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§1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

  1. §1.18(i) Hilbert spaces
  2. §1.18(ii) L2 spaces on intervals in
  3. §1.18(iii) Linear Operators on a Hilbert Space
  4. §1.18(iv) Formally Self-adjoint Linear Second Order Differential Operators
  5. §1.18(v) Point Spectra and Eigenfunction Expansions
  6. §1.18(vi) Continuous Spectra and Eigenfunction Expansions: Simple Cases
  7. §1.18(vii) Continuous Spectra: More General Cases
  8. §1.18(viii) Mixed Spectra and Eigenfunction Expansions
  9. §1.18(ix) Mathematical Background
  10. §1.18(x) Literature

A survey is given of the formal spectral theory of second order differential operators, typical results being presented in §1.18(i) through §1.18(viii). The various types of spectra and the corresponding eigenfunction expansions are illustrated by examples. These are based on the Liouville normal form of (1.13.29). A more precise mathematical discussion then follows in §1.18(ix).

§1.18(i) Hilbert spaces

A complex linear vector space V is called an inner product space if an inner product u,v is defined for all u,vV with the properties: (i) u,v is complex linear in u; (ii) u,v=v,u¯; (iii) v,v0; (iv) if v,v=0 then v=0. With norm defined by

1.18.1 v=v,v,

V becomes a normed linear vector space. If v=1 then v is normalized. Two elements u and v in V are orthogonal if u,v=0. A (finite or countably infinite, generalizing the definition of (1.2.40)) set {vn} is an orthonormal set if the vn are normalized and pairwise orthogonal.

An inner product space V is called a Hilbert space if every Cauchy sequence {vn} in V (i.e., limm,nvmvn=0) converges in norm to some vV, i.e., limnvvn=0. For an orthonormal set {vn} in a Hilbert space V Bessel’s inequality holds:

1.18.2 n|cn|2v2,

where vV and

1.18.3 cn=v,vn.

A Hilbert space V is separable if there is an (at most countably infinite) orthonormal set {vn} in V such that for every vV

1.18.4 n|cn|2=v2,

where cn is given by (1.18.3). Such orthonormal sets are called complete. By (1.18.4)

1.18.5 n=0|cn|2<.

Conversely, if complex numbers cn satisfy (1.18.5) then there is a unique vV such that (1.18.3) holds and v can be given by

1.18.6 v=n=0cnvn,

where the infinite sum means convergence in norm,

1.18.7 limNvn=0Ncnvn=0.

The standard example of an (infinite dimensional) separable Hilbert space is the space 2 with elements v=(c0,c1,c2,) such that

1.18.8 v2=n=0|cn|2<.

The inner product of v and w=(d0,d1,d2,) is

1.18.9 v,w=n=0cndn¯.

Every infinite dimensional separable Hilbert space V can be made isomorphic to 2 by choosing a complete orthonormal set {vn}n=0 in V. Then an isomorphism is given by

1.18.10 n=0cnvn(c0,c1,c2,):V2.

General references for this subsection include Friedman (1990, pp. 4–6), Shilov (2013, pp. 249–256), Riesz and Sz.-Nagy (1990, Ch. 5, §82).

§1.18(ii) L2 spaces on intervals in

Let X=[a,b] or [a,b) or (a,b] or (a,b) be a (possibly infinite, or semi-infinite) interval in . For a Lebesgue–Stieltjes measure dα on X let L2(X,dα) be the space of all Lebesgue–Stieltjes measurable complex-valued functions on X which are square integrable with respect to dα,

1.18.11 ab|f(x)|2dα(x)<.

Functions f,gL2(X,dα) for which fg,fg=0 are identified with each other. The space L2(X,dα) becomes a separable Hilbert space with inner product

1.18.12 f,g=abf(x)g(x)¯dα(x),

thus generalizing the inner product of (1.18.9). When α is absolutely continuous, i.e. dα(x)=w(x)dx, see §1.4(v), where the nonnegative weight function w(x) is Lebesgue measurable on X. In this section we will only consider the special case w(x)=1, so dα(x)=dx; in which case L2(X)L2(X,dx).

