A brief note on operator spectra

Ernesto

November 10, 2025

I keep forgetting basic definitions pertaining to the spectral theory of self-adjoint operators so I want to keep a few of them here as a reminder. I’ll illustrate the concepts with examples arising in the theory of Anderson localization.

A confusion that can occur especially when studying the physics literature, is that of mixing up the various definitions and nomenclature related to the spectrum of an operator. For example, the language usually used could leave one confused when meeting an operator $A$ with its spectrum equal to some continuous subset of the real line, yet having only a “pure point spectrum”. Additionally, it is common to say an operator has pure point spectrum for both the case in which it is entirely pure point or when it only has non-trivial pure point part (like the Hamiltonian of the hydrogen atom), but not much can be done about this except to pay attention to the context we’re in. This post is just a brief recollection of these concepts.

1 Bare minimum spectral theory

What follows can be found in detail in M. Aizenman and S. Warzel’s Random Operators. A lot of the following definitions apply to some broader class of linear operators, but we will implicitly assume that we deal with self-adjoint operators. Let $A : 𝒟(A) → ℋ$ be a densely defined self-adjoint operator on some Hilbert space $ℋ$ . The resolvent set is defined as

ρ(A ) := {z ∈ C : (A− zI) : 𝒟(A) → ℋ is bijective},

(1)

and the inverse $(A − zI)−1$ is called the resolvent. The spectrum of $A$ is $σ(A) := C ∖ρ(A )$ , which is a closed subset of the real line. It can be shown that for any $ψ ∈ ℋ$ the function defined by

Fψ(z) := ⟨ψ,(A− zI)−1ψ⟩

(2)

is a Herglotz-Pick function, which means that it is a holomorphic function $Fψ : C+ → C+$ from the upper-half plane into itself. There is a representation theorem for these types of functions which states that there exists a unique finite Borel measure $μ ψ$ on $R$ called the spectral measure of $A$ associated to $ψ$ , which satisfies

∫ Fψ (z) = -1--μ ψ(dλ). R λ− z

(3)

By polarization we obtain a complex finite Borel measure $μϕ,ψ$ , which can be used to define a functional calculus for bounded functions $f ∈ L ∞(R )$ satisfying

∫ ⟨ϕ,f(A )ψ ⟩ = f(λ)μ (dλ), ∀ϕ,ψ ∈ ℋ. R ϕ,ψ

(4)

Of course the characteristic functions $χB$ are bounded for any Borel set $B ∈ ℬ$ and can be used to define orthogonal projections $PA(B)$ . The map $PA : ℬ → ℒ(ℋ )$ defined in this manner is a projection-valued measure. The spectral theorem guarantees a correspondence between these projection-valued measures and self-adjoint operators on $ℋ$ . In particular we have $μϕ,ψ(B ) = ⟨ϕ,PA(B)ψ⟩$ .

Having such a Borel measure $μ$ , Lebesgue’s decomposition theorem states that $μ$ can be decomposed with respect to the Lebesgue measure into three mutually singular parts:

μ = μpp +μac +μsc,

(5)

known as the pure point part, the absolutely continuous part and the singular continuous part, respectively. Occasionally one calls these parts, spectral components. Using these components, we can define the closed subspaces

{ } ℋ# = ψ ∈ ℋ : μψ = μ#ψ ,

(6)

where $#$ is $pp,ac$ or $sc$ . It turns out that the Hilbert space can be decomposed as a direct sum of these subspaces:

ℋ = ℋpp ⊕ℋac ⊕ ℋsc,

(7)

with each summand being invariant under $A$ . If $A$ is bounded, then restricting to these subspaces defines the components of the spectrum

σ#(A ) = σ(A |ℋ# ),

(8)

where $σ#(A)$ is the pure point, absolutely continuous or singular continuous spectrum of $A$ .

From a slightly different perspective, we can define the discrete spectrum $σdis(A)$ as the set of isolated eigenvalues of finite multiplicity of $A$ . An isolated eigenvalue $λ$ of $A$ is one for which there exists $𝜖 > 0$ such that $σ (A )∩ (λ − 𝜖,λ + 𝜖) = {λ}$ and any isolated point in the spectrum is always an eigenvalue. The complement of $σdis(A )$ in $σ(A)$ , denoted by $σess(A)$ , is called the essential spectrum. It can easily be shown that the pure point spectrum is equal to the closure of the set of eigenvalues of $A$ and so $σdis(A) ⊂ σpp(A )$ .

We can produce a finer classification of spectra, and even more so for an unbounded operator, but we stop here since this is enough to already confuse oneself and is sufficient for the few examples that follow.

