Functional and Spectral Analysis

Infinite-dimensional quantum systems need more than larger matrices: continuity, domains, and spectral measures become part of the problem. This chapter explains those structures and also offers an independent finite-dimensional route through convex optimization and semidefinite certificates.

Enter this chapter

Begin with Banach and Hilbert Spaces, Completion, and Riesz Representation for operator theory. For feasibility and bootstrap bounds, start directly with Convex Cones, Separation, Conic Duality, and Semidefinite Programs.

Topics

1. Convex Cones, Separation, Conic Duality, and Semidefinite Programs — Distinguishes exact feasibility and separation results from numerical candidates.
2. Banach and Hilbert Spaces, Completion, and Riesz Representation — Builds complete normed spaces and the representation of continuous linear functionals.
3. Bounded, Compact, and Integral Operators — Separates continuity, compactness, and the conditions behind integral operators.
4. Unbounded Operators, Domains, Closure, and Adjoints — Treats an operator’s domain as part of its definition.
5. Self-Adjointness, Extensions, and Unitary Evolution — Determines when symmetric expressions generate unitary time evolution.
6. Spectra, Resolvents, Spectral Measures, and Functional Calculus — Replaces matrix diagonalization by resolvents and spectral measures.
7. Trace Ideals and Fredholm Determinants — States the summability conditions needed for traces and determinants.
8. Locally Convex, Nuclear, and Rigged Hilbert Spaces — Places generalized vectors in duals of suitably refined test spaces.
9. Operator Algebras and Positive Functionals: a Bridge — Connects abstract observables and positive states to Hilbert-space representations.

How the ideas fit together

The same differential expression can define different operators on different domains. Symmetry checks an integration-by-parts identity on the chosen domain; self-adjointness additionally matches that domain to the adjoint’s. Only the complete self-adjoint operator supplies the spectral calculus used to define unitary evolution. See Teschl 2014, §§ 2.2, 2.6, and 5.1.

A spectral value need not be an eigenvalue with a normalizable eigenvector. This is why generalized eigenfunctions, resolvents, and spectral measures complement rather than merely repeat finite-dimensional algebra. Likewise, an infinite sum of diagonal entries is not automatically a basis-independent trace.

Review the chapter

Does a real spectrum prove that an arbitrary operator generates unitary evolution in the specified inner product?

Solution

No. Even finite-dimensional non-normal matrices can have real eigenvalues without being self-adjoint. If their exponentials were unitary for every real time, differentiating the unitarity identity at zero would force self-adjointness. Infinite dimensions add domain and continuity requirements.

Where to go next

Continue to States, Observables, and Spectra for the physical information carried by spectral measures and quantum states.

References

• Gerald Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014. DOI.