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Oscillation Theory
In mathematics, in the field of ordinary differential equations, a nontrivial solution to an ordinary differential equation :F(x,y,y',\ \dots,\ y^)=y^ \quad x \in spectrum of associated boundary value problems. Examples The differential equation :y'' + y = 0 is oscillating as sin(''x'') is a solution. Connection with spectral theory Oscillation theory was initiated by Jacques Charles François Sturm in his investigations of Sturm–Liouville problems from 1836. There he showed that the n'th eigenfunction of a Sturm–Liouville problem has precisely n-1 roots. For the one-dimensional Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ... the question about oscillation/non-oscillation answers the question whether the eigenvalues accumulate at the bottom of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barry Simon
Barry Martin Simon (born 16 April 1946) is an American mathematical physicist and was the IBM professor of Mathematics and Theoretical Physics at Caltech, known for his prolific contributions in spectral theory, functional analysis, and nonrelativistic quantum mechanics (particularly Schrödinger operators), including the connections to atomic and molecular physics. He has authored more than 400 publications on mathematics and physics. His work has focused on broad areas of mathematical physics and analysis covering: quantum field theory, statistical mechanics, Brownian motion, random matrix theory, general nonrelativistic quantum mechanics (including N-body systems and resonances), nonrelativistic quantum mechanics in electric and magnetic fields, the semi-classical limit, the singular continuous spectrum, random and ergodic Schrödinger operators, orthogonal polynomials, and non- selfadjoint spectral theory. Early life Barry Simon's mother was a school teacher, his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the '' American Journal of Mathematics''. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sturm Separation Theorem
In mathematics, in the field of ordinary differential equations, Sturm separation theorem, named after Jacques Charles François Sturm, describes the location of roots of solutions of homogeneous second order linear differential equations. Basically the theorem states that given two linear independent solutions of such an equation the zeros of the two solutions are alternating. Sturm separation theorem If ''u''(''x'') and ''v''(''x'') are two non-trivial continuous linearly independent solutions to a homogeneous second order linear differential equation with ''x''0 and ''x''1 being successive roots of ''u''(''x''), then ''v''(''x'') has exactly one root in the open interval (''x''0, ''x''1). It is a special case of the Sturm-Picone comparison theorem. Proof Since \displaystyle u and \displaystyle v are linearly independent it follows that the Wronskian \displaystyle W ,v/math> must satisfy W ,vx)\equiv W(x)\neq 0 for all \displaystyle x where the differential equation is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sturm–Picone Comparison Theorem
In mathematics, in the field of ordinary differential equations, the Sturm–Picone comparison theorem, named after Jacques Charles François Sturm and Mauro Picone, is a classical theorem which provides criteria for the oscillation and non-oscillation of solutions of certain linear differential equations in the real domain. Let , for be real-valued continuous functions on the interval and let #(p_1(x) y^\prime)^\prime + q_1(x) y = 0 #(p_2(x) y^\prime)^\prime + q_2(x) y = 0 be two homogeneous linear second order differential equations in self-adjoint form with :0 < p_2(x) \le p_1(x) and : Let be a non-trivial solution of (1) with successive roots at and and let be a non-trivial solution of (2). Then one of the following properties holds. *There exists an in such that or *there exists a in R such that . The first part of the conclusion is due to Sturm (1836), while the second (alternative) part of the theorem is due to Picone (19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kneser's Theorem (differential Equations)
In mathematics, the Kneser theorem can refer to two distinct theorems in the field of ordinary differential equations: * the first one, named after Adolf Kneser, provides criteria to decide whether a differential equation is oscillating or not; * the other one, named after Hellmuth Kneser, is about the topology of the set of all solutions of an initial value problem with continuous right hand side. Statement of the theorem due to A. Kneser Consider an ordinary linear homogeneous differential equation of the form :y'' + q(x)y = 0 with :q: continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous .... We say this equation is ''oscillating'' if it has a solution ''y'' with infinitely many zeros, and ''non-oscillating'' otherwise. The theorem states that the equation is non-osci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wronski Determinant