2 Illustration

Consider the graph (also discrete, finite-difference) Laplacian $2 Δ : 𝒟 (Δ ) → ℓ(G)$ for $d G = Z$ . For a function $ψ ∈ 𝒟 (Δ )$ it is defined as

∑ (Δψ)(x) = (ψ(y)− ψ(x)). y:|x−y|=1

(9)

This is a bounded operator with $∥Δ∥ = 4d$ . We can compute its spectrum easily by using the Fourier transform $F : ℓ2(Zd) → L2([0,2π]d)$ which is defined as

−d∕2 ∑ −ik⋅x (Fψ)(k) = (2π) e ψ(x). x∈Zd

(10)

Then by direct computation we can show for any $ϕ ∈ L2 ([0,2π]d)$ that

Thus $Δ$ is unitarily equivalent to a multiplication operator on $L2 ([0,2π]d)$ . Bounded multiplication operators have as their spectrum the essential range, which in this case is the interval $[− 4d,0]$ . Thus

σ(− Δ) = [0,4d].

(15)

The Laplacian does not have eigenvectors (although the plane waves can be seen as “generalized” eigenvectors). In fact, with a bit of work, we can show that $μψ = μaψc$ for all $ψ ∈ ℓ2(Zd )$ . This implies that the whole spectrum is essential:

σ(− Δ) = σac(− Δ ) = σess(− Δ) = [0,4d].

(16)

Now consider a multiplication operator on $ℓ2(Zd)$ defined by some real-valued function $V ∈ L∞ (Zd)$ . The spectral measure only has pure point part, given by a sum of Dirac measures

μ = μpp = ∑ |ψ (x)|2δ . ψ ψ d V(x) x∈Z

(17)

Its eigenvectors consist of the localized functions $2 d δx ∈ ℓ (Z )$ with corresponding eigenvalues $V (x)$ with $d x ∈ Z$ . The closure of the eigenvalues gives the pure point spectrum. We require more information on the function $V$ to determine the nature of the discrete and essential spectrum. By definition we know that the essential spectrum will consist of all infinitely degenerate eigenvalues $V(x)$ along with the accumulation points of the full set of eigenvalues.

2.1 Anderson localization

Now consider the Schrödinger operator $H = − Δ + V$ , where $V$ is a multiplication operator whose action is to multiply a function $ψ$ by an IID random variable with density $ν$ at each point of $d Z$ . To be very precise requires defining the appropriate measure spaces but we will ignore this, and only note that with a little ergodic theory one can show that

σ(H ) = [0,4d]+ supp ν,

(18)

with probability one. The model described by the random Schrödinger operator $H$ is well-known as the Anderson tight-binding model on the “lattice” $Zd$ . It describes a simplified picture (no particle interactions) of the transport of an electron in a disordered medium, e.g. in an impure crystal. Originally introduced by P. W. Anderson in 1958, it has stimulated intense research in both solid state physics and more recently (through generalizations of course) to condensed matter theory. The physicists have “known” much about this model for a long time but somewhat surprisingly, there are still many outstanding mathematical conjectures. The main phenomenon studied in these models is that of localization. Roughly speaking, in a periodic crystal, one expects electronic conductance, but the inclusion of disorder through these random potentials can create an effect due to interference (also only conjectured), in which the eigenfunctions of $H$ become exponentially localized in the lattice. A common consequence of this is that diffusion (as measured by the spreading of the wave function) under the time evolution $e−itH$ is suppressed and conductance no longer occurs.

For the one dimensional lattice, $H$ has pure point spectrum with exponential localization occurring for all strengths of disorder. For $d ≥ 2$ , pure point spectra with exponential eigenfunctions have been proven to exist at the “band edges”, i.e., at the extremes of the spectrum, and an entirely pure point spectrum holds for sufficiently strong disorder. It is conjectured that for $d = 2$ there is no absolutely continuous part, but proof of the emptiness of $σac(H)$ is still wanting! Note that for the one-dimensional lattice, $σ(H) = [0,4]+ supp ν$ , and since $σ(H ) = σpp(H )$ , its eigenvalues form a dense subset of the spectrum, but no “extended states” exist. Physical theory (like the scaling theory) and substantial numerical evidence show that a “mobility edge”, signalling a phase transition from an insulator to a metal, exists for $d ≥ 3$ .

Things get much more complicated when considering interactions and relatively recent results indicating that (many-body) localization is an example of a failure of the eigenstate thermalization hypothesis have restored an intense interest in the field. As of now rigorous results exist only for the simplest of spin chains. Furthermore, the phenomenon of localization has ties to many other areas of research, including percolation theory and most interestingly, to the field of quantum chaology!

In Dirac's Sea