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About: Eigenfunction is a research topic. Over the lifetime, 11183 publications have been published within this topic receiving 216125 citations.

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01 Jan 1947
TL;DR: In this paper, the authors present an algebraic extension of LINEAR TRANSFORMATIONS and QUADRATIC FORMS, and apply it to EIGEN-VARIATIONS.
Abstract: Partial table of contents: THE ALGEBRA OF LINEAR TRANSFORMATIONS AND QUADRATIC FORMS. Transformation to Principal Axes of Quadratic and Hermitian Forms. Minimum-Maximum Property of Eigenvalues. SERIES EXPANSION OF ARBITRARY FUNCTIONS. Orthogonal Systems of Functions. Measure of Independence and Dimension Number. Fourier Series. Legendre Polynomials. LINEAR INTEGRAL EQUATIONS. The Expansion Theorem and Its Applications. Neumann Series and the Reciprocal Kernel. The Fredholm Formulas. THE CALCULUS OF VARIATIONS. Direct Solutions. The Euler Equations. VIBRATION AND EIGENVALUE PROBLEMS. Systems of a Finite Number of Degrees of Freedom. The Vibrating String. The Vibrating Membrane. Green's Function (Influence Function) and Reduction of Differential Equations to Integral Equations. APPLICATION OF THE CALCULUS OF VARIATIONS TO EIGENVALUE PROBLEMS. Completeness and Expansion Theorems. Nodes of Eigenfunctions. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS. Bessel Functions. Asymptotic Expansions. Additional Bibliography. Index.

7,426 citations

Journal ArticleDOI
TL;DR: In this article, the authors define a set of generalized density matrices for the Hermitean density matrix of order $k, which is further antisymmetric in each set of these indices.
Abstract: In order to calculate the average value of a physical quantity containing also many-particle interactions in a system of $N$ antisymmetric particles, a set of generalized density matrices are defined. In order to permit the investigation of the same physical situation in two complementary spaces, the Hermitean density matrix of order $k$ has two sets of indices of each $k$ variables, and it is further antisymmetric in each set of these indices.Every normalizable antisymmetric wave function may be expanded in a series of determinants of order $N$ over all ordered configurations formed from a basic complete set of one-particle functions ${\ensuremath{\psi}}_{k}$, which gives a representation of the wave function and its density matrices also in the discrete $k$-space. The coefficients in an expansion of an eigenfunction to a particular operator may be determined by the variation principle, leading to the ordinary secular equation of the method of configurational interaction. It is shown that the first-order density matrix may be brought to diagonal form, which defines the "natural spin-orbitals" associated with the system. The situation is then partly characterized by the corresponding occupation numbers, which are shown to lie between 0 and 1 and to assume the value 1, only if the corresponding spin-orbital occurs in all configurations necessary for describing the situation. If the system has exactly $N$ spin-orbitals which are fully occupied, the total wave function may be reduced to a single Slater determinant. However, due to the mutual interaction between the particles, this limiting case is never physically realized, but the introduction of natural spin-orbitals leads then instead to a configurational expansion of most rapid convergence.In case the basic set is of finite order $M$, the best choice of this set is determined by a form of extended Hartree-Fock equations. It is shown that, in this case, the natural spin-orbitals approximately fulfill some equations previously proposed by Slater.

2,724 citations

Journal ArticleDOI
TL;DR: In this paper, the spectral properties of solutions to the time-dependent Schrodinger equation were used to determine the eigenvalues and eigenfunctions of the Schrodings equation.
Abstract: A new computational method for determining the eigenvalues and eigenfunctions of the Schrodinger equation is described. Conventional methods for solving this problem rely on diagonalization of a Hamiltonian matrix or iterative numerical solutions of a time independent wave equation. The new method, in contrast, is based on the spectral properties of solutions to the time-dependent Schrodinger equation. The method requires the computation of a correlation function 〈ψ( r , 0)| ψ( r , t )〉 from a numerical solution ψ( r , t ). Fourier analysis of this correlation function reveals a set of resonant peaks that correspond to the stationary states of the system. Analysis of the location of these peaks reveals the eigenvalues with high accuracy. Additional Fourier transforms of ψ( r , t ) with respect to time generate the eigenfunctions. The effectiveness of the method is demonstrated for a one-dimensional asymmetric double well potential and for the two-dimensional Henon-Heiles potential.

2,646 citations

Journal ArticleDOI
Elliott H. Lieb1, Werner Liniger1
TL;DR: In this paper, the ground-state energy as a function of γ was derived for all γ, except γ = 0, and it was shown that Bogoliubov's perturbation theory is valid when γ is small.
Abstract: A gas of one-dimensional Bose particles interacting via a repulsive delta-function potential has been solved exactly. All the eigenfunctions can be found explicitly and the energies are given by the solutions of a transcendental equation. The problem has one nontrivial coupling constant, γ. When γ is small, Bogoliubov’s perturbation theory is seen to be valid. In this paper, we explicitly calculate the ground-state energy as a function of γ and show that it is analytic for all γ, except γ=0. In Part II, we discuss the excitation spectrum and show that it is most convenient to regard it as a double spectrum—not one as is ordinarily supposed.

2,230 citations

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