and pure half-spin particles, are also discussed. Received: 27 December 2000 / Accepted: 8 October 2001     

Separation of Variables in the Elliptic Gaudin Model

For the elliptic Gaudin model (a degenerate case of the XYZ integrable spin chain) a separation of variables is constructed in the classical case. The corresponding separated coordinates are obtained as the poles of a suitably normalized Baker-Akhiezer function. The classical results are generalized to the quantum case where the kernel of the separating integral operator is constructed. The simplest one-degree-of-freedom case is studied in detail. Received: 21 August 1998 / Accepted: 12 January 1999     

Light Scattering by Small Multilayer Particles: A Generalized Separation of Variables Method

A Rayleigh approximation is constructed for light scattering by small multilayer axisymmetric particles, in which their polarizability is determined by the generalized separation of variables method (SVM). In this method, scalar potentials, the gradients of which yield the electric-field strengths, are represented as expansions in spherical harmonics of the Laplace equation. Unknown coefficients of expansions are determined from the boundary conditions, which are reduced to infinite systems of linear algebraic equations (ISLAEs), since the separation of variables is incomplete. The T matrix of the electrostatic problem, principal element T₁₁ of which is proportional to the particle polarizability, is determined. The necessary condition for the ISLAEs solvability for the SVM coincides with the condition of the correct application of the extended boundary conditions method (EBCM). However, numerical calculations in which finite-dimensional (i.e., reduced) systems are solved, yield different results in ranges of variation of parameters that are close to the boundary of the range of applicability. An analysis of the numerical calculations of the scattering and absorption cross sections for two-layer confocal spheroids, an exact solution for which can be obtained using spheroidal harmonics, shows that the SVM is preferable to the EBCM. It turned out that the proposed method yields workable results in a wider range of variation of parameters. Even outside the range of applicability, in which it should be regarded as a certain approximate solution, its use in a number of cases is quite acceptable. Additional calculations for three-layer nonconfocal spheroids, as well as for three-layer similar pseudospheroids and Pascal’s snails, which can be obtained from spheroids as a result of the inversion with respect to the coordinate origin and one of the foci, respectively, confirm these inferences. We note that, for certain values of the parameters, the shapes of the latter particles are nonconvex.     

A Covariant Poisson Deformation Quantization with Separation of Variables up to the Third Order

We give a simple formula for the operator C ₃ of the standard deformation quantization with separation of variables on a Kähler manifold M. Unlike C ₁ and C ₂, this operator cannot be expressed in terms of the Kähler–Poisson tensor on M. We modify C ₃ to obtain a covariant deformation quantization with separation of variables up to the third order which is expressed in terms of the Poisson tensor on M and can thus be defined on an arbitrary complex manifold endowed with a Poisson bivector field of type (1,1).     

Renormalization Group and Asymptotic Spin-Charge Separation for Chiral Luttinger Liquids

The phenomenon of Spin-Charge separation in non-Fermi liquids is well understood only in certain solvable d=1 fermionic systems. In this paper we furnish the first example of asymptotic Spin-Charge separation in a d=1 nonsolvable model. This goal is achieved using Renormalization Group approach combined with Ward-Identities and Schwinger-Dyson equations, corrected by the presence of a bandwidth cut-offs. Such methods, contrary to bosonization, could be in principle applied also to lattice or higher dimensional systems.     

Asymptotic Behaviour of Spectral Functions and Singularities of Wightman Functions and Field Commutators

Singularties of Wightman functions and field commutators are studied in localizable and nonlocalizable cases including the spectral behaviour σ(q²) = exp (αq²), α > 0.     

Commuting Operators and Separation of Variables for Laplacians of Projectively Equivalent Metrics

Let two Riemannian metrics g and g on one manifold M ⁿ have the same geodesics (considered as unparameterized curves). Then we can construct invariantly n commuting differential operators of second order. The Laplacian _g of the metric g is one of these operators. For any x M ⁿ , consider the linear transformation G of T _x M ⁿ given by the tensor g ^Ig_j . If all eigenvalues of G are different at one point of the manifold then they are different at almost every point; the operators are linearly independent and their symbols are functionally independent. If all eigenvalues of G are different at each point of a closed manifold then it can be covered by the n-torus and we can globally separate the variables in the equation _g f = f on this torus.     

Star Products with Separation of Variables Admitting a Smooth Extension

Given a complex manifold M with an open dense subset ?? endowed with a pseudo-K?hler form ?? which cannot be smoothly extended to a larger open subset, we consider various examples where the corresponding K?hler?CPoisson structure and a star product with separation of variables on (??, ??) admit smooth extensions to M. We give a simple criterion of the existence of a smooth extension of a star product and apply it to these examples.