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  2. Eigenvalue perturbation - Wikipedia

    en.wikipedia.org/wiki/Eigenvalue_perturbation

    In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system = that is perturbed from one with known eigenvectors and eigenvalues =. This is useful for studying how sensitive the original system's eigenvectors and eigenvalues x 0 i , λ 0 i , i = 1 , … n {\displaystyle x_{0i},\lambda _{0i ...

  3. Perturbation theory (quantum mechanics) - Wikipedia

    en.wikipedia.org/wiki/Perturbation_theory...

    Perturbation theory is an important tool for describing real quantum systems, as it turns out to be very difficult to find exact solutions to the Schrödinger equation for Hamiltonians of even moderate complexity.

  4. Perturbation theory - Wikipedia

    en.wikipedia.org/wiki/Perturbation_theory

    Perturbation theory develops an expression for the desired solution in terms of a formal power series known as a perturbation series in some "small" parameter, that quantifies the deviation from the exactly solvable problem. The leading term in this power series is the solution of the exactly solvable problem, while further terms describe the ...

  5. Weyl's inequality - Wikipedia

    en.wikipedia.org/wiki/Weyl's_inequality

    Therefore, Weyl's eigenvalue perturbation inequality for Hermitian matrices extends naturally to perturbation of singular values. [1] This result gives the bound for the perturbation in the singular values of a matrix M {\displaystyle M} due to an additive perturbation Δ {\displaystyle \Delta } :

  6. Bauer–Fike theorem - Wikipedia

    en.wikipedia.org/wiki/Bauer–Fike_theorem

    In mathematics, the Bauer–Fike theorem is a standard result in the perturbation theory of the eigenvalue of a complex-valued diagonalizable matrix.In its substance, it states an absolute upper bound for the deviation of one perturbed matrix eigenvalue from a properly chosen eigenvalue of the exact matrix.

  7. Eigenvalues and eigenvectors - Wikipedia

    en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

    In the meantime, Joseph Liouville studied eigenvalue problems similar to those of Sturm; the discipline that grew out of their work is now called Sturm–Liouville theory. [14] Schwarz studied the first eigenvalue of Laplace's equation on general domains towards the end of the 19th century, while Poincaré studied Poisson's equation a few years ...

  8. Category:Perturbation theory - Wikipedia

    en.wikipedia.org/wiki/Category:Perturbation_theory

    This category deals with topics in perturbation theory and variational principles, which commonly occur in the theory of differential equations, with problems in quantum mechanics forming an important subset thereof.

  9. Two-state quantum system - Wikipedia

    en.wikipedia.org/wiki/Two-state_quantum_system

    One place where this occurs is in degenerate perturbation theory, where the off-diagonal elements are nonzero until the problem is solved by diagonalization. Because of the hermiticity of H {\displaystyle \mathbf {H} } the eigenvalues are real; or, rather, conversely, it is the requirement that the energies are real that implies the hermiticity ...