Eigenfunction

In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as D f = λ f , {displaystyle Df=lambda f,}     (1) H φ ( r ) = E φ ( r ) , {displaystyle Hvarphi (mathbf {r} )=Evarphi (mathbf {r} ),}     (2) i ℏ ∂ T ( t ) ∂ t = E T ( t ) . {displaystyle ihbar {frac {partial T(t)}{partial t}}=ET(t).}     (3) In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as for some scalar eigenvalue λ. The solutions to this equation may also be subject to boundary conditions that limit the allowable eigenvalues and eigenfunctions. An eigenfunction is a type of eigenvector. In general, an eigenvector of a linear operator D defined on some vector space is a nonzero vector in the domain of D that, when D acts upon it, is simply scaled by some scalar value called an eigenvalue. In the special case where D is defined on a function space, the eigenvectors are referred to as eigenfunctions. That is, a function f is an eigenfunction of D if it satisfies the equation where λ is a scalar. The solutions to Equation (1) may also be subject to boundary conditions. Because of the boundary conditions, the possible values of λ are generally limited, for example to a discrete set λ1, λ2, ... or to a continuous set over some range. The set of all possible eigenvalues of D is sometimes called its spectrum, which may be discrete, continuous, or a combination of both. 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. A widely used class of linear operators acting on infinite dimensional spaces are differential operators on the space C∞ of infinitely differentiable real or complex functions of a real or complex argument t. For example, consider the derivative operator d d t {displaystyle { frac {d}{dt}}} with eigenvalue equation This differential equation can be solved by multiplying both sides by d t f ( t ) {displaystyle { frac {dt}{f(t)}}} and integrating. Its solution, the exponential function is the eigenfunction of the derivative operator, where f0 is a parameter that depends on the boundary conditions. Note that in this case the eigenfunction is itself a function of its associated eigenvalue λ, which can take any real or complex value. In particular, note that for λ = 0 the eigenfunction f(t) is a constant.