Bernoulli polynomials

This solution of the vibrating drum problem is, at any point in time, an eigenfunction of the Laplace operator on a disk.

In mathematics, an eigenfunction of a linear operator, A, defined on some function space, is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has

{\mathcal {A}}f=\lambda f

for some scalar, λ, the corresponding eigenvalue. The solution of the differential eigenvalue problem also depends on any boundary conditions required of $f$ . In each case there are only certain eigenvalues $\lambda =\lambda _{n}$ ( $n=1,2,3,...$ ) that admit a corresponding solution for $f=f_{n}$ (with each $f_{n}$ belonging to the eigenvalue $\lambda _{n}$ ) when combined with the boundary conditions. Eigenfunctions are used to analyze $A$ .

For example, $f_{k}(x)=e^{kx}$ is an eigenfunction for the differential operator

{\mathcal {A}}={\frac {d^{2}}{dx^{2}}}-{\frac {d}{dx}}

for any value of $k$ , with corresponding eigenvalue $\lambda =k^{2}-k$ . If boundary conditions are applied to this system (e.g., $f=0$ at two physical locations in space), then only certain values of $k=k_{n}$ satisfy the boundary conditions, generating corresponding discrete eigenvalues $\lambda _{n}=k_{n}^{2}-k_{n}$ .

Specifically, in the study of signals and systems, the eigenfunction of a system is the signal $f(t)$ which when input into the system, produces a response $y(t)=\lambda f(t)$ with the complex constant $\lambda$ .^[1]

Examples

Derivative operator

A widely used class of linear operators acting on function spaces are the differential operators on function spaces. As an example, on the space $\mathbf {C^{\infty }}$ of infinitely differentiable real functions of a real argument $t$ , the process of differentiation is a linear operator since

\displaystyle {\frac {d}{dt}}(af+bg)=a{\frac {df}{dt}}+b{\frac {dg}{dt}},

for any functions $f$ and $g$ in $\mathbf {C^{\infty }}$ , and any real numbers $a$ and $b$ .

The eigenvalue equation for a linear differential operator $D$ in $\mathbf {C^{\infty }}$ is then a differential equation

Df=\lambda f

The functions that satisfy this equation are commonly called eigenfunctions. For the derivative operator $d/dt$ , an eigenfunction is a function that, when differentiated, yields a constant times the original function. That is,

\displaystyle {\frac {d}{dt}}f(t)=\lambda f(t)

for all $t$ . This equation can be solved for any value of $\lambda$ . The solution is an exponential function

f(t)=Ae^{\lambda t}.\

The derivative operator is defined also for complex-valued functions of a complex argument. In the complex version of the space $\mathbf {C^{\infty }}$ , the eigenvalue equation has a solution for any complex constant $\lambda$ . The spectrum of the operator $d/dt$ is therefore the whole complex plane. This is an example of a continuous spectrum.

Applications

Vibrating strings

Let $h(x,t)$ denote the sideways displacement of a stressed elastic chord, such as the vibrating strings of a string instrument, as a function of the position $x$ along the string and of time $t$ . From the laws of mechanics, applied to infinitesimal portions of the string, one can deduce that the function $h$ satisfies the partial differential equation

{\frac {\partial ^{2}h}{\partial t^{2}}}=c^{2}{\frac {\partial ^{2}h}{\partial x^{2}}},

which is called the (one-dimensional) wave equation. Here $c$ is a constant that depends on the tension and mass of the string.

This problem is amenable to the method of separation of variables. If we assume that $h(x,t)$ can be written as the product of the form $X(x)T(t)$ , we can form a pair of ordinary differential equations:

{\frac {d^{2}}{dx^{2}}}X=-{\frac {\omega ^{2}}{c^{2}}}X\quad \quad \quad

and

\quad \quad \quad \displaystyle {\frac {d^{2}}{dt^{2}}}T=-\omega ^{2}T.\

Each of these is an eigenvalue equation, for eigenvalues $-\omega ^{2}/c^{2}$ and $-\omega ^{2}$ , respectively. For any values of $\omega$ and $c$ , the equations are satisfied by the functions

X(x)=\sin \left({\frac {\omega x}{c}}+\phi \right)\quad \quad \quad

and

\quad \quad \quad T(t)=\sin(\omega t+\psi ),\

where $\phi$ and $\psi$ are arbitrary real constants. If we impose boundary conditions (that the ends of the string are fixed with $X(x)=0$ at $x=0$ and $x=L$ , for example) we can constrain the eigenvalues. For those boundary conditions, we find

\sin(\phi )=0\

, and so the phase angle

\phi =0\

and

\sin \left({\frac {\omega L}{c}}\right)=0.\

Thus, the constant $\omega$ is constrained to take one of the values $\omega _{n}=nc\pi /L$ , where $n$ is any integer. Thus, the clamped string supports a family of standing waves of the form

h(x,t)=\sin(n\pi x/L)\sin(\omega _{n}t).\

From the point of view of our musical instrument, the frequency $\omega _{n}\$ is the frequency of the $n$ th harmonic, which is called the $(n-1)$ th overtone.

Quantum mechanics

Eigenfunctions play an important role in many branches of physics. An important example is quantum mechanics, where the Schrödinger equation

{\mathcal {H}}\psi =E\psi

,

with

{\mathcal {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} ,t)

has solutions of the form

\psi (t)=\sum _{k}e^{-iE_{k}t/\hbar }\phi _{k},

where $\phi _{k}$ are eigenfunctions of the operator ${\mathcal {H}}$ with eigenvalues $E_{k}$ . The fact that only certain eigenvalues $E_{k}$ with associated eigenfunctions $\phi _{k}$ satisfy Schrödinger's equation leads to a natural basis for quantum mechanics and the periodic table of the elements, with each $E_{k}$ an allowable energy state of the system. The success of this equation in explaining the spectral characteristics of hydrogen is considered one of the greatest triumphs of 20th century physics.

Due to the nature of Hermitian Operators such as the Hamiltonian operator ${\mathcal {H}}$ , its eigenfunctions are orthogonal functions. This is not necessarily the case for eigenfunctions of other operators (such as the example $A$ mentioned above). Orthogonal functions $f_{i}$ , $i=1,2,\dots ,$ have the property that

0=\int f_{i}^{*}f_{j}

where $f_{i}^{*}$ is the complex conjugate of $f_{i}$

whenever $i\neq j$ , in which case the set $\{f_{i}\,|\,i\in I\}$ is said to be orthogonal. Also, it is linearly independent.

Notes

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References

Methods of Mathematical Physics by R. Courant, D. Hilbert ISBN 0-471-50447-5 (Volume 1 Paperback) ISBN 0-471-50439-4 (Volume 2 Paperback) ISBN 0-471-17990-6 (Hardback)

Bernoulli polynomials

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