Math 2.0

Operator theory Operator semigroups and groups Theorem

Lie–Trotter product formula

Builds on Hilbert spaceSelf-adjoint operatorEssentially self-adjoint operatorStone's theorem on one-parameter unitary groups

Statement

Let HH be a complex Hilbert space. Let AA and BB be self-adjoint operators on HH, with domains D(A)D(A) and D(B)D(B). Assume that A+BA + B is essentially self-adjoint on D(A)∩D(B)D(A) \cap D(B), and let CC be its closure. Then for every t∈ℝt \in \R and every x∈Hx \in H, eitCx=lim⁡n→∞(eitA/neitB/n)nx,e^{itC}x = \lim_{n \to \infty} \left( e^{itA/n} e^{itB/n} \right)^n x, where the unitary groups eitAe^{itA}, eitBe^{itB} and eitCe^{itC} are those of Stone's theorem.

Remarks

For n×nn \times n matrices this is the Lie product formula, and the proof idea is the same. The difference eisAeisB−eisCe^{isA}e^{isB} - e^{isC} is o(s)o(s) as s→0s \to 0 when applied to a vector of D(A)∩D(B)D(A) \cap D(B). A telescoping sum turns nn such errors of size o(1/n)o(1/n) into a total error that tends to 00. In infinite dimensions, the convergence is in the strong sense, vector by vector, and the essential self-adjointness of A+BA + B is used to pass from D(A)∩D(B)D(A) \cap D(B) to general xx. A similar formula holds for the semigroups e−tAe^{-tA} and e−tBe^{-tB} with t≥0t \ge 0, when AA and BB are bounded below and A+BA + B is essentially self-adjoint on D(A)∩D(B)D(A) \cap D(B).

Example: in quantum mechanics with A=−ΔA = -\Delta and B=VB = V a suitable potential, e−itA/ne^{-itA/n} acts by a Fourier multiplier and e−itV/ne^{-itV/n} by multiplication. The formula approximates the Schrödinger evolution by alternating free motion and potential kicks; this is the basis of split-step methods.

Going deeper

For a student

The result is due to Trotter, who extended Lie's formula for matrices to unbounded operators. It is used to build the Feynman–Kac formula: for −Δ+V-\Delta + V, the alternating products become integrals over paths of Brownian motion. In numerical analysis it justifies operator splitting, where a hard evolution is approximated by easy ones. The hypothesis that A+BA + B is essentially self-adjoint on D(A)∩D(B)D(A) \cap D(B) ensures that the limit is the evolution of a well-defined Hamiltonian.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

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Adjoint of an unbounded operator, concept, no evidence levelAdjoint of an unbound…conceptBorel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptContinuous functional calculus, concept, no evidence levelContinuous functional…conceptDensely defined operator, concept, no evidence levelDensely defined opera…conceptEssentially self-adjoint operator, concept, no evidence levelEssentially self-adjo…conceptHilbert space, concept, no evidence levelHilbert spaceconceptHilbert space adjoint, concept, no evidence levelHilbert space adjointconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionProjection-valued measure, concept, no evidence levelProjection-valued mea…conceptSelf-adjoint operator, concept, no evidence levelSelf-adjoint operatorconceptSpectrum of an operator, concept, no evidence levelSpectrum of an operat…conceptUnbounded operator, concept, no evidence levelUnbounded operatorconceptUnitary operator, concept, no evidence levelUnitary operatorconceptTranspose, definition, no evidence levelTransposedefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionNormal matrix, definition, no evidence levelNormal matrixdefinitionUnitary matrix, definition, no evidence levelUnitary matrixdefinitionSpectral theorem for normal matrices, statement, no evidence levelSpectral theorem for …statementSpectral theorem for bounded normal operators, statement, no evidence levelSpectral theorem for …statementSpectral theorem for unbounded self-adjoint operators, statement, no evidence levelSpectral theorem for …statementStone's theorem on one-parameter unitary groups, statement, no evidence levelStone's theorem on on…statementLie–Trotter product formula, statement, no evidence levelLie–Trotter product f…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Adjoint of an unbounded operator concept
  2. Borel sigma-algebra concept
  3. Continuous functional calculus concept
  4. Densely defined operator concept
  5. Essentially self-adjoint operator concept
  6. Hilbert space concept
  7. Hilbert space adjoint concept
  8. Identity matrix object
  9. Matrix definition
  10. Projection-valued measure concept
  11. Self-adjoint operator concept
  12. Spectrum of an operator concept
  13. Unbounded operator concept
  14. Unitary operator concept
  15. Transpose definition
  16. Conjugate transpose definition
  17. Normal matrix definition
  18. Unitary matrix definition
  19. Spectral theorem for normal matrices statement
  20. Spectral theorem for bounded normal operators statement
  21. Spectral theorem for unbounded self-adjoint operators statement
  22. Stone's theorem on one-parameter unitary groups statement
  23. Lie–Trotter product formula statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Lie_product_formula
About this page
TagKH7EPKDCH3
URIm2://KH7EPKDCH3
Kindstatement, theorem
WritingW0 stub
MSC47D03, 47B25, 81Q10
StatusThis copy of the site has no build data. Run m2 status KH7EPKDCH3.
AboutSelf-adjoint operator
UsesHilbert space, Self-adjoint operator, Essentially self-adjoint operator, Stone's theorem on one-parameter unitary groups
Changed2 November 2026

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