# (p.631) C Compact operators and eigenfunctions

# (p.631) C Compact operators and eigenfunctions

# C.1 Compact operators

Consider an abstract formulation of eigenvalue problems. Let

*A*:

*H*→

*H*is a linear operator on a Hilbert space

*H*. The nonzero solutions

*u*to (C.1) and the corresponding values λ are called eigenvectors and eigenvalues.

**Definition C.1** *An operator A on H is said to be bounded if there is a positive constant c such that*^{1}

The set of bounded linear operators is a linear space; the definition

**Proposition C.1** *A bounded linear operator is continuous, namely, if f _{n}* →

*f in H then Af*→

_{n}*Af in H. The converse also is true*.

*Proof*. Since *A* is bounded, there is a constant *M* > 0 such that

Hence, *f _{n}* →

*f*implies

*Af*→

_{n}*Af*. The converse is proved by showing that unbounded operators are not continuous. Let

*f*be a sequence in

_{n}*H*such that, for every integer

*n*, we have ║

*Af*║ >

_{n}*n*║

*f*║. Hence, the sequence given by

_{n}*g*║ is as small as we please, while ║

_{n}*Ag*║ > 1. □

_{n}**Proposition C.2** *For every bounded linear operator A on H there is a bounded linear operator A*′ *such that* ^{1}

*Proof*. By the Cauchy–Schwarz inequality, we have

*H*, the inner product (

*Au*, ν) is a bounded linear functional of

*u*. By the Riesz theorem, there is a vector

*w*∈

*H*such that

Accordingly, there is an operator *A*′ : *H* → *H* and *w* = *A*′ν. The operator *A*′ proves to be linear. Moreover, because

*A*′ν║ ≤ ║

*A*║ ║ν║ and hence

*A*′ is bounded. □

**Definition C.2** *The multiplicity of an eigenvalue* λ *is the number of independent eigenvectors u pertaining to that eigenvalue*.

**Definition C.3** *An operator A on H is said to be symmetric if*

Two properties follow at once. First, the eigenvalues of a symmetric linear operator are real. For, the inner product of *Au* = λ*u* with *u*, the symmetry of the inner product, and the symmetry of *A* yield

Inner multiplication of the first equation by *u* _{2} and use of the second one yield

By

_{1}≠ λ

_{2}, the conclusion (

*u*

_{1},

*u*

_{2}) = 0 follows.

**Definition C.4** *An operator A on H is said to be compact (or completely continuous) if A transforms bounded sets into compact sets*.

In fact, we use compactness through a consequence, namely, that if *A* is compact then, whenever {*u _{n}*} is a sequence in

*H*with ║

*u*║ <

_{n}*M*, the sequence {

*Au*} contains a subsequence $\left\{A{u}_{{n}_{k}}\right\}$ which converges to some vector in

_{n}*H*and hence

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**Proposition C.3** *A compact operator A is bounded and continuous*.

*Proof*. Since *A* is linear, if it is bounded then it is continuous. We prove the boundedness by contradiction. If *A* is unbounded, there is a sequence {*u _{n}*} with ║

*u*║ = 1 and ║

_{n}*Au*║ → ∞. We can then consider a subsequence $\left\{{u}_{{n}_{k}}\right\}$ such that

_{n}This sequence does not contain any Cauchy subsequence, which contradicts the compactness of *A*. □

**Proposition C.4** *A compact operator A on H transforms any bounded sequence* {*u _{n}*}

*into a sequence*{

*Au*},

_{n}*which contains a subsequence*$\left\{A{u}_{{n}_{k}}\right\}$

*such that*

*Proof*. To prove the result, we consider a bounded sequence, ║*u _{n}*║ <

*M*, and a Cauchy subsequence $\left\{{u}_{{n}_{k}}\right\}$. By means of the Cauchy–Schwarz inequality, we have

Conversely, let *A* be a linear operator which transforms each bounded sequence {*u _{n}*},

*u*∈

_{n}*H*, into a sequence {

*Au*} which contains a subsequence with the property (C.2). Hence (cf. [182]) the operator

_{n}*A*is compact.

**Proposition C.5** *If A is a bounded symmetric operator then the quadratic form (Au, u) is bounded in that there exist two finite numbers m, M such that*

*Proof*. By the boundedness of *A*, there are two numbers *m, M* such that

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**Proposition C.6** *Let A be a bounded linear symmetric operator on H and M, m be given by* (C.3). *If u* ∈ *H, with* ║*u*║ = 1, *satisfies* (*Au, u*) = *M, or* (*Au, u*) = *m, then M, or m, is an eigenvalue of A*.

