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.
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.