A generalization of Wigner's R-matrix method

A generalization of the Wigner's non-relativistic R-matrix theory of scattering by a central potential field is proposed. The idea is to use an R-matrix expansion basis generated by a Sturm-Liouville problem with an eigenparameter included both in a differential equation and in a boundary condition (in the standard theory an R-matrix basis is obtained by solving an eigenvalue problem with fixed boundary conditions). A partial fraction expansion of an R^(η)-matrix introduced is derived and shown to converge faster than a partial fraction expansion of Wigner's R-matrix used in the standard theory.     

A new proof of Wigner's theorem

In our paper we present a new, elementary proof for Wigner's famous unitary-antiunitary theorem.     

Study of Wigner's theorem on joint probabilities

The exact bearing of an important theorem proved by Wigner is established. The study brings out the fact that marginal conditions as well as mean conditions of a form currently required in joint probability attempts are in fact inadequate for the determination of a relevant concept of a joint probability. New vistas are thereby opened up.     

A generalization of the Jones theorem

The relationship is considered between the topology of a domain on which the boundary-value problem for the equation Δv + λv = f is stated and that of the corresponding Sobolev space. The Jones theorem is generalized so as to make it closer to the Weyl-von Neumann theorem.     

A generalization of Lancret’s theorem

General helices in a three dimensional Lie group with a bi-invariant metric are defined and a generalization of Lancret’s theorem is obtained. We conclude that the so-called spherical images of general helices are plane curves, and we obtain the so-called spherical general helices. We also give a relation between the geodesics of the so-called cylinders and general helices.     

A generalization of a theorem by Wightman

It is shown for some class of sets in the Minkowski space that the intersection of local Haag algebras assigned to open neighbourhoods of such a set contains only multiples of the identity.     

An elementary proof for the non-bijective version of Wigner's theorem

The non-bijective version of Wigner's theorem states that a map which is defined on the set of self-adjoint, rank-one projections (or pure states) of a complex Hilbert space and which preserves the transition probability between any two elements, is induced by a linear or antilinear isometry. We present a completely new, elementary and very short proof of this famous theorem which is very important in quantum mechanics. We do not assume bijectivity of the mapping or separability of the underlying space like in many other proofs.     

Quantum coherent operators: A generalization of coherent states

We introduce a technique to compare different, but related, quantum systems, thereby generalizing the way that coherent states are used to compare quantum systems to classical systems in semiclassical analysis. We then use this technique to estimate the dependence of the free energy of the quantum Heisenberg model on the spin value, and to estimate the relation between the ferromagnetic and antiferromagnetic free energies.Work supported in part by the U.S. National Science Foundation grant PHY-9019433.Work supported in part by the U.S. National Science Foundation grant DMS-9002416.     

A generalization of the spherical harmonic addition theorem

A generalization of the spherical harmonic addition theorem is proved. The resulting polynomial with four parameters, which corresponds to the Legendre polynomial for the usual spherical harmonic addition theorem, is expressed as four different but equivalent series. Each of them is a finite series of the Gegenbauer polynomials. Thereby the symmetry properties of this polynomial are clarified.     

A simple proof of the generalization of Israel's theorem

A simple proof of the generalization of the theorem of Israel concerning the uniqueness of the Schwarzschild black hole is presented.Dedicated to Achille Papapetrou on the occasion of his retirement.     

A generalization of Rayleigh's theorem for the infinite harmonic crystal

We show that if one replacesN masses in an infinite harmonic, perfect-dimensional crystal byN ₁ lighter andN ₂ heavier ones such thatN ₁+ N₂ =N is finite, then one introduces at most vN₁ (isolated) bound states. This can be considered as an extension of the results of Romerio and Wreszinski.Supported by the Netherlands Organization for the Advancement of Pure Research (Z.W.O.).     

A generalization of Weyl's theorem on projectively related affine connections

From a physical point of view, the geodesics in a four-dimensional Lorentzian spacetime are really significant only as point sets. In 1921 Weyl proved that two torsion-free covariant derivative operators D_M and on a manifold M have the same geodesics with possibly different parametrizations if and only if there is a 1-form α on M such that , where 1 is the identity (1,1) tensor on M. By a theorem of Ambrose, Palais and Singer [1], torsion-free covariant derivative operators are generated by affine sprays, and vice versa. More generally, any (not necessarily affine) spray induces a number of covariant derivatives in the tangent bundle τ of M, or in the pull-back bundle τ∗τ. We show that in the context of sprays, similarly to Weyl's relation, a correspondence between the Yano derivatives can be detected.     

A generalization of the virial theorem for strongly singular potentials

Using scale transformations we prove a generalization of the virial theorem for the eigenfunctions of Schrödinger Hamiltonians which are defined as the Friedrichs extension of strongly singular differential operators. Our theorem also applies to situations where the ground state has divergent kinetic and potential energy and thus the usual version of the virial theorem becomes meaningless.     

A generalization of the Gel’fond theorem concerning number systems

The distribution law for the remainders in the expansion of reals in a special number system is obtained.     

A generalization of the fluctuation-dissipation theorem for the case of excess noise

Electrical fluctuations proportional to the square of the external field are investigated in the case of quasi-equilibrium conditions. The binary correlation function, proportional to the square of the external field, is expressed in terms of the zero-field quaternary correlation function for both quantum and classical cases. This result can be considered as a generalization of the fluctuation-dissipation theorem. The 1/f spectrum is obtained as a consequence of the symmetry properties of the quaternary correlation function without consideration of the physical mechanisms responsible for the noise. The method of investigation of the 1/f spectrum through zero-field quaternary correlation function is discussed.     

A generalization of the fluctuation-dissipation theorem for the case of excess noise

Electrical fluctuations proportional to the square of the external field are investigated in the case of quasiequilibrium conditions. The binary correlation fluctuation proportional to the square of the external field is expressed in terms of the zero-field quaternary correlation function for both quantum and classical cases. This result can be considered as a generalization of the fluctuation-dissipation theorem. As applied to the electrical $\text{1}\text{f}$ noise the novel method investigation both in theory and in experiment is presented. The frequency dependence of zero-field quaternary potential correlation function is suggested to be investigated instead of binary correlation function proportional to the square of external field.