discrete spectrum

In mathematics and physics, continuous spectrum is, roughly speaking, a non-countable set of eigenvalues of an operator. An operator acting on a Hilbert space is said to have a continuous spectrum if its eigenvalues can be changed continuously. If the spectrum of an operator is not continuous, we say that it is has discrete spectrum. Some of the basic questions in spectral theory are to characterise the discrete spectrum and purely continuous spectrum, just as a measure, such as a probability measure, can typically ...

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Obviously, if we were to vary over all possible states with norm 1 trying to minimize the expectation value of H, the lowest value would be E0 and the corresponding state would be an eigenstate of E0. Varying over the entire Hilbert space is usually too complicated for physical calculations, and a subspace of the entire Hilbert space is chosen, parametrized by some (real) differentiable parameters αi (i=1,2..,N). The choice of the subspace is called the ansatz. Some choices of ansatzes lead to better approxima ...

See also:

Variational method quantum mechanics, Variational method quantum mechanics - Introduction, Variational method quantum mechanics - Ansatz

Read more here: » Variational method quantum mechanics: Encyclopedia II - Variational method quantum mechanics - Ansatz

The starting point of the program may be seen as the Artin reciprocity law which generalizes quadratic reciprocity. The Artin reciprocity law applies to an algebraic number field whose Galois group over Q is abelian, assigns L-functions to the one-dimensional representations of this Galois group; and states that these L-functions are identical to certain Dirichlet L-series (that is, the analogues of the Riemann zeta function constructed from Dirichlet characters). The precise correspondence between these ...

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Langlands program, Langlands program - Connection with number theory, Langlands program - The setting of automorphic representations, Langlands program - A general principle of functoriality, Langlands program - Ideas leading up to the Langlands program, Langlands program - Prizes

Read more here: » Langlands program: Encyclopedia II - Langlands program - Connection with number theory

Suppose we are given a Hilbert space and a Hermitian operator over it called the Hamiltonian H. Ignoring complications about continuous spectra, we look at the discrete spectrum of H and the corresponding eigenspaces of each eigenvalue λ (see spectral theorem for Hermitian operators for the mathematical background): with and . Physical states are normalized, meaning that their norm is equal to 1. Once again ignoring complica ...

See also:

Variational method quantum mechanics, Variational method quantum mechanics - Introduction, Variational method quantum mechanics - Ansatz

Read more here: » Variational method quantum mechanics: Encyclopedia II - Variational method quantum mechanics - Introduction

Langlands then generalized things further: instead of using the general linear group GLn, other connected reductive groups can be used. Furthermore, given such a group G, Langlands constructs a complex Lie group LG, and then, for every automorphic cuspidal representation of G and every finite-dimensional representation of LG, he defines an L-function. One of his conjectures states that these L-functions satisfy a ...

See also:

Langlands program, Langlands program - Connection with number theory, Langlands program - The setting of automorphic representations, Langlands program - A general principle of functoriality, Langlands program - Ideas leading up to the Langlands program, Langlands program - Prizes

Read more here: » Langlands program: Encyclopedia II - Langlands program - A general principle of functoriality

The insight of Langlands was to find the proper generalization of Dirichlet L-functions which would allow the formulation of Artin's statement in this more general setting. Hecke had earlier related Dirichlet L-functions with automorphic forms (holomorphic functions on the upper half plane of C that satisfy certain functional equations). Langlands then generalized these to automorphic cuspidal representations, which are certain infinite dimensional irreducible representations of the general linear group GLn over the adele ring of Q. (This ring simultaneo ...

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Langlands program, Langlands program - Connection with number theory, Langlands program - The setting of automorphic representations, Langlands program - A general principle of functoriality, Langlands program - Ideas leading up to the Langlands program, Langlands program - Prizes

Read more here: » Langlands program: Encyclopedia II - Langlands program - The setting of automorphic representations

In a very broad context, the program built on existing ideas: the philosophy of cusp forms formulated a few years earlier by Israel Gelfand, the work and approach of Harish-Chandra on semisimple Lie groups, and in technical terms the trace formula of Selberg and others. What initially was very new in Langlands' work, besides technical depth, was the proposed direct connection to number theory, together with t ...

See also:

Langlands program, Langlands program - Connection with number theory, Langlands program - The setting of automorphic representations, Langlands program - A general principle of functoriality, Langlands program - Ideas leading up to the Langlands program, Langlands program - Prizes

Read more here: » Langlands program: Encyclopedia II - Langlands program - Ideas leading up to the Langlands program