*Proof*. Let *u* _{M} ∈ *H* satisfy (*Au* _{M}, *u* _{M}) = *M* and hence

By the definition of *M*, for every *u* ∈ *H*, we have

Accordingly, the functional (*Mu - Au, u*) is minimal at *u* = *u* _{M}. For every φ ∈ *H* let *u* = *u* _{M} + *t*φ. We can write

The arbitrariness of φ ∈ *H* implies that

The proof for *m* proceeds along the same lines. □

**Theorem C.1** *For any compact symmetric operator A, the numbers m, M of* (C.3) *are eigenvalues of A*.

*Proof*. By Proposition C.6, we only need to show that there is *u* _{M} ∈ *H* such that (*Mu* _{M} - *Au* _{M}, *u* _{M}) = *M*║*u* _{M}║^{2} -(*Au* _{M}, *u* _{M}) = 0. Assume, by contradiction, that no nonzero vector exists with such a property. Then

*Mu - Au, u*) is positive definite. We can then view (

*M*ν -

*A*ν,

*w*) as an inner product in

*H*; hence the Cauchy–Schwarz inequality yields

By the definition of *M*, there is a sequence {*u _{n}*}, with ║

*u*║ = 1, such that

_{n}*n*→ ∞. Accordingly, letting ν =

*u*and

_{n}*w*=

*Mu*, we have

_{n}- Au_{n}*Mu*→ 0. Because

_{n}- Au_{n}*A*is compact, there is a subsequence $\left\{{u}_{{n}_{k}}\right\}$ of {

*u*} which converges to a vector,

_{n}*u*say, of

*H*, that is,

Since *A* is bounded, and hence continuous,

Accordingly, since *Au _{n} - Mu_{n}* → 0, we have

The analogous proof holds for *m*. □

Let λ_{0} be the eigenvalue satisfying

Let *u* _{0} be an eigenvector associated with λ_{0}; for simplicity, let ║*u* _{0}║ = 1. Consider the space *H* _{0} = {*u* ∈ *H*; (*u, u* _{0}) = 0}. The operator *A*, restricted to *H* _{0}, is still compact. We can then argue as with *m* and *M*. Thus, there exists an eigenvalue λ_{1} such that

Of course, |λ_{0}| ≥ |λ_{1}|. Denote by *u* _{1}, ║*u* _{1}║ = 1, the eigenvector (or one of the eigenvectors) associated with λ_{1}. By iterating the procedure, we set up a sequence of eigenvalues {λ_{n}} with

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**Lemma C.1** *Let A be a symmetric compact operator and let* |λ_{0}| = sup_{u∈H} |(*Au, u*)|/║*u*║^{2} *be the eigenvalue whose absolute value is the maximum in question. Then*

*Proof*. By the definition of ║*A*║ we have

For any two vectors *u* _{1}, *u* _{2}, it follows that

For any *u* ∈ *H*, let *u* _{1} = |λ_{0}|^{1/2} *u, u* _{2} = |λ_{0}|^{-1/2} *Au*. We have

Comparison with (C.4) yields the conclusion. □

**Proposition C.7** *If H is an infinite-dimensional space then the sequence* {λ_{n}} *converges to zero*.

*Proof*. We prove the result by contradiction. If λ_{n} → λ ≠ 0, then the sequence {*u _{n}*/λ

_{n}} is bounded. Since

*A*is compact, the sequence {

*Au*/λ

_{n}_{n}} should contain a converging subsequence in

*H*. Now,

*Au*/λ

_{n}_{n}=

*u*and

_{n}Hence, no subsequence is convergent and the contradiction is found. □

**Theorem C.2** *The eigenvectors u _{n} of a compact operator A constitute an orthogonal system in H such that, for every w* ∈

*H*,

*Further, if Aw=0 ⇔ w=0 then* {*u _{n}*}

*is a basis for H*.

