The n-Category Café

Typo: “Ricatti” instead of “Riccati” in the first three instances.

The first physics topic where I’d seen Riccati equations invoked was supersymmetric quantum mechanics. In this subject, one studies a Hamiltonian like $$H = \frac{p^2}{2m} + V(x) = -\frac{\hbar^2}{2m} \partial_x^2 + V(x)$$ by believing really hard that it can be treated like the simple harmonic oscillator. Because if we could factor $H$ into the product of an operator $A$ and its adjoint, then we could find the ground state of $H$ by figuring out what $A$ annihilates, thereby turning a second-order differential equation into a first-order one. Moreover, if $H_1 = A^\dagger A$, then its partner Hamiltonian $H_2 = A A^\dagger$ will be isospectral with $H_1$, except for a zero-energy ground state. In cases of interest, it turns out that $H_2$ has the same functional form as $H_1$ up to some parameter adjustments, so we can iterate this trick, building up the spectrum and the eigenstates of a physically meaningful Hamiltonian. This even works for the hydrogen atom, once you factor apart the radial and angular dependences.

If we want to split a Hamiltonian into an operator and its adjoint, we should probably start with an operator which is linear in the derivative of $x$, thus: $$A = \frac{\hbar}{\sqrt{2m}} \partial_x + W(x),$$ where $W(x)$ is called the superpotential. Taking the adjoint of $A$ flips the sign on the derivative (recall that $p$ must be self-adjoint): $$A^\dagger = -\frac{\hbar}{\sqrt{2m}} \partial_x + W(x).$$ The partner potentials $V_1(x)$ and $V_2(x)$ are, in terms of the superpotential, $$V_1(x) = W^2(x) - \frac{\hbar}{\sqrt{2m}} \partial_x W(x)$$ and $$V_2(x) = W^2(x) + \frac{\hbar}{\sqrt{2m}} \partial_x W(x).$$ We have the first derivative of $W$ depending on $W^2$ and another function of $x$, which means that these are Riccati equations without the linear term.

I think the answer to Question 1 is that this theory must exist, at least in Platonic heaven — so I just want to know if it’s made its way down here to Earth.

Here’s how the theory starts:

If $M = G/H$ then we get a map from the Lie algebra of $G$ to vector fields on $M$:

$$V \colon \mathfrak{g} \to Vect(M)$$

in fact a Lie algebra homomorphism. Then, given any smooth function

$$f \colon \mathbb{R} \to \mathfrak{g}$$

we get a differential equation describing the motion of a point $y(t)$ on $M$:

$$\frac{d}{d t} y(t) = V(f(t))_{y(t)}$$

If $M$ is compact this will have solutions for all smooth $f$ and all initial data $y(0) \in M$, and the solutions will be of the form

$$y(t) = T(t) y(0)$$

where

$$T \colon \mathbb{R} \to G$$

is some smooth function obeying

$$T(0) = 1 \in G,$$

$$\frac{d}{d t} T(t) = f(t) T(t)$$

The space $C^\infty(\mathbb{R},\mathfrak{g})$ of smooth functions $f \colon \mathbb{R} \to \mathfrak{g}$ serves as a Lie algebra for the infinite-dimensional Lie group $C^\infty(\mathbb{R},G)$, so this group has an action — the adjoint action — on $C^\infty(\mathbb{R},\mathfrak{g})$. Thus, this group acts on the space of ‘generalized Riccati equations’

$$\frac{d}{d t} v(t) = V(f(t))_{y(t)}$$

where $f \in C^\infty(\mathbb{R},\mathfrak{g})$.

Note also that all the solutions of a generalized Riccati equation are encapsulated in a function $T \in C^\infty(\mathbb{R}, G)$.

Someone must have investigated this theory.

On Twitter, Francis has a nice remark about the connection between Riccati equations and 2nd-order equations:

The way I understand this is that we are projectivizing a second order linear differential equation. Namely, if $u$ satisfies a 2nd order ODE, then $y=u/u'$ satisfies Riccati. We could therefore view $y$ as the point $[u : u']$ in projective space.

He means a 2nd-order linear ODE. All this makes tons of sense to me now — it’s really nice!

If you have a 2nd-order linear ODE with smooth coefficients it has a 2d vector space of solutions $u$. For each $t \in \mathbb{R}$, the time evolution map

$$T(t) : (u(0), u'(0)) \mapsto (u(t),u'(t))$$

is a linear map from $\mathbb{C}^2$ to itself. Projectivizing we get a smooth one-parameter family of projective linear maps

$$P T(t) :\mathbb{C}\mathrm{P}^1 \to \mathbb{C}\mathrm{P}^1$$

But this precisely a Ricatti equation!

But remember, given a point $[a : b] \in \mathbb{C}\mathrm{P}^1$ we can treat it as a point in $\mathbb{C}$, namely $a/b$, as long as $b \ne 0$. So, in low-brow terms, the ratio $u(t)/u'(t)$ obeys a Ricatti equation.

The converse works just as easily.

As a first step, the relations among a Riccati equation with constants multiplying the sl(2) diff op generators, the Schwarzian derivative, and the KdV equation are informative and rife with special number arrays in combinatorics. Look at g(h(x)) in my answer to the MO-Q “Is there an underlying explanation for the magical powers of the Schwarzian derivative?”

A natural extension of the Riccati equation is the general Lie infinitesimal generator/vector related to the inviscid Burgers-Hopf equation and solutions of autonomous ODEs for evolution/flow equations. See the MO-Q “Why is there a connection between enumerative geometry and nonlinear waves?”

On the Bernoullis and an early (maybe first) Riccati equation, see the MO-Q (and linked MSE-Q) “Link of a power series by the Bernoullis for a Riccati equation to zonotopes?”

Carinena and Lucas have written extensively on this topic (see arXiv). Their paper “Lie systems: theory, generalizations, and applications” has a nice summary of the history of these ideas. Lucas et al. even introduce quaternionic and octonionic Riccati equations in “Geometry of Riccati equations over normed division algebras.” (I just revisited these ideas after updating my MO-Q “Geometric picture of invariant differential of an elliptic curve” with some remarks on the constant coefficient Riccati equation and relations to formal groups.)

The extension of the sl(2) algebra connections of the constant coefficient Riccati equation is the Witt-Virasoro algebra, related to some interesting combinatorics of polytopes and rooted trees (see, e.g., MO-Q “Motivation of Virasoro algebra” and “Important formulas in combinatorics” as well as the MO-Qs noted above.)