Christoffel-Darboux type identity

The classical Christoffel-Darboux identity for Hermite polynomials reads n∑k=0Hk(x)Hk(y)2kk!=12n+1n!Hn+1(x)Hn(y)−Hn(x)Hn+1(y)x−y. I am interested in a similar identity, where one of the indices is shifted by one, explicitly Fn(x,y):=n∑k=0Hk+1(x)Hk(y)2k+1(k+1)!=xn∑k=01k+1Hk(x)Hk(y)2kk!−n∑k=0kk+1Hk−1(x)Hk(y)2kk!, where the recursion relation Hk+1(x)=2xHk(x)−2kHk−1(x) was used. In the first term Christoffel-Darboux cannot be applied due to the 1/(k+1) prefactor and the second term is not quite Fn−1(y,x) … Read more

Dynamics of pairwise distances in the $n$-body problem

Disclaimer: I have asked this question on Physics SE a week ago, but got no answers. I know that some MO users are interested in the $n$-body problem, so I decided to cross post here as well. Consider the $n$-body problem where we are interested in describing the time evolution of $n$ masses interacting through … Read more

Why do we care about simplicity of the spectrum in Oseledets’ theorem?

Oseledets’ theorem is a fundamental result in Ergodic theory (see for example here, or Chapter 4 of Lectures on Lyapunov Exponents by Marcelo Viana). The simplicity of the spectrum has been studied by Guivarc’h and Raugi ’86, Gol’dsheid and Margulis ’89, Bonatti-Viana ’04, Avila-Viana ’07 and very recently by Mauricio Poletti. My* questions are: What … Read more

KAM stable orbits are smooth

I’m in my final year of my undergraduate studies doing work on modelling the n-body problem numerically and I also have some interest in theoretical guarantees. Now, I’ve been looking for a theorem that guarantees the following: KAM stable⟹smooth orbits To be precise I mean that if we decompose a KAM stable solution to the … Read more

How to find a geometric construction of a TQFT

Assume that the surface ∑ is equipped with the structure of a smooth algebraic curve over C. We denote by H0(M∑,L⊗k) the space of sections of L⊗k on M∑, where M∑ is the moduli space of semi-stable rank N bundles with trivial determinant over ∑ , and L is the determinant line bundle on M∑. … Read more

Estimate the composition of a bounded multiplier with a trace class operator

Let $T$ be a trace class operator on $\ell^2 (\mathbb{N})$. Let $A$ be a multiplier on $\ell^2 (\mathbb{N})$ defined by a sequence $a=(a_n)_{n\in\mathbb{N}}$ in $\ell^{\infty} (\mathbb{N})$. That is, for $u=(u_n)_{n\in\mathbb{N}}$ in $\ell^{2} (\mathbb{N})$, $(Au)_n=a_nu_n$ for all $n$ in $\mathbb{N}$. I would like to control the Schatten 1-norm of the compositions $AT$ and $TA$. I know … Read more

Asymptotic expansion of a Gaussian integral and heat kernel

When considering the heat kernel of a Schr\”odinger operator −Δ+V(x) where Δ is the standard Laplacian on Rn and V is a nonnegative potential function that has nice behavior at infinity (proper, grows polynomially), one usually sees the term e−tV(x) and the asymptotic expansion of the Gaussian integral ∫Rne−tV(x), t→0. If V(x) is homogeneous, namely, V(rx)=rαV(x), … Read more

What is known about the projective representations of $\mathrm{SO}(n_1,n_2)$?

He${}$llo MO. Let $\mathrm{O}(n_1,n_2)$ be the pseudo-orthogonal group. I am interested in its (continuous, not necessarily unitary, finite-dimensional) irreducible projective representations, for arbitrary $n_1,n_2$ (or, at least, the special cases $n_1=0,1$). I am having a hard time to find good characterisations thereof, or even a complete classification (some particular representations are discussed in some physics … Read more

Topological field theories and their path integrals

Examples of topological field theories are the cohomological field theories as they were initially defined by Witten [1]. Such examples include the Donaldson-Witten theory in 4d or the Gromo-Witten theory in 2d. These theories contain a bunch of fields and usually one wants to consider the partition function $$ Z_{CohTQFT} := \int DX ~ e^{-S_{CohTQFT}[X]} … Read more

Langlands dual and integrable representations

Assume I successfully classified the integrable representations of a certain semi-simple Lie group $G$. Given this information, what do I know about the integrable representations of $G^\vee$, the Langlans dual of $G$? More generally, is there any direct information I can learn about $G^\vee$ given the integrable reps. of $G$? Or is there no straightforward … Read more