Assume that {ϕn}n=0 is an orthonormal basis of L2(X). The formulas in §1.18(i) are then:

1.18.13 cn=f,ϕn=abf(x)ϕn(x)¯dx,
1.18.14 ab|f(x)|2dx=n=0|cn|2,
1.18.15 f(x)=limmn=0mcnϕn(x),

where the limit has to be understood in the sense of L2 convergence in the mean:

1.18.16 limmab|f(x)n=0mcnϕn(x)|2dx=0.

Often circumstances allow rather stronger statements, such as uniform convergence, or pointwise convergence at points where f(x) is continuous, with convergence to (f(x0)+f(x0+))/2 if x0 is an isolated point of discontinuity.

We can rewrite (1.18.15), together with (1.18.13), formally as

1.18.17 f(x)=n=0f,ϕnϕn(x)=abK(x,y)f(y)dy,

where the integral kernel is given by

1.18.18 K(x,y)=n=0ϕn(x)ϕn(y)¯.

Thus, in the notation of §1.17, we have an expansion

1.18.19 δ(xy)=n=0ϕn(x)ϕn(y)¯,

of the Dirac delta distribution. Equation (1.18.19) is often called the completeness relation. The analogous orthonormality is

1.18.20 δn,m=abϕn(x)ϕm(x)¯dx.

§1.18(iii) Linear Operators on a Hilbert Space

Bounded and Unbounded Linear Operators

A linear operator T on a (complex) linear vector space V is a map T:VV such that

1.18.21 T(αv+βw)=αTv+βTw,
v,wV, α,β.

In the following let V be a Hilbert space. A linear operator T on V is bounded with norm T if

1.18.22 TsupvV,v=1Tv<.

More generally, a linear operator T on V needs not be defined on all of V, but only on a linear subspace 𝒟(T) of V which is called the domain of T. Then T:𝒟(T)V is a linear map. Assume that 𝒟(T) is dense in V, i.e., for each vV there is a sequence {vn} in 𝒟(T) such that vnv0 as n. If supv𝒟(T),v=1Tv is finite then T is bounded, and T extends uniquely to a bounded linear operator on V. If the supremum is , then T is an unbounded linear operator on V.

Self-Adjoint and Symmetric Operators

If T is a bounded linear operator on V then its adjoint is the bounded linear operator T such that, for v,wV,

1.18.23 Tv,w=v,Tw.

The operator T is called self-adjoint if T=T, and referred to as symmetric if (1.18.23) holds for v,w in the dense domain 𝒟(T) of T. There is also a notion of self-adjointness for unbounded operators, see §1.18(ix). One then needs a self-adjoint extension of a symmetric operator to carry out its spectral theory in a mathematically rigorous manner.

An essential feature of such symmetric operators is that their eigenvalues λ are real, and eigenfunctions

1.18.24 Tuλ=λuλ,

uλ𝒟(T), corresponding to distinct eigenvalues, are orthogonal: i.e., uλ,uλ=0, for λλ. If an eigenvalue has multiplicity >1, the eigenfunctions may always be orthogonalized in this degenerate sub-space.

Formally Self-Adjoint and Self-Adjoint Differential Operators: Self-Adjoint Extensions

Focus is now placed on second order differential operators as these are the subject of the remainder of §1.18.

Consider the second order differential operator acting on real functions of x in the finite interval [a,b]

1.18.25 T=d2dx2,

and functions f(x),g(x)C2(a,b), assumed real for the moment. The adjoint T of T does satisfy Tf,g=f,Tg where f,g=abf(x)g(x)dx. We integrate by parts twice giving:

1.18.26 abf′′(x)g(x)dx=f(x)g(x)|abf(x)g(x)|ab+abf(x)g′′(x)dx.

Ignoring the boundary value terms it follows that

1.18.27 T=T=d2dx2,

and thus T is said to be formally self adjoint.

For T to be actually self adjoint it is necessary to also show that 𝒟(T)=𝒟(T), as it is often the case that T and T have different domains, see Friedman (1990, p 148) for a simple example of such differences involving the differential operator ddx.

This question may be rephrased by asking: do f(x) and g(x) satisfy the same boundary conditions which are needed to fully specify the solutions of a second order linear differential equation? A simple example is the choice f(a)=f(b)=0, and g(a)=g(b)=0, this being only one of many. This insures the vanishing of the boundary terms in (1.18.26), and also is a choice which indicates that 𝒟(T)=𝒟(T), as f(x) and g(x) satisfy the same boundary conditions and thus define the same domains. Thus T is indeed self adjoint.