In mathematics, the Wronskian of ''n'' differentiable functions is the determinant formed with the functions and their derivatives up to order . It was introduced in 1812 by the Polish mathematician Józef Wroński, and is used in the study of differential equations, where it can sometimes show the linear independence of a set of solutions. Definition The Wrońskian of two differentiable functions and is W(f,g)=f g' - g f' . More generally, for real- or complex-valued functions , which are times differentiable on an interval , the Wronskian W(f_1,\ldots,f_n) is a function on x\in I defined by W(f_1, \ldots, f_n) (x)= \det \begin f_1(x) & f_2(x) & \cdots & f_n(x) \\ f_1'(x) & f_2'(x) & \cdots & f_n' (x)\\ \vdots & \vdots & \ddots & \vdots \\ f_1^(x)& f_2^(x) & \cdots & f_n^(x) \end. This is the determinant of the matrix constructed by placing the functions in the first row, the first derivatives of the functions in the second row, and so on through the (n-1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gerald Teschl
Gerald Teschl (born 12 May 1970 in Graz) is an Austrian mathematical physicist and professor of mathematics. He works in the area of mathematical physics; in particular direct and inverse spectral theory with application to completely integrable partial differential equations (soliton equations). Career After studying physics at the Graz University of Technology (diploma thesis 1993), he continued with a PhD in mathematics at the University of Missouri. The title of his thesis supervised by Fritz Gesztesy was ''Spectral Theory for Jacobi Operators'' (1995). After a postdoctoral position at the Rheinisch-Westfälischen Technische Hochschule Aachen (1996/97), he moved to Vienna, where he received his Habilitation at the University of Vienna in May 1998. Since then he has been a professor of mathematics there. In 1997 he received the Ludwig Boltzmann Prize from the Austrian Physical Society, 1999 the Prize of the Austrian Mathematical Society. In 2006 he was awarded with the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fritz Gesztesy
Friedrich "Fritz" Gesztesy (born 5 November 1953 in Austria) is a well-known Austrian-American mathematical physicist and Professor of Mathematics at Baylor University, known for his important contributions in spectral theory, functional analysis, nonrelativistic quantum mechanics (particularly, Schrödinger operators), ordinary and partial differential operators, and completely integrable systems (soliton equations). He has authored more than 300 publications on mathematics and physics. Career After studying physics at the University of Graz, he continued with his PhD in theoretical physics. The title of his dissertation 1976 with Heimo Latal and Ludwig Streit was ''Renormalization, Nelson's symmetry and energy densities in a field theory with quadratic interaction''. After working at the Institut for Theoretical Physics of the University of Graz (1977–82) and several stays abroad at the Bielefeld University (Alexander von Humboldt Scholarship 1980–81 and 1983–84) an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematics), function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equation, ''partial'' differential equations (PDEs) which may be with respect to one independent variable, and, less commonly, in contrast with stochastic differential equations, ''stochastic'' differential equations (SDEs) where the progression is random. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where a_0(x),\ldots,a_n(x) and b(x) are arbitrary differentiable functions that do not need to be linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schrödinger Equation
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after Erwin Schrödinger, an Austrian physicist, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of the wave function, the quantum-mechanical characterization of an isolated physical system. The equation was postulated by Schrödinger based on a postulate of Louis de Broglie that all matter has an associated matter wave. The equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sturm–Liouville Theory
In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form \frac \left (x) \frac\right+ q(x)y = -\lambda w(x) y for given functions p(x), q(x) and w(x), together with some boundary conditions at extreme values of x. The goals of a given Sturm–Liouville problem are: * To find the for which there exists a non-trivial solution to the problem. Such values are called the ''eigenvalues'' of the problem. * For each eigenvalue , to find the corresponding solution y = y(x) of the problem. Such functions y are called the '' eigenfunctions'' associated to each . Sturm–Liouville theory is the general study of Sturm–Liouville problems. In particular, for a "regular" Sturm–Liouville problem, it can be shown that there are an infinite number of eigenvalues each with a unique eigenfunction, and that these eigenfunctions form an orthonormal basis of a certain Hilbert space of functions. This theory is important ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
