Continuous Spectrum Without Eigenvectors

A continuous spectral value need not belong to any normalizable eigenvector. It is detected instead by bounded-inverse failure, nonzero spectral windows, and normalized states whose eigenvalue residual becomes arbitrarily small. Plane waves encode useful generalized eigenfunctions, but physical Hilbert-space states are wave packets.

Helpful background. Spectra of Operators states the Weyl criterion. Spectral Projections relates approximate eigenvectors to spectral windows.

Spectral values and exact eigenvectors differ

For self-adjoint $A$,

$$E_A(\{\lambda\})\neq0 \quad\Longleftrightarrow\quad \lambda\text{ is an eigenvalue}.$$

By contrast, $\lambda\in\sigma(A)$ means that every open neighborhood of $\lambda$ has nonzero spectral projection. These statements differ because a countably additive measure can assign zero mass to every singleton while assigning positive mass to intervals.

For position multiplication on $L^2(\mathbb R)$, an exact eigenvector at $x_0$ would have to vanish outside the singleton $\{x_0\}$. Such a vector is zero in $L^2$. Nevertheless every interval around $x_0$ supports nonzero wavefunctions, so $x_0$ belongs to the spectrum.

It is therefore incorrect to infer an empty spectrum from the absence of normalizable solutions to the eigenvalue equation.

Normalized approximate eigenvectors

For a self-adjoint operator, $\lambda$ is spectral precisely when unit vectors $\psi_n\in D(A)$ can satisfy

$$\|(A-\lambda)\psi_n\|\longrightarrow0.$$

If $\lambda$ is not an eigenvalue, these vectors cannot converge in norm to a nonzero limit. Otherwise closedness of $A$ would produce an actual eigenvector. The limiting spectral accuracy is gained while compactness of the vector sequence is lost.

For position, take normalized packets supported in $(x_0-\varepsilon,x_0+\varepsilon)$. Their position residual is at most $\varepsilon$. As $\varepsilon\to0$, they become more localized and can converge weakly to zero, but they do not converge in $L^2$ to a delta function. A delta distribution is outside that Hilbert space.

For momentum, improving spectral accuracy instead requires a broad spatial packet. The two constructions illustrate different routes by which normalized approximate eigenvectors fail to have a norm limit.

A free-particle Weyl sequence with an error bound

On $L^2(\mathbb R)$, let $H_0=-(\hbar^2/2m)d^2/dx^2$ with domain $H^2(\mathbb R)$. Choose $\chi\in C_c^\infty(\mathbb R)$ with $\|\chi\|_2=1$, and for a real wave number $k$ define

$$\psi_R(x)=R^{-1/2}\chi(x/R)e^{ikx}, \qquad E=\frac{\hbar^2k^2}{2m}.$$

Every $\psi_R$ is normalized and lies in $D(H_0)$. Differentiating the smooth envelope gives

$$(H_0-E)\psi_R =-\frac{\hbar^2e^{ikx}}{2m\sqrt R} \left[\frac{\chi''(x/R)}{R^2} +\frac{2ik\,\chi'(x/R)}{R}\right].$$

After rescaling the integrals,

$$\|(H_0-E)\psi_R\| \leq\frac{\hbar^2}{2m} \left(\frac{\|\chi''\|_2}{R^2} +\frac{2|k|\|\chi'\|_2}{R}\right).$$

The residual tends to zero, proving that every $E\geq0$ is spectral. At the threshold $k=0$, the leading bound is of order $R^{-2}$.

The packets converge weakly to zero: their overlap with any compactly supported smooth test vector tends to zero because of the factor $R^{-1/2}$, and density extends the result to every fixed $L^2$ vector. Thus they also diagnose essential spectrum. No finite box or delta-normalized state is needed in this proof.

Generalized eigenfunctions organize the representation

The formal momentum eigenfunction $e^{ipx/\hbar}$ satisfies the differential equation but has infinite $L^2(\mathbb R)$ norm. With the convention

$$\widehat\psi(p)=\frac{1}{\sqrt{2\pi\hbar}} \int e^{-ipx/\hbar}\psi(x)\,dx,$$

the Fourier transform extends by Plancherel to a unitary map on $L^2$. Momentum becomes multiplication by $p$, and free energy becomes multiplication by $p^2/(2m)$.

The expansion into plane waves is rigorously an $L^2$ transform identity, with stronger pointwise claims requiring extra assumptions. Individual plane waves can also be interpreted as continuous functionals on Schwartz test functions. That extension is the subject of Rigged Hilbert Spaces.

Delta normalization expresses orthogonality at the distributional level. It is not normalization to unit Hilbert norm, and it does not create exact-momentum state vectors in the original space.

Continuous spectrum is broader than free propagation

A general self-adjoint operator may have singular continuous spectral measures, for which there is no density with respect to Lebesgue measure. It may also have thresholds, channel multiplicity, and embedded eigenvalues. A free plane-wave expansion is therefore a model-specific representation, not the definition of continuous spectrum.

Moreover, the range-based continuous spectral set can contain accumulation points of a pure point operator. The distinction from measure-theoretic continuous subspaces is explained in Point, Continuous, and Residual Spectrum.

Scattering theory adds comparison with free dynamics and proves which subspace is described by asymptotic scattering states. The spectral theorem alone does not establish asymptotic completeness.

Exercises

Why does a periodic box of length $L$ have normalizable momentum eigenvectors although the whole-line operator does not?

Solution

The box eigenfunctions have constant modulus $L^{-1/2}$ and discrete momenta $2\pi\hbar n/L$. Their normalization uses a finite integration interval. As $L$ grows, the allowed momenta become dense, but the individual normalized vectors do not converge in whole-line Hilbert norm to normalized plane waves. The limiting spectral representation changes from a discrete sum to a continuum measure.

Use the free-particle packet above to estimate its momentum residual.

Solution

For $P=-i\hbar\,d/dx$, $(P-\hbar k)\psi_R=-i\hbar R^{-3/2}\chi'(x/R)e^{ikx}$. Hence $\|(P-\hbar k)\psi_R\|=\hbar\|\chi'\|_2/R$. Broadening the spatial envelope narrows its momentum content while maintaining unit norm.

References

• B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013, doi:10.1007/978-1-4614-7116-5 — Fourier representation and continuous spectra.
• K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics 265, Springer, 2012, doi:10.1007/978-94-007-4753-1 — spectral measures and approximate eigenvectors.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., American Mathematical Society, 2014, author’s book page — Weyl criteria, free Schrödinger operators, and scattering.