Other choices of boundary conditions, identical for f(x) and g(x), and which also lead to the vanishing of the boundary terms in (1.18.26), each lead to a distinct self adjoint extension of T. The nature of these extensions for unbounded intervals such as [0,), and unbounded operators on them, are the subject of §1.18(ix).

§1.18(iv) Formally Self-adjoint Linear Second Order Differential Operators

Let X=(a,b) be a finite or infinite open interval in . Consider on X the linear formally self-adjoint second order differential operator

1.18.28 =d2dx2+q(x),

with q(x) real and continuous, unless otherwise noted.

Eigenvalues and eigenfunctions of T, self-adjoint extensions of with well defined boundary conditions, and utilization of such eigenfunctions for expansion of wide classes of L2 functions, will be the focus of the remainder of this section.

The special form of (1.18.28) is especially useful for applications in physics, as the connection to non-relativistic quantum mechanics is immediate: d2dx2 being proportional to the kinetic energy operator for a single particle in one dimension, q(x) being proportional to the potential energy, often written as V(x), of that same particle, and which is simply a multiplicative operator. The sum of the kinetic and potential energies give the quantum Hamiltonian, or energy operator; often also referred to as a Schrödinger operator. Other applications follow from the fact that is suitable for describing vibrations, especially standing waves, which arise in many parts of engineering and the physical sciences, see Birkhoff and Rota (1989, §§10.3 and 10.16). See §18.39(i).

In what follows T will be taken to be a self adjoint extension of following the discussion ending the prior sub-section. For 𝒟(T) we can take C2(X), with appropriate boundary conditions, and with compact support if X is bounded, which space is dense in L2(X), and for X unbounded require that possible non-L2 eigenfunctions of (1.18.28), with real eigenvalues, are non-zero but bounded on open intervals, including ±.

Stated informally, the spectrum of T is the set of it’s eigenvalues, these being real as T is self-adjoint. These sets may be discrete, continuous, or a combination of both, as discussed in the following three subsections. Should an eigenvalue correspond to more than a single linearly independent eigenfunction, namely a multiplicity greater than one, all such eigenfunctions will always be implied as being part of any sums or integrals over the spectrum.

§1.18(v) Point Spectra and Eigenfunction Expansions

General Results

Let T be a self-adjoint extension of differential operator of the form (1.18.28) and assume T has a complete set of L2 eigenfunctions, {ϕλn(x)}n=0 , xX=[a,b] this latter being an appropriate sub-set of , or, in some cases X= itself, with real eigenvalues λn. These eigenvalues will be assumed distinct, i.e., of unit multiplicity, unless otherwise stated. The point, or discrete spectrum of T is then given by 𝝈p={λ0,λ1,}. The eigenfunctions form a complete orthogonal basis in L2(X), and we can take the basis as orthonormal:

1.18.29 abϕλn(x)ϕλm(x)¯dx=δn,m,

and completeness implies

1.18.30 n=0ϕλn(x)ϕλn(y)¯=δ(xy).

Now formulas (1.18.13)–(1.18.20) apply. For f(x)C(X)L2(X)𝒟(T), f(x) has the eigenfunction expansion, following directly from (1.18.17)–(1.18.19),

1.18.31 f(x)=n=0ϕλn(x)abf(y)ϕλn(y)¯dy=n=0f^(λn)ϕλn(x)


1.18.32 f^(λn)=f,ϕλn.


1.18.33 ab|f(x)|2dx=n=0|f^(λn)|2.

Spectral expansions of T, and of functions F(T) of T, these being expansions of T and F(T) in terms of the eigenvalues and eigenfunctions summed over the spectrum, then follow:

1.18.34 (Tf)(x)=n=0λnf^(λn)ϕλn(x)=ab(n=0λnϕλn(x)ϕλn(y)¯)f(y)dy,
1.18.35 (F(T)f)(x)=ab(n=0F(λn)ϕλn(x)ϕλn(y)¯)f(y)dy.

