Question: In my spare time, I have been studying and analysing continued fractions. 

I was having a conversation with someone on Discord in a Mathematics server and he was telling me that continued fractions can be related to quantum physics. He didn't go into it too much and the concept he was describing seemed a little vague to me. I am aware that Pincherle's theorem[1] asserts there is an intimate relationship with three-term recurrence relations of the form $x\_{n+1}=b\_nx\_n+a\_nx\_{n-1}$ and continued fractions, more accurately with their partial convergents, given that this recurrence relation has a minimum if a related cfrac converges. But I am not quite the physicist myself to compare physics with the properties of cfracs.

Although I can go on Google and do some research on this, I figured it might serve useful to have a post on this in this beta community, but I do apologise if it is too broad or open-ended and thus upsets any regulations here.

Any thoughts on this?

### References

[1] Pincherle, S. (1894). Delle funzioni ipergeometriche e di varie questioni ad esse attinenti. *Giorn. Mat. Battaglini*. 32:209–29

[2] Parusnikov, V. I. A Generalization of Pincherle's Theorem to k-Term Recursion Relations. *Math Notes* **2005,** *78* (5-6), 827–840. [DOI: 10.1007/s11006-005-0188-7](https://doi.org/10.1007/s11006-005-0188-7).

Answer: 
In the paper "[A Continued-Fraction Representation of the Time-Correlation Functions](https://academic.oup.com/ptp/article/34/3/399/1943170)", generalized susceptibilities and transport coefficients for materials are obtained using a continued-fraction expansion of the Laplace transform of the time-correlation functions.

This was the precursor to what is now called the "[hierarchical equations of motion](https://en.wikipedia.org/wiki/Hierarchical_equations_of_motion)" which are used to study the dynamics of a quantum system (such as an electron) coupled to a bosonic bath (for example the vibrations of the lattice in a GaAs semi-conductor quantum dot). This area is called "dissipative quantum dynamics" or "open quantum systems" and is used to study for example, the decoherence of qubits in solid-state quantum computers.
You may take a look at the [Method of Continued Fractions](https://en.wikipedia.org/wiki/Method_of_continued_fractions) used in quantum scattering theory—this was only [formed in 1983](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.28.2151)1 so is rather recent. Related is the PhD thesis by [Kónya (2000)](https://arxiv.org/pdf/quant-ph/0101040.pdf)2; §3.3 onwards. 

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 Reference 

 [1] Horáček, J., Sasakawa, T. (1983). Method of continued fractions with application to atomic physics. *Physical Review A.* 28(4):2151–2156. 

 [2] Kónya, B. (2000). Continued fraction representation of quantum mechanical Green's operators. PhD thesis. arXiv:0101040 [quant-ph].