Rigorous QM References

Rigorous quantum mechanics is where notation that is harmless in finite dimensions becomes a theorem with hypotheses. The central questions are domains, self-adjointness, spectra, unitary groups, quadratic forms, scattering theory, and the relation between generalized eigenvectors and actual Hilbert-space vectors.

Use this page with the Mathematical Quantum Mechanics Roadmap, symmetric versus self-adjoint operators, and unbounded operators.

Main Mathematical Physics References

Reed and Simon, Methods of Modern Mathematical Physics.

Best for: the standard multi-volume reference on functional analysis, Fourier analysis, self-adjointness, scattering, and Schrödinger operators.

Watch for: it is not a physics textbook. Use it to support theorem-level statements and pair it with physical examples elsewhere.

Teschl, Mathematical Methods in Quantum Mechanics.

Best for: a focused and accessible rigorous route through Hilbert spaces, self-adjoint operators, spectral theory, and one-dimensional Schrödinger operators.

Watch for: its strengths are mathematical structure and one-dimensional operator theory; many-body physics is not the focus.

Hall, Quantum Theory for Mathematicians.

Best for: finite-dimensional quantum theory, spectral theorem, Lie groups, angular momentum, and the mathematical organization of standard quantum examples.

Watch for: it is a mathematical introduction, so physical and experimental motivations should be supplied from physics sources.

Operator Theory and Schrödinger Operators

Cycon, Froese, Kirsch, and Simon, Schrödinger Operators.

Best for: Schrödinger operator techniques, spectra, bound states, and mathematical methods for quantum Hamiltonians.

Watch for: it assumes functional-analysis background.

Kato, Perturbation Theory for Linear Operators.

Best for: analytic perturbation theory, forms, self-adjoint operator perturbations, and stability of spectra.

Watch for: this is a mathematics monograph; cite it when perturbation-theory hypotheses matter, not for ordinary first-order formulas.

Simon, Quantum Mechanics for Hamiltonians Defined as Quadratic Forms.

Best for: quadratic-form methods and Hamiltonian construction beyond naive differential expressions.

Watch for: it is specialized; most readers should first learn the domain problem through a modern mathematical quantum text.

Scattering and Spectral Analysis

Yafaev, Mathematical Scattering Theory.

Best for: rigorous scattering theory, wave operators, completeness, and spectral assumptions.

Watch for: it is more mathematical than the scattering material in standard graduate quantum mechanics.

Reed and Simon, Vol. III, Scattering Theory.

Best for: the classic mathematical physics treatment of scattering.

Watch for: use physics texts for cross sections and experimental interpretation, and rigorous texts for existence and completeness statements.

Quantum Statistical Mechanics and Operator Algebras

Bratteli and Robinson, Operator Algebras and Quantum Statistical Mechanics.

Best for: $C^*$ -algebras, KMS states, thermodynamic limits, and mathematical quantum statistical mechanics.

Watch for: this is beyond ordinary nonrelativistic quantum mechanics and should be cited only when algebraic or infinite-system structure is actually needed.

Convention Checks

Mathematicians often take the inner product linear in the first argument; physicists usually take it linear in the second.
A symmetric operator is not automatically self-adjoint.
A formal differential expression is not a Hamiltonian until the domain and boundary conditions are specified.
Generalized eigenvectors are distributional tools, not elements of the Hilbert space in the usual continuous-spectrum cases.
Spectral theorem statements differ for bounded, unbounded, and normal operators.

Cross-Links

References

M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.
M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, 1975.
M. Reed and B. Simon, Methods of Modern Mathematical Physics III: Scattering Theory, Academic Press, 1979.
M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, 1978.
G. Teschl, Mathematical Methods in Quantum Mechanics, 2nd ed., American Mathematical Society, 2014.
B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger Operators, Springer, 1987.
T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, 1995.
B. Simon, Quantum Mechanics for Hamiltonians Defined as Quadratic Forms, Princeton University Press, 1971.
D. R. Yafaev, Mathematical Scattering Theory: General Theory, American Mathematical Society, 1992.
O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, 2nd ed., Springer, 1987-1997.