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*Proof*. Given any *w* ∈ *H*, we consider the sequence

By the orthonormality condition (*u _{j}, u_{k}*) = δ

_{jk}, we have

Hence, the function φ_{n} = *w - w _{n}* is a vector in

*H*= {φ ∈

_{n}*H*; (φ,

*u*

_{1}) = 0, …, (φ,

*u*) = 0}. The operator

_{n}*A*is well defined on

*H*and the norm ║

_{n}*A*║

_{n}in

*H*equals |λ

_{n}_{n}|. Hence, for any φ

_{n}∈

*H*, we have

_{n}In view of Proposition C.7, it follows that

By use of Bessel's inequality, we have

Hence, it follows that $\sum}_{k=m}^{n}\left(w,{u}_{k}\right){u}_{k$ is a Cauchy sequence and there is a vector ν ∈ *H* such that

Now suppose that

*A*ν =

*Aw*, we have ν =

*w*. Hence the eigenvectors {

*u*} are a basis in that, for every

_{n}*w*∈

*H*, we have

□

# (p.638) C.2 Eigenfunctions in spatially homogeneous resonators

Consider a resonator occupying a domain Ω ⊂ ℝ^{3} which is bounded and endowed with a Lipschitzian boundary. We look for time-harmonic solutions *E* = *E* _{0}(*x*) exp(iω*t*), *H* = *H* _{0}(*x*) exp(iω*t*). Hence, *E* _{0}, *H* _{0} are required to satisfy the boundary-value problem

The solutions for (*E* _{0}, *H* _{0}) : Ω → ℂ^{3} x ℂ^{3} are considered in the space ${\mathscr{H}}_{0}^{1}\left(\Omega \right)\times \mathscr{H}\left(\Omega \right)$. Application of the curl operator and comparison yield

If the permittivity ε and the permeability μ are constant, eqns (C.5) and (C.6) take the form

Now, since ▿ · *E* _{0} = 0 (no free charges) and ▿ · *H* _{0} = 0, the identity ▿ x (▿ x *w*) = ▿(▿ · *w*) - ▿*w* yields

In addition,

*w*stands for

*E*

_{0}or

*H*

_{0}. With this in mind, we revisit well-known eigenfunctions associated with various domains Ω (cf. [139–141, 148]).

## C.2.1 The one-dimensional interval

Let Ω = {*x; x* ∈ (0, *a*)}. The divergence-free condition implies that the *x*-component of *w* is constant, possibly zero. Hence, apart from an inessential constant, the vector *w* is transverse. Let *w* be the pertinent component and let the subscript *x* stand for d/d*x*; the differential equation reads

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The boundary conditions become *w* = 0 or *w*′ = 0. Any function *w* ∈ *L* ^{2}(0, *a*) has the expansion

If *w* = 0 at the end points then *b _{n}* = 0,

*n*= 0, 1, …. If, instead,

*w*′ = 0 then

*c*= 0,

_{n}*n*= 1, 2, …. In both cases, we have an infinity of eigenvalues

*k*, that is, ${k}_{n}^{2}={n}^{2}{\pi}^{2}/{a}^{2}$, whence

_{n}## C.2.2 The rectangle

Let Ω = {(*x, y*); *x* ∈ (0, *a*), *y* ∈ (0, *b*)}. Any component satisfies

To fix ideas, let *w* = 0 on the boundary. We start from the expansion

_{m}(

*y*). We obtain that the admissible values of

*k*are given by

Hence, the eigenfunctions are sin(*m*π*x*/*a*) sin(*n*π*y*/*b*) and

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If ∂*w*/∂*n* = 0 on the boundary then constants, cos(*n*π*x/a*) and cos(*m*π*y/b*) are eigenfunctions, so that the solution is expressed as

If, instead, the boundary conditions are

The solution for the case when *w* = 0 at *y* = 0, *a* and ∂*w*/∂*x* at *x* = 0, *b* is obtained by interchanging *x* and *y*.

In all cases

The eigenvalues may be multiple; certainly each eigenvalue, with *m* ≠ *n*, is double if *a* = *b* because the pairs *m, n* and *n, m* correspond to the same eigenvalue.

The eigenvalues change if the boundary conditions are, for example, of the mixed type in the form

The generalization to the rectangular parallelepiped is purely formal. Of course, the eigenfunctions depend on the boundary conditions. If, for
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example, *w* vanishes on the boundary then the eigenfunctions take the form sin(*m*π*x/a*) sin(*n*π*y/b*) sin(*p*π*z/c*). The eigenvectors *k _{mnp}* are given by

## C.2.3 The circle

Let Ω be the circle of radius *a* and use the polar coordinates *r*, φ so that Ω = {(*r*, φ); *r* ∈ [0, *a*], φ ∈ [0, 2π]}. The unknown function *w*(*r*, φ) satisfies

*u*(

*r*, φ) in a Fourier series with respect to φ, parametrized by

*r*,

In fact, substitution in the differential equation shows that α_{m} satisfies the differential equation

Letting ξ = *kr*, we have

*m*, for the unknown function α

_{m}in the variable

*kr*. Hence,

Since α_{m} must be bounded when *r* = 0, we take *d _{m}* = 0.