Example 1: Three Simple Cases where q(x)=0, X=[0,π]

Possible eigenfunctions of d2dx2 being sin(kx), cos(kx), e±ikx, consider three cases, which illustrate the importance of boundary conditions.

Case 1: ϕ(0)=ϕ(π)=0. The L2 normalized eigenfunctions on [0,π] are

1.18.36 ϕsin(n,x)=2πsin(nx),

with λn=n2, n=1,2,3,.

Case 2: ϕ(0)=ϕ(π)=0. The L2 normalized eigenfunctions on [0,π] are

1.18.37 ϕcos(0,x)=1π,ϕcos(n,x)=2πcos(nx),

with λn=n2, n=0,1,2,.

Case 3: Periodic Boundary Conditions: ϕ(0)=ϕ(π) and ϕ(0)=ϕ(π). The L2 normalized eigenfunctions on [0,π] are

1.18.38 ϕexp(±n,x)=1πe±2inx,

with λ±n=4n2, n=0,1,2,, with all eigenvalues, for |n|>0, having multiplicity two, as changing the sign of n changes the eigenfunction but not the eigenvalue, and multiplicity one for n=0. Letting n run from to this multiplicity change is automatically included:

1.18.39 δ(xy)=1πn=e2in(xy).

This may be compared to (1.17.21), the resulting Fourier, or eigenfunction, expansion

1.18.40 f(x)=1πn=e2inxf^(λn),

where f^(λn)= 1π0πf(y)e2inydy=f,ϕexp(n), being that of (1.8.3) and (1.8.4). The eigenfunction expansions of (1.8.1) and (1.8.2) follow from Cases 1, 2, above.

Hermite’s Differential Equation, X=(,)

The space X is now the full real line, (,). Writing Hermite’s differential equation (see Tables 18.3.1 and 18.8.1) in the form above, the eigenfunctions are ex2/2Hn(x) (Hn a Hermite polynomial, n=0,1,2,), with eigenvalues λn=2n+1𝝈p, for the differential operator

1.18.41 Hermite=d2dx2+x2,

Applying equations (1.18.29) and (1.18.30), the complete set of normalized eigenfunctions being

1.18.42 ϕn(x)=1π122nn!ex2/2Hn(x),

(1.18.31) becomes

1.18.43 f(x)=n=0e(x2+y2)/2π122nn!Hn(x)Hn(y)f(y)dy,

for f(x)L2 and piece-wise continuous, with convergence as discussed in §1.18(ii).

See Titchmarsh (1962a, pp. 73–75). The implicit boundary conditions taken here are that the ϕn(x) and ϕn(x) vanish as x±, which in this case is equivalent to requiring ϕn(x)L2(X), see Pauling and Wilson (1985, pp. 67–82) for a discussion of this latter point.

§1.18(vi) Continuous Spectra and Eigenfunction Expansions: Simple Cases

General Results

Eigenfunctions corresponding to the continuous spectrum are non-L2 functions. Let T be the self adjoint extension of a formally self-adjoint differential operator of the form (1.18.28) on an unbounded interval X, which we will take as X=[0,+), and assume that q(x)0 monotonically as x, and that the eigenfunctions are non-vanishing but bounded in this same limit. Assume T has no point spectrum, i.e., T has no eigenfunctions in L2(X), then the spectrum 𝝈 of T consists only of a continuous spectrum, referred to as 𝝈c. In this subsection it is assumed that 𝝈c=[0,). This will be generalized, along with the choice of X, in §1.18(vii).

Orthogonality and normalization may then be chosen such that analogous to (1.18.19) and (1.18.20), we have

1.18.44 0ϕλ(x)ϕλ(x)¯dx=δ(λλ),

and completeness relation

1.18.45 δ(xy)=0ϕλ(x)ϕλ(y)¯dλ.

See Friedman (1990, pp. 233–252) for elementary discussions of both equations and the normalization process; and also the references in §1.18(ix).

Now formulas (1.18.13)–(1.18.20) apply. For f(x)C(X)L2(X)𝒟(T), f(x) has the eigenfunction expansion, analogous to that of (1.18.33),

1.18.46 f(x)=0ϕλ(x)f^(λ)dλ,


1.18.47 f^(λ)=f,ϕλ.


1.18.48 0|f(x)|2dx=0|f^(λ)|2dλ.

