Mathematical Modeling | PDE Theory | Spectral Theory | Evolution Equations |
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Spectral analysis is a central method in studies of linear systems with the superposition principle. Fourier spectral representations are used in quantum mechanics, wave propagation, harmonic analysis, and signal processing. If f(x) is a square integrable function on real line, it can be represented in a dual (spectral) space with the integral transforms:

f(x) = (2π)
The Fourier transform F(k)
corresponds to a spectral
density of the harmonic oscillations
e^{ikx} between
the wavenumbers k and
k+dk. The inverse Fourier
transform f(x) corresponds to a spectral decomposition
of a function f(x) over a continuous linear combination of the
harmonic oscillations e^{ikx}.
If the function f(x) is C^{1}
continuous everywhere in x,
the Fourier transform F(k)
is C^{1} continuous everywhere
in k, and the Fourier integrals converge.

Linear differential equations, especially initial-value problems for wave equations, can be easily solved with the use of spectral decompositions such as the Fourier transforms. For example, small-amplitude long water waves are described by the linear Korteweg--de Vries equation:

uThe Fourier transform solves the problem on real line of x in a spectral k space as

u(x,t) = (2 π)
Since the Fourier transform F(k)
has the property: F'(k) = ik F(k)
for a square integrable function f(x),
the time evolution of the spectral density U(k,t)
is trivial, U_{t} = i k^{3} U(k,t),
with the straightforward solution: U(k,t) = F(k)
exp(ik^{3}t).
As a result, the exact solution u(x,t)
of the initial-value problem for
the linear KdV equation is a spectral superposition with density
F(k) of linear waves
exp(i(kx - ω(k) t)),
where ω(k) = -k^{3}
is the dispersion relation.

Inverse scattering transform generalizes spectral analysis for solution of initial-value problems for nonlinear differential equations. For example, the inverse scattering transform is applied to solve the nonlinear KdV equation:

uThe KdV equation is related to the spectrum of the stationary Schrodinger equation on a real line of x with a t-dependent potential u(x,t):

- w''(x) + u(x,t) w(x) = λ w(x)
Quantum theory develops spectral analysis of the stationary
Schrodinger equation. If initial data f(x)
is a square integrable function on real line, the spectral
data consists of the continuous spectrum at
λ = k^{2} > 0
with eigenfunctions w(x;k)
and a finite-dimensional discrete
spectrum at λ = - p^{2}_{n} < 0
with eigenfunctions w_{n}(x).
The spectral decomposition
of the potential u(x,t) over
eigenfunctions of the continuous and discrete spectrum takes the form:

where ρ(k,t) and
ρ_{n}(t) are coefficients of the
spectral decomposition. The time-evolution of the spectral data is
trivial with the simple solution:

The initial values for the spectral data have to be found from spectral analysis of the stationary Schrodinger equation with the initial potential u(x,0) = f(x).

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