Let now *w* = 0 at *r* = *a* and let ξ_{mn}, *n* = 1, 2, …, be the zeroes of *J _{m}*(ξ), that is,

*kr*, admissible values of

*k*are given by

*k*= ξ

_{mn}a_{mn}. Hence, the eigenfunctions are exp(i

*m*φ)

*J*(ξ

_{m}_{mn}

*r/a*) and, if

*w*belongs to

*L*

^{2}(Ω) then

The eigenvalues *k* ^{2} are then the squares of *k _{mn}* = ξ

*, namely, the squares of 1/*

_{mn}/a*a*times the zeroes of

*J*.

_{m}A series of the type ${\sum}_{n=1}^{\infty}{c}_{n}{J}_{m}\left({\xi}_{mn}r/a\right)$ where

*w*holds if

*w*∈

*L*

^{2}(Ω) then, by identifying

*w*with

*f*(

*r*) exp(i

*m*φ), it follows that the Fourier–Bessel expansion of a function

*f*holds when ${\int}_{0}^{a}{\left|f\right|}^{2}r\text{d}r<\infty$.

If, instead, the boundary condition is ∂*w*/∂*r* = 0, at *r* = *a*, then the functions *J _{m}*(

*kr*) must satisfy

Hence the values *k _{mn}* of

*k*are allowed such that

*k*= η

_{mn}_{mn}/

*a*and η

_{mn}are the zeroes of d

*J*/d

_{mn}*r*; for most values of

*m*, η

_{m1}= 0. We have

*d*

_{m1}= 0 unless

*m*= 0. A series of the form ${\sum}_{m=1}^{\infty}{d}_{n}{J}_{m}\left({\eta}_{mn}r/a\right)$, where

*d*

_{1}= 0 when

*m*≠ 0, is called a Dini series. The Dini expansion of a function

*f*holds when

*m*is an integer and ${\int}_{0}^{a}{\left|f\right|}^{2}r\text{d}r$ exists.

## (p.643) C.2.4 The sphere

Let Ω be the sphere of radius *a* and describe Ω through the spherical polar coordinates *r*, θ, φ with *r* ∈ [0, *a*], θ ∈ [0, π], φ ∈ [0, 2π). The equation ▵*w* + *k* ^{2} *w* = 0 takes the form

As with the circle, we start from a Fourier series in φ, namely,

Substitution shows that α_{m} must satisfy the differential equation

This shows that α_{m} may conveniently be determined by separation of variables, α_{m}(*r*, θ) = *R _{m}*(

*r*)Θ

_{m}(θ), thus obtaining

This equation holds if and only if there is a constant, λ say, such that

Consider the last equation and observe first that d/d cos θ = -(1/sin θ)d/dθ hence, letting *x* = cos θ, we have the form

The unknown function Θ_{m} is required to be bounded at *x* = ±1.

The Legendre polynomials

*x*= ±1 and satisfying

*P*(1) = 1, and hence they are eigenfunctions of

_{n}*n*(

*n*+ 1), for every

*n*∈ N. We then recognize that

_{m}, whereas the constant λ takes the values

*n*(

*n*+ 1),

*n*∈ N. The functions ${P}_{n}^{\left(m\right)}$ are the associated Legendre functions of order

*m*; of course, ${P}_{n}^{\left(0\right)}\left(x\right)={P}_{n}\left(x\right)\text{}\text{and}\text{}{P}_{n}^{\left(m\right)}\left(x\right)\ne 0$ only for

*m*≤

*n*.

Upon the substitution, *R _{m}*(

*r*) =

*r*

^{-1/2}

*Y*(

_{m}*r*), since λ =

*n*(

*n*+ 1), we have

*r*. The boundedness at

*r*= 0 implies that

*Y*is the Bessel function of the first kind of order

_{m}*n*+ 1/2,

*J*+1/2(

_{n}*kr*). Hence, we see that

*R*is in fact parametrized by

*m*through

*n, n*≥

*m*, and write

In conclusion, the eigenfunctions have the form

To determine *k*, we consider the boundary condition. Let *w* = 0 at *r* = *a*. If ζ_{n+1/2,ν} are the zeroes of *j _{n}*, the admissible values of

*k*are given by

*ka*= ζ

_{n+1/2,ν}and hence we have found the eigenvalues and the eigenfunctions of the Helmholtz equation for the sphere as

## Notes:

(1)
The operator *A*′ is called adjoint operator of *A*.