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: 
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].
The continued fraction expansion is the most common way to calculate real-frequency dynamical Green's functions using [Lanczos exact diagonalization](https://en.wikipedia.org/wiki/Lanczos_algorithm). The method was introduced in this setting in [Gagliano, E. R., and Balseiro, C. A., "*Dynamical Properties of Quantum Many-Body Systems at Zero Temperature*," Physical Review Letters **59**, 2999 (1987)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.59.2999), but a more common reference these days is [Dagotto, E., "*Correlated electrons in high-temperature superconductors*," Reviews of Modern Physics **66**, 763 (1994)](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.66.763).

The same expansion was also used in the first attempts to calculate dynamical correlation functions using the density-matrix renormalization group, see [K. A. Hallberg, "*Density-matrix algorithm for the calculation of dynamical properties of low-dimensional systems*," Physical Review B **52**, R9827(R) (1995)](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.52.R9827). However, it was discovered that the method has some issues, especially with higher frequencie, see [Kühner, T. D., and White, S. R., White"*Dynamical correlation functions using the density matrix renormalization group*," Physical Review B **60**, 335 (1999)](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.60.335). Today, it's mostly been displaced by other methods in the DMRG setting.