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.

Version: v1 · 2026-09-28 · History
Domain-specific #: 10701
Domain group: Natural Sciences
Origin domain: Physics
Subdomain: Quantum Scattering Theory → Physics

Core Idea¶

Method of continued fractions is treated here as the recurring natural_sciences_engineering_health 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. The goal of the method is to solve the integral equation.

|\psi\rangle = |\phi\rangle + G_0 V|\psi\rangle. iteratively and to construct convergent continued fraction for the T-matrix. In the first one (denoted as MCFV) we construct approximations of the potential energy operator V in the form of separable function of rank 1, 2, 3 ...

For Method of continued fractions, the abstraction is narrower than the article's general subject matter: a positive case must preserve 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. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in natural_sciences_engineering_health, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — In practical calculation the infinite chain fraction is replaced by finite one assuming that.
Constitutive relation — 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.
Operating condition — The method can thus be understood as resummation of (in general divergent) Born series by Padé approximants.
Recognition evidence — In general the method requires similar amount of numerical work as calculation of terms of Born series, but it provides much faster convergence of the results.
Admissible variation — It was invented by Horáček and Sasakawa in 1983.
Characteristic consequence — V = \frac{V|\phi\rangle\langle\phi|V}{\langle\phi|V|\phi\rangle} + V_1 .
Failure boundary — The integral equation for the rank-one part of potential is easily soluble.

What It Is Not¶

Not the whole field of natural_sciences_engineering_health. The node requires the specific identity stated by 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.
Not an over-broad reading. The chain fraction for T-matrix now also holds, with little bit different definition of coefficients \beta_i, \gamma_i .
Not an over-broad reading. V = \frac{V|\phi\rangle\langle\phi|V}{\langle\phi|V|\phi\rangle} + V_1 .
Not an over-broad reading. The integral equation for the rank-one part of potential is easily soluble.
Not automatically Continued Fraction. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Method of continued fractions applies literally inside natural_sciences_engineering_health wherever the source-defined carrier and relation can be established. Its documented habitats include:

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 .
Properties and relation to other methods. The method has been successfully used for solution of problems in both nuclear and molecular physics.
Algorithm of MCFG. The second variant of the method construct the approximations to the Green's operator.
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.
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.
Properties and relation to other methods. The method was tested on calculation of collisions of electrons from hydrogen atom in static-exchange approximation.

Outside natural_sciences_engineering_health, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.

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. The strongest recognition evidence in the frozen account is: In general the method requires similar amount of numerical work as calculation of terms of Born series, but it provides much faster convergence of the results. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification The chain fraction for T-matrix now also holds, with little bit different definition of coefficients \beta_i, \gamma_i . so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Method of continued fractions compresses multiple natural_sciences_engineering_health 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} + V_1 . This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the natural_sciences_engineering_health 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. In general the method requires similar amount of numerical work as calculation of terms of Born series, but it provides much faster convergence of the results.
Test variation. Change an implementation or setting while preserving it was invented by Horáček and Sasakawa in 1983.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.

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. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

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 . This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → 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; recognition evidence → In general the method requires similar amount of numerical work as calculation of terms of Born series, but it provides much faster convergence of the results

Applied / In Practice¶

In this case the method reproduces exact results for scattering cross-section on 6 significant digits in 4 iterations. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Properties and relation to other methods; invariant → 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; boundary → the case exits the class when the chain fraction for T-matrix now also holds, with little bit different definition of coefficients \beta_i, \gamma_i

Structural Tensions¶

T1 — Stable identity versus admissible variation. The chain fraction for T-matrix now also holds, with little bit different definition of coefficients \beta_i, \gamma_i . The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. V = \frac{V|\phi\rangle\langle\phi|V}{\langle\phi|V|\phi\rangle} + V_1 . The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. The integral equation for the rank-one part of potential is easily soluble. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. The full solution of the original problem can therefore be expressed as. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. In practical calculation the infinite chain fraction is replaced by finite one assuming that. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Method of continued fractions literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. 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. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Method of continued fractions distinguish that the broader parent Pattern leaves together?

Structural–Framed Character¶

Method of continued fractions is structural-leaning. Its structural side is the repeatable organization summarized by 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. Its framed side is the natural_sciences_engineering_health vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: The method can thus be understood as resummation of (in general divergent) Born series by Padé approximants. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. 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 stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: In practical calculation the infinite chain fraction is replaced by finite one assuming that. 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. It further constrains recognition and variation through: The method can thus be understood as resummation of (in general divergent) Born series by Padé approximants. In general the method requires similar amount of numerical work as calculation of terms of Born series, but it provides much faster convergence of the results.

What is domain-bound. natural sciences engineering health supplies the operative entities, technical vocabulary, warrants, and exceptions that make Method of continued fractions literal. Its documented scope includes the condition that 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 . Another bounded application condition is that The method has been successfully used for solution of problems in both nuclear and molecular physics. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—It was invented by Horáček and Sasakawa in 1983.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry is a kind of Algorithm.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Method of continued fractions. The reviewed identity 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. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Relationships to Other Abstractions¶

Current abstraction Method of continued fractions Domain-specific

Parents (1) — more general patterns this builds on

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)

Show alternative path (1)

Neighborhood in Abstraction Space¶

Method of continued fractions sits in a moderately populated region (57^th 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

Not to Be Confused With¶

Pattern. The parent omits the specialist differentia. Tell: Can the case establish 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?
Continued Fraction. Represent a number or function by recursively nesting fractional terms in successive denominators, with finite truncations forming convergents governed by two-step recurrences. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Beam Propagation Method. Approximate predominantly forward optical-wave evolution by factoring out a carrier, reducing the Helmholtz or Maxwell problem to a one-way propagation equation, and marching its transverse field through longitudinal steps. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Holstein–Herring method. Extract the exponentially small exchange splitting between asymptotically degenerate molecular states from probability-current flux through a median surface, avoiding cancellation of nearly equal total energies. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Method of continued fractions remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside natural_sciences_engineering_health lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Method_of_continued_fractions (revision 1335351013).

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.