The analog of (1.18.34) is

1.18.49 (Tf)(x)=0(0λϕλ(x)ϕλ(y)¯dλ)f(y)dy,

and that of (1.18.35) is

1.18.50 (F(T)f)(x)=0(0F(λ)ϕλ(x)ϕλ(y)¯dλ)f(y)dy.

This implies

1.18.51 F(T)f,f=0F(λ)|f^(λ)|2dλ.

In particular, this holds for F(λ)=(zλ)1, z𝝈c

1.18.52 (zT)1f,f=𝝈|f^(λ)|2dλzλ,
fL2(X), z𝝈c,

this being a matrix element of the resolvent F(T)=(zT)1, this being a key quantity in many parts of physics and applied math, quantum scattering theory being a simple example, see Newton (2002, Ch. 7). Then

1.18.53 limν0+12πi((μiνT)1f,f(μ+iνT)1f,f)

gives |f^(μ)|2 for μ𝝈c, and 0 otherwise. This is the discontinuity across the branch cut in (1.18.52) 𝝈c, from z below to above the cut, divided by 2πi. See Newton (2002, §§7.1 and 7.3).

More generally, for fC(X), xX, see (1.4.24),

1.18.54 limϵ0+Xf(y)x±iϵydy=PXf(y)xydyiπf(x).

Example 1: Bessel–Hankel Transform, X=[0,)

By Bessel’s differential equation in the form (10.13.1) we have the functions xJν(xλ) (λ0, for Jν see §10.2(ii)) as eigenfunctions with eigenvalue λ of the self-adjoint extension of the differential operator

1.18.55 Bessel=d2dx2+ν214x2,

Applying the representation (1.17.13), now symmetrized as in (1.17.14), as 1xδ(xy)=1xyδ(xy),

1.18.56 δ(xy)=0xtJν(xt)ytJν(yt)dt,
ν>1, x,y0.

For f(x) piecewise continuously differentiable on [0,)

1.18.57 limR0f(y)(0RxtJν(xt)ytJν(yt)dt)dy=12(f(x+)+f(x)),
x>0, ν>1,

provided that:

1.18.58 1y1|f(y)|dy <,
01(1+yν+12)|f(y)|dy <.

(1.18.57) is the Hankel transform (10.22.76)–(10.22.77). See Titchmarsh (1962a, pp. 87–90) for a first principles derivation for the case ν1.

For generalizations see the Weber transform (10.22.78) and an extended Bessel transform (10.22.79).

Example 2: Sine and Cosine Transforms, X=[0,)

The Fourier cosine and sine transform pairs (1.14.9) & (1.14.11) and (1.14.10) & (1.14.12) can be easily obtained from (1.18.57) as for ν=±12 the Bessel functions reduce to the trigonometric functions, see (10.16.1).

More generally,

1.18.59 f(x)=1π(0cos(xt)cos(yt)dt)f(y)dy+1π(0sin(xt)sin(yt)dt)f(y)dy,

For f(x) even in x this yields the Fourier cosine transform pair (1.14.9) & (1.14.11), and for f(x) odd the Fourier sine transform pair (1.14.10) & (1.14.12). These latter results also correspond to use of the δ(xy) as defined in (1.17.12_1) and (1.17.12_2).

§1.18(vii) Continuous Spectra: More General Cases

More generally, continuous spectra may occur in sets of disjoint finite intervals [λa,λb](0,), often called bands, when q(x) is periodic, see Ashcroft and Mermin (1976, Ch 8) and Kittel (1996, Ch 7). Should q(x) be bounded but random, leading to Anderson localization, the spectrum could range from being a dense point spectrum to being singular continuous, see Simon (1995), Avron and Simon (1982); a good general reference being Cycon et al. (2008, Ch. 9 and 10). For example, replacing 2qcos(2z) of (28.2.1) by λcos(2παn+θ), n gives an almost Mathieu equation which for appropriate α has such properties.

