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Marchenko Equation
In mathematical physics, more specifically the one-dimensional inverse scattering problem, the Marchenko equation (or Gelfand-Levitan-Marchenko equation or GLM equation), named after Israel Gelfand, Boris Levitan and Vladimir Marchenko, is derived by computing the Fourier transform of the scattering relation: : K(r,r^\prime) + g(r,r^\prime) + \int_r^ K(r,r^) g(r^,r^\prime) \mathrmr^ = 0 Where g(r,r^\prime)\,is a symmetric kernel, such that g(r,r^\prime)=g(r^\prime,r),\,which is computed from the scattering data. Solving the Marchenko equation, one obtains the kernel of the transformation operator K(r,r^\prime) from which the potential can be read off. This equation is derived from the Gelfand–Levitan integral equation, using the Povzner–Levitan representation. Application to scattering theory Suppose that for a potential u(x) for the Schrödinger operator L = -\frac + u(x), one has the scattering data (r(k), \), where r(k) are the reflection coefficients from continuous s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics, known as physical mathematics. Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical parts of our world. Classical mechanics Applying the techniques of mathematical physics to classical mechanics typically involves the rigorous, abstract, and advanced reformulation of Newtonian mechanics in terms of Lagrangian mechanics and Hamiltonian mechanics (including both approaches in the presence of constraints). Both formulations are embodied in analytical mechanics and lead ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Scattering Problem
In mathematics and physics, the inverse scattering problem is the problem of determining characteristics of an object, based on data of how it scatters incoming radiation or particles. It is the inverse problem to the direct scattering problem, which is to determine how radiation or particles are scattered based on the properties of the scatterer. Soliton equations are a class of partial differential equations which can be studied and solved by a method called the inverse scattering transform, which reduces the nonlinear PDEs to a linear inverse scattering problem. The nonlinear Schrödinger equation, the Korteweg–de Vries equation and the KP equation are examples of soliton equations. In one space dimension the inverse scattering problem is equivalent to a Riemann-Hilbert problem. Inverse scattering has been applied to many problems including radiolocation, echolocation, geophysical survey, nondestructive testing, medical imaging, and quantum field theory In theoretical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Israel Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand (, , ; – 5 October 2009) was a prominent Soviet and American mathematician, one of the greatest mathematicians of the 20th century, biologist, teacher and organizer of mathematical education. He made significant contributions to many branches of mathematics, including group theory, representation theory and functional analysis. The recipient of many awards, including the Order of Lenin and the first Wolf Prize, he was a Foreign Fellow of the Royal Society and professor at Moscow State University and, after immigrating to the United States shortly before his 76th birthday, at Rutgers University. Gelfand is also a 1994 MacArthur Fellow. His legacy continues through his students, who include Endre Szemerédi, Alexandre Kirillov, Edward Frenkel, Joseph Bernstein, David Kazhdan, as well as his own son, Sergei Gelfand. Early years A native of Kherson Governorate, Russian Empire (now, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boris Levitan
Boris Levitan (7 June 1914 – 4 April 2004) was a mathematician who worked on almost periodic functions, Sturm–Liouville operators and inverse scattering. Levitan was born in Berdyansk (southeastern Ukraine), and grew up in Kharkiv. He graduated from Kharkov University in 1936. In 1938, he submitted his PhD thesis "Some Generalization of Almost Periodic Function" under the supervision of Naum Akhiezer. He then defended the habilitation thesis "Theory of Generalized Translation Operators". Levitan was drafted into the army at the beginning of World War II in 1941, and served until 1944. From 1944 to 1961, he worked at the Dzerzhinsky Military Academy, and from 1961 until about 1992 at Moscow University. In 1992, he emigrated to the United States. During the last years of his life, he worked for the University of Minnesota The University of Minnesota Twin Cities (historically known as University of Minnesota) is a public university, public Land-grant universit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Marchenko
Volodymyr Oleksandrovych Marchenko (; born 7 July 1922) is a Ukrainian mathematician who specialises in mathematical physics. Biography Volodymyr Oleksandrovych Marchenko was born in Kharkiv on 7 July 1922. He defended his PhD thesis in 1948 under the supervision of Naum Landkof, and in 1951, he defended his DSc thesis. He worked in Kharkiv University until 1961. For 4 decades, he headed the Mathematical Physics Department at the Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine. Marchenko was awarded the Lenin Prize in 1962, the N. N. Krylov Prize in 1980, the State Prize of Ukraine in Science and Technology in 1989, and the N. N. Bogolyubov prize in 1996. Since 1969 he is a member of the National Academy of Sciences of Ukraine, since 1987 of the Russian Academy of Sciences and since 2001 of the Royal Norwegian Society of Sciences and Letters The Royal Norwegian Society of Sciences and Letters (, DKNVS) is a Norway, Nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Kernel
In mathematics, an integral transform is a type of transform that maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the ''inverse transform''. General form An integral transform is any transform ''T'' of the following form: :(Tf)(u) = \int_^ f(t)\, K(t, u)\, dt The input of this transform is a function ''f'', and the output is another function ''Tf''. An integral transform is a particular kind of mathematical operator. There are numerous useful integral transforms. Each is specified by a choice of the function K of two variables, that is called the kernel or nucleus of the transform. Some kernels have an associated ''inverse kernel'' K^( u,t ) which (roughly speaking) yields an inverse transform: :f( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schrödinger Operator
In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's ''energy spectrum'' or its set of ''energy eigenvalues'', is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory. The Hamiltonian is named after William Rowan Hamilton, who developed a revolutionary reformulation of Newtonian mechanics, known as Hamiltonian mechanics, which was historically important to the development of quantum physics. Similar to vector notation, it is typically denoted by \hat, where the hat indicates that it is an operator. It can also be written as H or \check. Introduction The Hamiltonian of a system represents the total energy of the system; that is, the sum of the kinetic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scattering
In physics, scattering is a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including particles and radiation) in the medium through which they pass. In conventional use, this also includes deviation of reflected radiation from the angle predicted by the law of reflection. Reflections of radiation that undergo scattering are often called ''diffuse reflections'' and unscattered reflections are called ''specular'' (mirror-like) reflections. Originally, the term was confined to light scattering (going back at least as far as Isaac Newton in the 17th century). As more "ray"-like phenomena were discovered, the idea of scattering was extended to them, so that William Herschel could refer to the scattering of "heat rays" (not then recognized as electromagnetic in nature) in 1800. John Tyndall, a pioneer in light scattering research, noted the connecti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lax Pair
A lax is a salmon. LAX as an acronym most commonly refers to Los Angeles International Airport in Southern California, United States. LAX or Lax may also refer to: Places Within Los Angeles * Union Station (Los Angeles), Los Angeles' main train depot, whose Amtrak station code is "LAX" * The Port of Los Angeles, whose port identifier code is "LAX" Other * Lax, Switzerland, a municipality of the canton of Valais * Lax Lake (other) * La Crosse, Wisconsin, a city on the Mississippi River Sports * Los Angeles Xtreme, a former American football team * Lacrosse, a sport * The Latin American Xchange, a professional wrestling stable Media and entertainment * ''LAX'' (album), the third studio album from rapper The Game * ''LAX'' (TV series), a 2004–05 television series set in Los Angeles International Airport * " LA X", the two-part sixth season 2010 premiere of the television show ''Lost'' * LAX, a night club at Luxor Las Vegas * "LAX", a song by the rapper X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
