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Resonances, vibrations, together with natural frequencies, occur everywhere in nature. For example, one associates natural frequencies with musical instruments, with response to dynamic loading of flexible structures, and with spectral lines present in the light originating in a distant part of the Universe.

The simplest case of natural frequencies is illustrated by the vibration of a string. Its deflection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n1300301.png" /> satisfies boundary conditions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n1300302.png" />, and an initial condition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n1300303.png" />. Its motion is described by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n1300304.png" />. Separation of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n1300305.png" /> leads to a pair of equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n1300306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n1300307.png" />.

In equations of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n1300308.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n1300309.png" /> is an operator whose domain is a certain class of functions, the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003010.png" /> is called an eigenvalue (cf. [[Eigen value|Eigen value]]), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003011.png" /> is the corresponding eigenfunction. A (possibly complex) number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003012.png" /> is said to belong to the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003014.png" /> (cf. also [[Spectrum of an operator|Spectrum of an operator]]) if the  "resolvent"  operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003015.png" /> does not exist (cf. also [[Resolvent|Resolvent]]). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003016.png" /> is an eigenvalue if it is a pole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003018.png" /> denotes the identity operator. In the equation of a vibrating string, the boundary conditions are satisfied only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003019.png" /> is the square of a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003022.png" />. The natural frequencies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003023.png" /> are square roots of the eigenvalues: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003024.png" />. The corresponding natural modes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003025.png" /> are the trigonometric functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003026.png" />. The [[Euler formula|Euler formula]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003027.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003028.png" />, simplifies many arguments and offers a better insight into vibration and resonance, among others. The eigenvalues are real because in this case the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003029.png" /> is self-adjoint, meaning that for any pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003030.png" /> in the domain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003031.png" />, the inner product has the following symmetry: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003032.png" /> (cf. also [[Self-adjoint operator|Self-adjoint operator]]). In simple cases these products can be written as integrals.

As an example of this abstract theory, consider a free vibration of a membrane occupying a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003033.png" />. It is modelled by an eigenvalue equation (cf. also [[Neumann eigenvalue|Neumann eigenvalue]]; [[Rayleigh–Faber–Krahn inequality|Rayleigh–Faber–Krahn inequality]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003034.png" /> be denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003035.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003036.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003037.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003038.png" /> is the [[Laplace operator|Laplace operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003039.png" />. The deflection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003040.png" /> on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003041.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003042.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003043.png" />. (Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003044.png" /> denotes the uniform membrane tension, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003045.png" /> is mass per unit length.) Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003046.png" /> is the disguised equation of motion, with eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003047.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003048.png" /> is the natural frequency of vibration.

One can introduce the following product for arbitrary functions satisfying boundary conditions whose gradients are square integrable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003049.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003050.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003051.png" /></td> </tr></table>

Putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003052.png" />, one obtains an energy equation for a freely vibrating membrane. It connects the two basic energy forms: potential and kinetic. Rayleigh's principle relates the value of the smallest (fundamental) natural frequency of the system to the minimum, attained over all possible forms of vibration, of the ratio of the average kinetic energy over average potential energy, computed over a single cycle of vibration.

Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003053.png" /> being self-adjoint implies conservation of energy. So the problem with self-adjoint operators is not very realistic: The vibrating string does not know how to stop vibrating. It will go on forever with the same frequency and the same amplitude. However, a real string will insist on dissipating some of its energy and this has to be reflected in the properties of the operator.

Generally, a correction is made by inserting first-order differential terms into the differential equation, but not always.

In the example of a vibrating elastic shaft one has (with suitable boundary conditions) a self-adjoint Euler–Bernoulli equation:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003054.png" /></td> </tr></table>

S. Timoshenko suggested a fifth-order derivative correction term: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003055.png" /> or similar, taking care of small dissipative effects caused by rotational inertia deforming the shape of the cross-sectional area. With this term, and also possible first-order derivative terms included, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003056.png" /> is no longer self-adjoint, and the eigenvalues become complex numbers. Ignoring small damping terms, the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003057.png" /></td> </tr></table>

can be solved. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003058.png" /> is an eigenvalue of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003059.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003060.png" /> is the real part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003061.png" />. Then the approximate dissipation-free solution can be written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003062.png" />. Observe that the indicated inverse exists, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003063.png" /> is not an eigenvalue. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003064.png" /> is close to an eigenfunction corresponding to the  "true"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003066.png" /> is very close to the pole of the resolvent, the response <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003067.png" /> may become very large. This is a classical example of natural frequency resonance.

In the energy-conserving problems described above, the domain of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003068.png" /> is compact, the inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003069.png" /> is a [[Compact operator|compact operator]], the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003070.png" /> consists of real eigenvalues only, and the only accumulation point for the eigenvalues is at infinity.

Complications arise in quantum physics, where, in general, the domain of the operator is not compact and a continuous spectrum is superimposed on the true eigenvalues. Consider the Schrödinger operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003071.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003072.png" /> is a potential (cf. also [[Schrödinger equation|Schrödinger equation]]). Since boundary values are absent, the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n130/n130030/n13003073.png" /> is the positive part of the real line. Since solutions cannot be contained in a compact set, barriers set by the potential which produce a well of minimum energy surrounded by  "hills"  are not respected. In fact, there is a unique (meromorphic) continuation of the resolvent, whereby the solutions tunnel through the obstacles. This refutes the classical laws of physics, under which particles can be trapped at the bottom of a potential well (corresponding to a minimal energy level).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Courant,   D. Hilbert,   "Methods of mathematical physics" , Interscience  (1953)  (particularly Chapts. 6–7)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.D. Landau,   E.M. Lifshitz,   "A course in theoretical physics" , '''1: Mechanics''' , Pergamon &amp; Addison-Wesley  (1960)  (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Goldstein,   "Classical mechanics" , Addison-Wesley  (1950)  (particularly Chapt. 10)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E.C. Titchmarsh,   "Eigenfunction expansions associated with second order differential equations" , '''1''' , Oxford Univ. Press  (1958)  (Edition: Second)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.B. Keller,   S. Antman,   "Bifurcation theory and nonlinear eigenvalue problesm" , ''Lecture Notes Courant Inst. Math. Sci.'' , New York Univ.  (1968)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  G. Arfken,   "Mathematical methods for physicists" , Acad. Press  (1985)  (Edition: Third)</TD></TR></table>