§1.18(viii) Mixed Spectra and Eigenfunction Expansions

In general, operators T being formally self-adjoint second order differential operators of the form (1.18.28), with X unbounded, will have both a continuous and a point spectrum, and thus, correspondingly, nonL2(X) eigenfunctions as in §1.18(vi) and L2(X) eigenfunctions as in §1.18(v). We assume a continuous spectrum λ𝝈c=[0,), and a finite or countably infinite point spectrum 𝝈p with elements λn. In what follows, integrals over the continuous parts of the spectrum will be denoted by 𝝈c, and sums over the discrete spectrum by 𝝈p, with 𝝈=𝝈c𝝈p denoting the full spectrum. It is to be noted that if any of the λ𝝈 have degenerate sub-spaces, that is subspaces of orthogonal eigenfunctions with identical eigenvalues, that in the expansions below all such distinct eigenfunctions are to be included. Then orthogonality and normalization relations are

1.18.60 Xϕλ(x)ϕλ(x)¯dx=δ(λλ),
1.18.61 Xϕλn(x)ϕλm(x)¯dx=δn,m,
1.18.62 Xϕλn(x)ϕλ(x)¯dx=0,
λn𝝈p, λ𝝈c,

compare (1.18.29) and (1.18.44). The formal completeness relation is now

1.18.63 δ(xx)=𝝈pϕλn(x)ϕλn(x)¯+𝝈cϕλ(x)ϕλ(x)¯dλ,

compare (1.18.30) and (1.18.45), and the eigenfunction expansions are of the form

1.18.64 f(x)=𝝈cf^(λ)ϕλ(x)dλ+𝝈pf^(λn)ϕλn(x),

Note that the notations of (1.18.32) and (1.18.47) are used to distinguish the contributions from the discrete and continuous parts of the spectrum. Then

1.18.65 X|f(x)|2dx=𝝈c|f^(λ)|2dλ+𝝈p|f^(λn)|2,

The analogs of (1.18.49)–(1.18.52) may be written in a similar fashion each now including contributions from both the discrete and continuous parts of the spectrum, as in (1.18.65). Showing one, representative, example: the analog of (1.18.52) is now

1.18.66 (zT)1f,f=𝝈p|f^(λn)|2zλn+𝝈c|f^(λ)|2dλzλ,
fL2(X), z𝝈.

This representation has poles with residues |f^(λn)|2 at the discrete eigenvalues and a branch cut along [0,) with discontinuity, from below to above the cut, 2πi|f^(λ)|2, as in (1.18.53), see Newton (2002, §7.1.1).

Note that the integral in (1.18.66) is not singular if approached separately from above, or below, the real axis: in fact analytic continuation from the upper half of the complex plane, across the cut, and onto higher Riemann Sheets can access complex poles with singularities at discrete energies λresiΓres/2 corresponding to quantum resonances, or decaying quantum states with lifetimes proportional to 1/Γres. For this latter see Simon (1973), and Reinhardt (1982); wherein advantage is taken of the fact that although branch points are actual singularities of an analytic function, the location of the branch cuts are often at our disposal, as they are not singularities of the function, but simply arbitrary lines to keep a function single valued, and thus only singularities of a specific representation of that analytic function. This is accomplished by the variable change xxeiθ, in , which rotates the continuous spectrum 𝝈c𝝈ce2iθ and the branch cut of (1.18.66) into the lower half complex plain by the angle 2θ, with respect to the unmoved branch point at λ=0; thus, providing access to resonances on the higher Riemann sheet should θ be large enough to expose them. This dilatation transformation, which does require analyticity of q(x) in (1.18.28), or an analytic approximation thereto, leaves the poles, corresponding to the discrete spectrum, invariant, as they are, as is the branch point, actual singularities of (zT)1f,f.

Example 1: In one and two dimensions any q(x) with a ‘Dip, or Well’ has a partly discrete spectrum

Suppose that X is the whole real line in one dimension, and that q(x), in (1.18.28) has (non-oscillatory) limits of 0 at both ±, and thus a continuous spectrum on 𝝈0. What then is the condition on q(x) to insure the existence of at least a single eigenvalue in the point spectrum? The discussions of §1.18(vi) imply that if q(x)0 then there is only a continuous spectrum. Surprisingly, if q(x)<0 on any interval on the real line, even if positive elsewhere, as long as Xq(x)dx0, see Simon (1976, Theorem 2.5), then there will be at least one eigenfunction with a negative eigenvalue, with corresponding L2(X) eigenfunction. Thus, and this is a case where q(x) is not continuous, if q(x)=αδ(xa), α>0, there will be an L2 eigenfunction localized in the vicinity of x=a, with a negative eigenvalue, thus disjoint from the continuous spectrum on [0,). Similar results hold for two, but not higher, dimensional quantum systems. See Brownstein (2000) and Yang and de Llano (1989) for numerical examples, based on variational calculations, and Simon (1976) and Chadan et al. (2003) for rigorous mathematical discussion.

