Spectra of Operators

The spectrum records where an operator cannot be inverted by a bounded operator defined on the whole Hilbert space. Eigenvalues are one source of this failure, but continuous spectrum is equally important in quantum mechanics. A free particle has a nonempty energy spectrum without normalizable energy eigenvectors.

Helpful background. Self-Adjoint Operators gives the real-spectrum theorem. Unbounded Operators explains why the domain must accompany the action.

Spectrum is failure of bounded inversion

Let $A$ be a closed densely defined operator. A complex number $z$ belongs to the resolvent set $\rho(A)$ when

$$A-zI:D(A)\longrightarrow\mathcal H$$

is bijective and its inverse is bounded as an operator on $\mathcal H$. The spectrum is $\sigma(A)=\mathbb C\setminus\rho(A)$.

Closedness implies that a bijective $A-zI$ already has a bounded inverse by the closed graph theorem. Keeping boundedness visible in the definition emphasizes the relevant stability: a small forcing term should not require a large solution.

For bounded operators, the Neumann series shows that $|z|>\|A\|$ lies in $\rho(A)$, so the spectrum is bounded. For unbounded self-adjoint operators the spectrum can extend to arbitrarily large real values. Self-adjointness ensures that every nonreal $z$ lies in the resolvent set.

A spectral value may correspond to a nontrivial kernel, to failure of surjectivity, or to both. The precise point/continuous/residual classification is developed in Point, Continuous, and Residual Spectrum.

Approximate eigenvectors detect self-adjoint spectrum

For a self-adjoint $A$ and real $\lambda$, the Weyl criterion says

$$\lambda\in\sigma(A) \quad\Longleftrightarrow\quad \exists\psi_n\in D(A),\ \|\psi_n\|=1,\ \|(A-\lambda)\psi_n\|\to0.$$

A proof of the difficult direction can be phrased contrapositively. If there were a uniform lower bound $\|(A-\lambda)\psi\|\geq c\|\psi\|$ with $c>0$, the range would be closed. The kernel would vanish, and self-adjointness would make the range dense by the range-kernel identity. Thus the range would be all of $\mathcal H$ and the inverse bounded, placing $\lambda$ in the resolvent.

The approximating vectors need not converge in norm. If they did converge to a unit vector, closedness would make that limit an eigenvector. Continuous-spectrum sequences instead escape any such conclusion.

Multiplication gives a worked spectral calculation

Take $Q\psi(x)=x\psi(x)$ on its maximal domain in $L^2(\mathbb R)$. For nonreal $z$, multiplication by $(x-z)^{-1}$ is bounded, giving the resolvent. For any real $\lambda$, choose a normalized vector supported in $(\lambda-\varepsilon,\lambda+\varepsilon)$. Then

$$\|(Q-\lambda)\psi_\varepsilon\|\leq\varepsilon.$$

Thus every real $\lambda$ is spectral. But $Q\psi=\lambda\psi$ would force $\psi$ to be supported on a singleton, which has measure zero. Therefore $\sigma(Q)=\mathbb R$ and its point spectrum is empty.

In momentum representation, the free Hamiltonian is multiplication by $p^2/(2m)$. The same support argument gives $\sigma(H_0)=[0,\infty)$, with no $L^2$ eigenvectors in infinite Euclidean space. This calculation uses the full function space, not a box-discretized approximation.

Discrete and essential spectrum

For self-adjoint operators, the discrete spectrum consists of isolated eigenvalues of finite multiplicity. Define the essential spectrum by removing those points:

$$\sigma_{\mathrm{ess}}(A)=\sigma(A)\setminus\sigma_{\mathrm{disc}}(A).$$

This convention includes continuous spectrum, eigenvalues of infinite multiplicity, and accumulation points of eigenvalues. Thus “essential” and “continuous” are not synonyms.

A refinement of the Weyl criterion says that $\lambda$ belongs to the essential spectrum precisely when approximate eigenvectors can be chosen to converge weakly to zero. Such a sequence cannot remain concentrated in a finite-dimensional eigenspace.

For the harmonic oscillator on the line, eigenvalues are isolated, have finite multiplicity, and tend to infinity. The resolvent is compact and the essential spectrum is empty. The free particle instead has essential spectrum $[0,\infty)$. These structural conclusions explain why confinement and infinite-volume propagation require different spectral language.

Gaps and spectral stability

An open interval disjoint from $\sigma(A)$ is a spectral gap. In a gap the resolvent exists, and for self-adjoint $A$,

$$\|(A-z)^{-1}\| =\frac{1}{\operatorname{dist}(z,\sigma(A))}.$$

This equality links distance from the spectrum to conditioning of the inverse equation. It is special to normal operators; for a nonnormal operator the resolvent norm can be much larger than inverse spectral distance.

If $V$ is a bounded self-adjoint perturbation and $\operatorname{dist}(z,\sigma(A))>\|V\|$, then the Neumann-series factorization shows $z\in\rho(A+V)$. This provides a concrete stability estimate, although it does not resolve fine spectral type.

Exercises

On $\ell^2(\mathbb N)$ let $Ae_n=n^{-1}e_n$. Determine its spectrum and decide which points are discrete.

Solution

The spectrum is $\{1,1/2,1/3,\ldots\}\cup\{0\}$. Each $1/n$ is an isolated simple eigenvalue, hence discrete. Zero is not an eigenvalue, but $\|Ae_n\|=1/n\to0$, so it is spectral. The vectors $e_n$ converge weakly to zero, placing zero in the essential spectrum. A limit point need not itself be an eigenvalue.

Can a finite-dimensional matrix approximation prove that an infinite-dimensional operator has only point spectrum?

Solution

No. Every finite matrix has only eigenvalues, while its limiting operator may have continuous spectrum. One needs a convergence theorem and control of the relevant spectral projections or resolvents. The multiplication and free-particle examples show how a sequence of discrete approximations can describe a continuum.

References

• K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics 265, Springer, 2012, doi:10.1007/978-94-007-4753-1 — closed-operator spectra and discrete spectral theory.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., American Mathematical Society, 2014, author’s book page — resolvents, Weyl criteria, and Schrödinger spectra.