Method of continued fractions¶
The method of continued fractions is a method developed specifically for solution of integral equations of quantum scattering theory like Lippmann–Schwinger equation or Faddeev equations.
Core Idea¶
Method of continued fractions is treated here as the recurring naturalsciencesengineeringhealth identity summarized by this source-grounded definition: The method of continued fractions is a method developed specifically for solution of integral equations of quantum scattering theory like Lippmann–Schwinger equation or Faddeev equations. The method of continued fractions is a method developed specifically for solution of integral equations of quantum scattering theory like Lippmann–Schwinger equation or Faddeev equations. It was invented by Horáček and Sasakawa in 1983.
Scope of Application¶
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Properties and relation to other methods. In the case of first iteration of MCFV method we get the same result as from Schwinger variational principle with trial function |\psi\rangle = |\phi\rangle .
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Properties and relation to other methods. The method has been successfully used for solution of problems in both nuclear and molecular physics.
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Algorithm of MCFG. The second variant of the method construct the approximations to the Green's operator.
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Properties and relation to other methods. The expressions for the T-matrix resulting from both methods can be related to certain class of variational principles.
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Properties and relation to other methods. The higher iterations with N-terms in the continuous fraction reproduce exactly 2N terms (2N + 1) of Born series for the MCFV (or MCFG) method respectively.
Clarity¶
A clear use of Method of continued fractions names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is The method of continued fractions is a method developed specifically for solution of integral equations of quantum scattering theory like Lippmann–Schwinger equation or Faddeev equations.
Manages Complexity¶
Method of continued fractions compresses multiple naturalsciencesengineeringhealth details into a stable diagnostic relation. The source shows both the central mechanism—it can also be shown that both methods reproduce exactly the solution of the Lippmann-Schwinger equation with the potential given by finite-rank operator.—and the practical consequence—v = \frac{V|\phi\rangle\langle\phi|V}{\langle\phi|V|\phi\rangle} + V1 .
Abstract Reasoning¶
- Type the carrier. Identify the naturalsciencesengineeringhealth entities to which the claim applies.
- State the relation. Use the source-grounded identity: The method of continued fractions is a method developed specifically for solution of integral equations of quantum scattering theory like Lippmann–Schwinger equation or Faddeev equations.
- Check operation and conditions. The method can thus be understood as resummation of (in general divergent) Born series by Padé approximants.
- Demand recognition evidence.
Knowledge Transfer¶
Within the home domain. Knowledge about Method of continued fractions transfers literally when a new case preserves the same carrier type, relation, and recognition test. In the case of first iteration of MCFV method we get the same result as from Schwinger variational principle with trial function |\psi\rangle = |\phi\rangle . The method has been successfully used for solution of problems in both nuclear and molecular physics. Beyond the home domain. No canonical parent is asserted for Method of continued fractions.
Relationships to Other Abstractions¶
Current abstraction Method of continued fractions Domain-specific
Parents (1) — more general patterns this builds on
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Method of continued fractions is a kind of Algorithm Prime
The continued-fractions method is a repeatable computational procedure for solving declared integral equations.
Hierarchy paths (2) — routes to 2 parentless roots
- Method of continued fractions → Algorithm → Function (Mapping)
Neighborhood in Abstraction Space¶
Method of continued fractions sits in a moderately populated region (57th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Named Analytic Theorems & Operators (39 abstractions)
Nearest neighbors
- Mehler Kernel — 0.86
- Translation operator (quantum mechanics) — 0.85
- False position method — 0.85
- Linear elasticity — 0.85
- Functional determinant — 0.85
Computed from structural-signature embeddings · 2026-10-08