Wigner’s infinite-product representation of theRmatrix: Rederivation, interpretation, and potential applications

1986 ◽  
Vol 34 (3) ◽  
pp. 1896-1904 ◽  
Author(s):  
E. J. Robinson
Keyword(s):  
Infinite Product ◽  
Product Representation ◽  
Potential Applications ◽  
Infinite Product Representation

scholarly journals DUAL EQUATION AND INVERSE PROBLEM FOR AN INDEFINITE STURM–LIOUVILLE PROBLEM WITH M TURNING POINTS OF EVEN ORDER

2012 ◽  
Vol 17 (5) ◽  
pp. 618-629
Author(s):  
Hamidreza Marasi ◽  
Aliasghar Jodayree Akbarfam
Keyword(s):  
Inverse Problem ◽  
Infinite Product ◽  
Turning Points ◽  
Liouville Problem ◽  
Product Representation ◽  
Dual Equation ◽  
Number Of Zeros ◽  
Even Order ◽  
Sturm Liouville ◽  
Infinite Product Representation

In this paper the differential equation y″ + (ρ 2 φ 2 (x) –q(x))y = 0 is considered on a finite interval I, say I = [0, 1], where q is a positive sufficiently smooth function and ρ 2 is a real parameter. Also, [0, 1] contains a finite number of zeros of φ 2 , the so called turning points, 0 < x 1 < x 2 < … < x m < 1. First we obtain the infinite product representation of the solution. The product representation, satisfies in the original equation. As a result the associated dual equation is derived and then we proceed with the solution of the inverse problem.

On the Canonical Solution of the Sturm–Liouville Problem with Singularity and Turning Point of Even Order

2011 ◽  
Vol 54 (3) ◽  
pp. 506-518 ◽  
Author(s):  
A. Neamaty ◽  
S. Mosazadeh
Keyword(s):  
Liouville Equation ◽  
Turning Point ◽  
Infinite Product ◽  
Liouville Problem ◽  
Asymptotic Estimates ◽  
Product Representation ◽  
Representation Of Solutions ◽  
Even Order ◽  
Sturm Liouville ◽  
Infinite Product Representation

AbstractIn this paper, we are going to investigate the canonical property of solutions of systems of differential equations having a singularity and turning point of even order. First, by a replacement, we transform the system to the Sturm–Liouville equation with turning point. Using of the asymptotic estimates provided by Eberhard, Freiling, and Schneider for a special fundamental system of solutions of the Sturm–Liouville equation, we study the infinite product representation of solutions of the systems. Then we transform the Sturm–Liouville equation with turning point to the equation with singularity, then we study the asymptotic behavior of its solutions. Such representations are relevant to the inverse spectral problem.

scholarly journals Partition functions for supersymmetric gauge theories on spheres

2018 ◽  
Vol 33 (30) ◽  
pp. 1850179
Author(s):  
A. A. Bytsenko ◽  
M. Chaichian ◽  
A. E. Gonçalves
Keyword(s):  
Gauge Theories ◽  
Infinite Product ◽  
Main Idea ◽  
Gauge Field Theory ◽  
Partition Functions ◽  
Bell Polynomials ◽  
Product Representation ◽  
Spectral Functions ◽  
Supersymmetric Gauge ◽  
Infinite Product Representation

In this paper we briefly review the main idea of the localization technique and its extension suitable in supersymmetric gauge field theory. We analyze the partition function of the vector multiplets with supercharges and its blocks on the even- and odd-dimensional spheres and squashed spheres. We exploit the so-called Faà di Bruno’s formula and show that multipartite partition functions can be written in the form of expansion series of the Bell polynomials. Applying the restricted specialization argument we show that q-infinite-product representation of partition functions admits presentation in terms of the Patterson–Selberg (or the Ruelle-type) spectral functions.

A new infinite product representation for real numbers

1987 ◽  
Vol 104 (1) ◽  
pp. 29-44 ◽  
Author(s):  
Arnold Knopfmacher ◽  
John Knopfmacher
Keyword(s):  
Infinite Product ◽  
Real Numbers ◽  
Product Representation ◽  
Infinite Product Representation

Bessel function Jν(z) of complex order and its zeros

1967 ◽  
Vol 63 (1) ◽  
pp. 141-146 ◽  
Author(s):  
Laurence S. Hall
Keyword(s):  
Complex Number ◽  
Numerical Evaluation ◽  
Bessel Functions ◽  
Explicit Knowledge ◽  
Infinite Product ◽  
Product Representation ◽  
Complex Order ◽  
Arbitrary Complex ◽  
Complex Zeros ◽  
Infinite Product Representation

AbstractMethods are developed for the computation of the complex zeros of (½z)−νJν(z) when the index ν is an arbitrary complex number. These methods, which do not require an explicit knowledge of the Jv(z), are susceptible to rapid numerical evaluation on a computer. Beyond the interest in the zeros in their own right, these methods now make feasible the use of the infinite product representation of Jν(z) for the rapid computation of Bessel functions of complex order.

scholarly journals Turán-Type Inequalities for Some Lommel Functions of the First Kind

2015 ◽  
Vol 59 (3) ◽  
pp. 569-579 ◽  
Author(s):  
Árpád Baricz ◽  
Stamatis Koumandos
Keyword(s):  
Classical Result ◽  
Entire Functions ◽  
Infinite Product ◽  
Product Representation ◽  
Lommel Functions ◽  
Turán Type Inequalities ◽  
Infinite Product Representation

AbstractIn this paper certain Turán-type inequalities for some Lommel functions of the first kind are deduced. The key tools in our proofs are the infinite product representation for these Lommel functions of the first kind, a classical result of Pólya on the zeros of some particular entire functions, and the connection of these Lommel functions with the so-called Laguerre–Pólya class of entire functions. Moreover, it is shown that in some cases Steinig's results on the sign of Lommel functions of the first kind combined with the so-called monotone form of l’Hospital's rule can be used in the proof of the corresponding Turán-type inequalities.

An in situ and ex situ TEM study of the formation of thin cobalt silicide films by molecular beam epitaxy on the silicon (100) surface

1988 ◽  
Vol 46 ◽  
pp. 884-885
Author(s):  
D. Loretto ◽  
J. M. Gibson ◽  
S. M. Yalisove ◽  
R. T. Tung
Keyword(s):  
Molecular Beam Epitaxy ◽  
Molecular Beam ◽  
Defect Density ◽  
High Vacuum ◽  
Ultra High Vacuum ◽  
Ex Situ ◽  
Metal Base ◽  
In Situ Experiments ◽  
Potential Applications

The cobalt disilicide/silicon system has potential applications as a metal-base and as a permeable-base transistor. Although thin, low defect density, films of CoSi2 on Si(111) have been successfully grown, there are reasons to believe that Si(100)/CoSi2 may be better suited to the transmission of electrons at the silicon/silicide interface than Si(111)/CoSi2. A TEM study of the formation of CoSi2 on Si(100) is therefore being conducted. We have previously reported TEM observations on Si(111)/CoSi2 grown both in situ, in an ultra high vacuum (UHV) TEM and ex situ, in a conventional Molecular Beam Epitaxy system.The procedures used for the MBE growth have been described elsewhere. In situ experiments were performed in a JEOL 200CX electron microscope, extensively modified to give a vacuum of better than 10-9 T in the specimen region and the capacity to do in situ sample heating and deposition. Cobalt was deposited onto clean Si(100) samples by thermal evaporation from cobalt-coated Ta filaments.

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