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Each value of λ corresponds to one or more eigenfunctions. If multiple linearly independent eigenfunctions have the same eigenvalue, the eigenvalue is said to be degenerate and the maximum number of linearly independent eigenfunctions associated with the same eigenvalue is the eigenvalue's degree of degeneracy or geometric multiplicity. [4] [5]
where , the Hamiltonian, is a second-order differential operator and , the wavefunction, is one of its eigenfunctions corresponding to the eigenvalue , interpreted as its energy. However, in the case where one is interested only in the bound state solutions of the Schrödinger equation, one looks for ψ E {\displaystyle \psi _{E}} within the ...
These formulas are used to derive the expressions for eigenfunctions of Laplacian in case of separation of variables, as well as to find eigenvalues and eigenvectors of multidimensional discrete Laplacian on a regular grid, which is presented as a Kronecker sum of discrete Laplacians in one-dimension.
Such values λ are called the eigenvalues of the problem. For each eigenvalue λ, to find the corresponding solution = of the problem. Such functions are called the eigenfunctions associated to each λ. Sturm–Liouville theory is the general study of Sturm–Liouville problems. In particular, for a "regular" Sturm–Liouville problem, it can ...
It can be shown using the self-adjointness proved above that the eigenvalues are real. The compactness of the manifold M {\displaystyle M} allows one to show that the eigenvalues are discrete and furthermore, the vector space of eigenfunctions associated with a given eigenvalue λ {\displaystyle \lambda } , i.e. the eigenspaces are all finite ...
Indeed, the uniqueness of the fundamental eigenfunctions implies that its zeros are precisely the eigenvalues of D and that each non-zero eigenspace is one-dimensional. In particular there are at most countably many eigenvalues of D and, if there are infinitely many, they must tend to infinity.
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Since T K is a linear operator, we can talk about eigenvalues and eigenfunctions of T K. Theorem. Suppose K is a continuous symmetric positive-definite kernel. Then there is an orthonormal basis {e i} i of L 2 [a, b] consisting of eigenfunctions of T K such that the corresponding sequence of eigenvalues {λ i} i is nonnegative.