Spectral Projections

A spectral projection selects the part of a state whose observable value lies in a chosen Borel set. It provides a precise version of an energy cutoff, a bound-state subspace, or an approximate eigenvalue window. Its usefulness comes from being bounded even when the observable itself is unbounded.

Helpful background. Projection-Valued Measures develops projection algebra and strong additivity. Resolvent Operators supplies the sign convention for the contour formulas below.

Spectral windows select reducing subspaces

Let $A$ be self-adjoint and $P=E_A(\Delta)$. Spectral calculus gives

$$P D(A)\subseteq D(A), \qquad AP\psi=PA\psi\quad(\psi\in D(A)).$$

The first assertion follows because restricting a finite second-moment integral to $\Delta$ cannot make it infinite. The same holds for $I-P$, so both subspaces reduce $A$.

The restricted operator on $P\mathcal H$ has domain $D(A)\cap P\mathcal H$ and is self-adjoint there. Reduction is stronger than observing that a formal commutator vanishes on a convenient test space: it controls the full domain and the orthogonal complement.

If $\Delta\subseteq[-M,M]$, then $P\mathcal H\subseteq D(A)$ and

$$\|AP\psi\|\leq M\|P\psi\|.$$

Thus an unbounded observable becomes bounded on a finite spectral window. This is the mathematical basis for energy-cutoff approximations.

Narrow windows produce approximate eigenvectors

For $P_\varepsilon=E_A((\lambda-\varepsilon,\lambda+\varepsilon))$, every $\phi\in P_\varepsilon\mathcal H$ satisfies

$$\|(A-\lambda)\phi\|\leq\varepsilon\|\phi\|.$$

The proof integrates $(a-\lambda)^2$ over the selected window. If $\lambda\in\sigma(A)$, every such open neighborhood has nonzero projection, so normalized vectors in its range give approximate eigenvectors.

Conversely, if $\lambda$ lies outside the spectrum, a sufficiently small window has zero projection. The spectrum is therefore the support of the operator-valued measure, not the set of points with nonzero singleton projections.

The singleton identity is

$$\operatorname{Ran}E_A(\{\lambda\})=\ker(A-\lambda).$$

A continuous spectral value has zero singleton projection even though all its neighborhoods have nonzero projections. Shrinking a window can therefore reduce the selected norm to zero in every fixed state while still allowing a normalized, window-dependent approximate eigenvector.

Isolated eigenvalues can be extracted by a contour

Suppose an isolated spectral component is enclosed by a positively oriented contour $\Gamma$ lying in $\rho(A)$. With $R_A(z)=(A-z)^{-1}$, its spectral projection is

$$P=-\frac{1}{2\pi i}\oint_\Gamma R_A(z)\,dz.$$

The minus sign comes from the scalar identity $(\lambda-z)^{-1}=-(z-\lambda)^{-1}$. Applying the spectral calculus reduces the contour integral to the scalar residue theorem: enclosed spectral values contribute one and exterior values contribute zero.

For an isolated eigenvalue $\lambda_0$, the resolvent contains

$$R_A(z)=\frac{P_{\lambda_0}}{\lambda_0-z} +\text{a term analytic near }\lambda_0.$$

The projector can have rank greater than one or even infinite rank. An isolated spectral point is discrete spectrum only when its eigenspace has finite dimension. A contour cannot isolate a point embedded in continuous spectrum because no surrounding contour can remain in the resolvent set while separating it from nearby spectral values.

Endpoint conventions matter in limits

For a decreasing sequence of Borel sets $\Delta_n\downarrow\Delta$, the projections converge strongly: $E_A(\Delta_n)\to E_A(\Delta)$. In particular, closed windows shrinking to $\lambda$ converge to the eigenprojection at $\lambda$.

The distribution family

$$F_A(t)=E_A((-\infty,t])$$

is strongly right-continuous. Its jump at $t=\lambda$ is the singleton projection $E_A(\{\lambda\})$. Open and closed endpoint choices therefore differ at eigenvalues, even though they agree for purely absolutely continuous position distributions.

These distinctions matter when a cutoff passes through a bound state: the projected state can change by a finite component rather than by an arbitrarily small tail.

Energy windows and numerical interpretation

A small spectral window can group several nearby eigenvalues without selecting a preferred eigenbasis inside that subspace. The projection is basis-independent, making it more stable than individual eigenvectors near degeneracies.

For numerical work, compare subspaces or projectors when levels nearly cross. A well-converged total projection does not necessarily fix a unique eigenvector orientation within a degenerate space. Likewise, a projection onto a finite energy band does not mean the band has finite dimension; a continuous spectral interval typically has infinite-dimensional range.

Exercises

For normalized $\psi$ and $P=E_A(\Delta)$, find the distance from $\psi$ to the selected subspace and the probability retained by the cutoff.

Solution

The orthogonal projection minimizes distance to its range, so the distance is $\|\psi-P\psi\|=\sqrt{1-\|P\psi\|^2}$. The retained probability is $\|P\psi\|^2=\mu_\psi^A(\Delta)$. If this is nonzero, normalized conditioning uses $P\psi/\|P\psi\|$; the cutoff itself is an unnormalized orthogonal component.

For position on $\mathbb R$, explain how $E_Q([-\varepsilon,\varepsilon])\to0$ strongly while each nonzero projection has norm one.

Solution

For fixed $\psi$, the selected norm squared is $\int_{-\varepsilon}^{\varepsilon}|\psi(x)|^2\,dx\to0$. For every positive $\varepsilon$ there is a normalized vector supported inside the interval, so the projection norm is one. The maximizing vector changes with $\varepsilon$; strong and norm convergence test different quantifiers.

References

• B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013, doi:10.1007/978-1-4614-7116-5 — spectral subspaces and continuous spectrum.
• 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 projections and isolated spectrum.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., American Mathematical Society, 2014, author’s book page — resolvents and spectral-window estimates.