As far I can see this is essentially a rewriting of the Schrödinger equation, a common starting point for WKB approximations. They do keep the time dependence, but they also seem to make approximations.  And they work with 1-dimensional problems, which (in principle) can be solved very fast to satisfactory accuracy on your smartphone; I think extensions to higher dimensions, if possible, will be interesting. 

Anyway, I don't think they really want to replace Quantum Mechanics by something else. By transforming to a Riccati equation you hide the underlying superposition principle. But it is still there, only hidden.

As far I can see this is essentially a rewriting of the Schrödinger equation, a common starting point for WKB approximations. They do keep the time dependence, but they also seem to make approximations.  And they work with 1-dimensional problems, which (in principle) can be solved very fast to satisfactory accuracy on your smartphone; I think extensions to higher dimensions, if possible, will be interesting. 

Anyway, I don't think they really want to replace Quantum Mechanics by something else. By transforming to a Riccati equation you hide the underlying superposition principle. But it is still there, only hidden.

totally agree with Kåre Olaussen.

Classic nonlinear systems with undefined initial conditions (random  i.c.) are described by linear equations for correlation functions. Hence, the nonlinearity does not deny linearity:-)

In fact classical dynamics doesn't have anything to do with it; this is just a change of variables, that may be convenient for certain problems. When the wavefunction satisfies the Schrödinger equation, its logarithmic derivative satisfies an equation that may be technically easier to solve, sometimes, e.g. http://journals.aps.org/pra/abstract/10.1103/PhysRevA.43.1169

One shouldn't confuse mathematics-of which this is an example-with physics-which is what is done with the solution, namely the calculation of transition probabilities. That's what matters. 

I just want to add that  I have estimated Dynamic Ricatti equations of dynamical financial market systems and later tested that they satisfy certain Physics Laws like "Arrow of Time" or Conservation of Linear Momentum or some systems Laws like Classification Laws developed by me for such econophysical systems using String Theory. I think this can also be a valid line of approach. Earl Prof (University Institute Chair) Dr. (D.Phil.) SKM QC EPS Fellow (Indirect) MRES MES MAICTE

Dear Victor,

It is no secret that you can formulate QM via the Riccati equation. Actually our resonance group in Uppsala and Stockholm analysed and reformulated scattering theory from the classical Weyl's theory, where the Riccati equation played a major role.

Our aim was to rigorously incorporate complex resonance energies with uniquely defined life times. Unfortunately the interesting branch of the solution to Riccati's equation was unstable. The remedy is, however, quite simple by e.g. employing the so-called exterior complex scaling.

I enclose our paper from the eighties, where we calculated and analysed the method and particularly explained why the numerical behaviour needed to be understood in detail. Those parts particularly related "Riccati" are marked with yellow.

The Riccati method was easily extended to coupled equations and applied to various atomic and molecular systems.

As I know Dieter Schuch I am sure he is aware of our work.  

I would like to argue a general point concerning linearity vs. non-linearity. It may come as a surprise that the distinction is not so clear cut as many might think. There is in particular a tendency to identify ``nonlinear'' with ``complex, chaotic and generally mysterious'' which I cannot agree with.

In particular, QM is undoubtedly a linear theory, at least in its ordinary Schroedinger formulation. But then, so is classical mechanics, in the formulation Koopman gave to it (see https://en.wikipedia.org/wiki/Koopman%E2%8%93von_Neumann_classical_mechanics). The Hamiltonian approach is better known, but less suitable for comparisons with QM. In any case, QM can also be given a non-linear appearance, by looking at the equations of motion for various observables. Indeed, these equations are far worse than non-linear, they are usually not even closed, that is, they lead to infinite hierarchies of equations.

Summarising, the nature of the theory is determined by its physics, not by the linear vs. non-linear appearance of its equations. When one theory becomes equivalent to another in some limit, as is the case for quantum mechanics and classical, we must assume that the encompassing theory is at least as complex as the limiting theory, irrespective of issues of linearity.

Probably I should end by saying that this is a long-winded way of agreeing with Kare.

There is no contradiction here, i.e. to reformulate a linear theory, like QM, to a nonlinear equation, like the Riccati equation – the latter translating the boundary condition (the logarithmic derivative) from one point in the interval to another one, i.e. to the singular point(s) (the limit point case).

The approach depends on what you want to know. In many cases a particular eigenfunction is not enough – one might want to know the spectral function, the total- or differential cross sections, S- or K-matrix etc.

These questions were displayed and answered in the enclosed article in my earlier posting. In order to reveal the simple relation between the logarithmic derivative and the spectra function, see the example of the Stark effect, enclosed.  

It is interesting how the Riccati equation solves the optimal control problem in the control theory for a time variable linear system (Kwakernaak and Sivan) according to the theory of Kalman.

The variational techique which solves the optimal control problem finding the optimal path for the variable of the time variable system is very similar to the path integral approach of Feynman. The propagators and the way the matrixes are combined seems to  have a close link to Feynman's approach.

Dear Profs. Brandas and Quatrinni,

                                                                I would like to draw your attention to a Econophysical piece of research we had undertaken on Asset Management of Banks and its ability to String Theoretically integrate Banking Systems across the world and improve upon the Ba Loan portfolio situations and adhere to the Basle Norms more strictly. Incidentally, this looks very much like Nuclear Engineering or a method of solving the Classical Alfred Nobel problem of harnessing mechanical energy of dynamites. It is uses the properties of Econophysics Equilibrium actions on Banking Sector Asset Market Systems and experimentally solves the Ricatti Equations, which are hidden, by converting into linear resultants by normalising the market potential forces of Banking Sector Industries. It uses diagrams similar to Feinman diagramms and Quantum Scaling and Computer Learning Algorithms which makes it partly genetic. If you care to take alook at my HAAG theorem extension paper to Stock Markets and Asset Markets on my RG page it will become clear. All actions and String Actions are in Space X Time demensions and Entropy is Controlled Naturally and Systemically by learning. Ever since this research was published 15 years ago in the first stage the Banking Sector in India has grown to almost  four times, interest rate controls have come down by over 30% but the NPA ratios have gone down four ten years by about 50% and then have started increasing but not so far by more than 10 % over the original figures. Earl Prof. (Sir Ashutosh Mukherjee University Institute Chair Professor) Dr. (D.Phil.) SKM QC EPS Fellow (Indirect) MES MRES MAICTE

Please, keep in mind that Ricatti equations are being used in the theory of exactly integrable systems  https://books.google.com/books?id=sL0p1CoVJC0C&pg=PA282&lpg=PA282&dq=ricatti+equation+and+exacrtly+integrable+systems&source=bl&ots=YNHExPdGcN&sig=4ddkD09AIyvzay25FCPdORBHD6w&hl=en&sa=X&ved=0ahUKEwjt1eLS0IfLAhWFth4KHdUhBBoQ6AEIUzAF#v=onepage&q=ricatti%20equation%20and%20exacrtly%20integrable%20systems&f=false    Accordingly, all these systems admit easy quantization .Complex nonlinear Ricatti is still Ricatti equation

Dear all,

Riccati is mathematician  who lived in Italy, Ricatti I don't know who he is.

Dear Prof. Quattrini and others,

                                                        Sorry for mispelling the Riccati equation name, just an oversight. I stand corrected. SKM