Example 2: Radial 3D Schrödinger operators, including the Coulomb potential

Consider formally self-adjoint operators of the form

1.18.67 =12d2dr2+(+1)2r2+V(r),
=0,1,2,, V(r)0 as r,

which appear in the quantum theory of binding or scattering of a particle in a spherically symmetric potential V(r) in three dimensions, and where r[0,). The bound states are in the negative energy discrete spectrum, and the scattering states are in the positive energy continuous spectrum, 𝝈c=[0,), or, said more simply, in the continuum. See §18.39 for discussion of Schrödinger equations and operators. For fixed angular momentum the appropriate self-adjoint extension of the above operator may have both a discrete spectrum of negative eigenvalues λn,n=0,1,,N1, with corresponding L2([0,),r2dr) eigenfunctions ϕn(r), and also a continuous spectrum λ[0,), with Dirac-delta normalized eigenfunctions ϕλ(r), also with measure r2dr. Unlike in the example in the paragraph above, in 3-dimensions a “dip below zero, or a potential well” in V(r) does not always correspond to the existence of a discrete part of the spectrum. The well must be deep and broad enough to allow existence of such L2 discrete states. The number, N, of discrete states depends on the nature of V(r), as well as , and, again, V(r) must vanish as r, corresponding to the traditionally assumed start of the energy continuum at λ=0. In unusual cases N=, even for all , such as in the case of the Schrödinger–Coulomb problem (V=r1) discussed in §18.39 and §33.14, where the point spectrum actually accumulates at the onset of the continuum at λ=0, implying an essential singularity, as well as a branch point, in matrix elements of the resolvent, (1.18.66). See Bethe and Salpeter (1977, Ch. 1, (4.12)–(4.13)) for the resulting transform pair in this case.

§1.18(ix) Mathematical Background

Self-Adjoint Operators

If T is an unbounded linear operator on a Hilbert space V with dense domain 𝒟(T) then the adjoint T of T is the linear operator with domain

1.18.68 𝒟(T)={wV|supv𝒟(T),v=1|Tv,w|<},

such that

1.18.69 Tv,w=v,Tw,
v𝒟(T), w𝒟(T).

A linear operator T with dense domain is called symmetric if

1.18.70 Tv,w=v,Tw,

If T is symmetric then TT, i.e., 𝒟(T)𝒟(T) and Tv=Tv for v𝒟(T). Then also TTT, where T=(T). If TT=T then T is essentially self-adjoint and if T=T then T is self-adjoint.

Spectrum of an Operator

Let T be a linear operator on V with dense domain 𝒟(T) and with range (T)={Tvv𝒟(T)}. Such an operator T is called injective if, for any u,v in its domain, Tu=Tv implies that u=v. The resolvent set ρ(T) consists of all z such that (i) zT is injective, (ii) (zT) is dense in V, (iii) the resolvent (zT)1 is bounded. The spectrum 𝝈(T) is the complement in of ρ(T). The spectrum 𝝈(T) is the disjoint union of three sets:

  1. 1.

    The point spectrum 𝝈p. It consists of all z for which zT is not injective, or equivalently, for which z is an eigenvalue of T, i.e., Tv=zv for some v𝒟(T)\{0}.

  2. 2.

    The continuous spectrum 𝝈c. It consists of all z for which zT is injective and has dense range, but (Tz)1 is not bounded.

  3. 3.

    The residual spectrum. It consists of all z for which zT is injective, but does not have dense range.

If T is a bounded operator then its spectrum is a closed bounded subset of . If T is self-adjoint (bounded or unbounded) then σ(T) is a closed subset of and the residual spectrum is empty. Note that eigenfunctions for distinct (necessarily real) eigenvalues of a self-adjoint operator are mutually orthogonal. If an eigenvalue is of multiplicity greater than 1 then an orthonormal basis of eigenfunctions can be given for the eigenspace.

Self-adjoint extensions of a symmetric Operator

Let T be a symmetric operator on a Hilbert space V, so 𝒟(T) is dense in V and TTT. For z\ let Nz be the z-eigenspace of T, i.e., Nz is the linear subspace of 𝒟(T) consisting of all v for which Tv=zv. Then dimNz is constant for z>0 and also constant for z<0. Put n+=dimNz (z>0) and n=dimNz (z<0), the deficiency indices for T. Then T has self-adjoint extensions iff n+=n. We have a direct sum of linear spaces: 𝒟(T)=𝒟(T)+Ni+Ni. Assume n+=n. Then any self-adjoint extension of T is determined by a linear isometry U:NiNi and it is the restriction of T to {v+w+Uwv𝒟(T),wNi}.

Self-adjoint extensions of (1.18.28) and the Weyl alternative

For a formally self-adjoint second order differential operator , such as that of (1.18.28), the space 𝒟() can be seen to consist of all fL2(X) such that the distribution f can be identified with a function in L2(X), which is the function f. Then, for z\, fNz iff f is an ordinary solution (i.e., fC2(X)) of Lf=zf which is moreover in L2(X). Thus Nz has dimension 0, 1 or 2. Also, because q is real-valued, fNz iff f¯Nz¯. So has self-adjoint extensions with deficiency indices n+=n=0, or 1 or 2. Pick c(a,b). Let n1,n1 be the deficiency indices for restricted to (a,c), and n2,n2 the ones for restricted to (c,b). Then n1 and n2 are independent of c. By Weyl’s alternative n1 equals either 1 (the limit point case) or 2 (the limit circle case), and similarly for n2. The two (equal) deficiency indices of are then equal to n1+n22. A boundary value for the end point a is a linear form on 𝒟() of the form

1.18.71 (f)=limxa+(α(x)f(x)+β(x)f(x)),

where α and β are given functions on X, and where the limit has to exist for all f. Then, if the linear form is nonzero, the condition (f)=0 is called a boundary condition at a. Boundary values and boundary conditions for the end point b are defined in a similar way. If n1=1 then there are no nonzero boundary values at a; if n1=2 then the above boundary values at a form a two-dimensional class. Similarly at b. Any self-adjoint extension of can be obtained by restricting to those f𝒟() for which, if n1=2, 1(f)=0 for a chosen 1 at a and, if n2=2, 2(f)=0 for a chosen 2 at b.

Integral transforms (10.22.78) and (10.22.79) are examples of the utility of these extensions.

Spectral expansions and self-adjoint extensions

The above results, especially the discussions of deficiency indices and limit point and limit circle boundary conditions, lay the basis for further applications. The reader is referred to Coddington and Levinson (1955), Friedman (1990, Ch. 3), Titchmarsh (1962a), and Everitt (2005b, pp. 45–74) and Everitt (2005a, pp. 272–331), for detailed methods and results.

§1.18(x) Literature

The materials developed here follow from the extensions of the Sturm–Liouville theory of second order ODEs as developed by Weyl, to include the limit point and limit circle singular cases. This work is well overviewed by Coddington and Levinson (1955, Ch. 9), and then applied in detail by Titchmarsh (1946), Titchmarsh (1962a), Titchmarsh (1958), and Levitan and Sargsjan (1975) which also connects the Weyl theory to the relevant functional analysis. In parallel, similar, and more general formulations have grown out of functional analysis itself, as in the work of Stone (1990), Rudin (1973), Reed and Simon (1980), Reed and Simon (1975), Reed and Simon (1978), Reed and Simon (1979), Cycon et al. (2008), Dunford and Schwartz (1988, Ch. XIII), Hall (2013, pp. 127-223). Friedman (1990) provides a useful introduction to both approaches; as does the conference proceeding Amrein et al. (2005), overviewing the combination of Sturm–Liouville theory and Hilbert space theory. See, in particular, the overview Everitt (2005b, pp. 45–74), and the uniformly annotated listing of 51 solved Sturm–Liouville problems in Everitt (2005a, pp. 272–331), each with their limit point, or circle, boundary behaviors categorized.