Conjectured regimes for the decay of the smallest nonzero eigenvalues of Toeplitz operators

Establish the three asymptotic regimes for the smallest nonzero eigenvalues of the squared Toeplitz operators T^2_{f,p} on a compact Hermitian complex manifold with a positive line bundle, depending on the vanishing order \kappa(f) and whether the support of f is full: prove that min Spec^*(T^2_{f,p}) behaves like p^{-\kappa(f)} if \kappa(f)<\infty; like exp(−c_f\sqrt{p}) if \kappa(f)=\infty and supp f=X; and like exp(−c_f p) if \kappa(f)=\infty and X\setminus supp f\neq\varnothing; and characterize the corresponding eigensections.

Background

Motivated by explicit computations on \mathbb{CP}1, the paper exhibits three distinct asymptotic behaviors for minimal Toeplitz eigenvalues: polynomial decay p{−k} for finite vanishing order, stretched-exponential e{−2\sqrt{p}} when the vanishing order is infinite but the support is full, and exponential e{−c p} when the support is not full. These examples suggest a general classification.

The authors present partial upper and lower bounds (e.g., polynomial upper bounds for finite vanishing order; lower bounds of the form e{−c p} when the symbol is supported on a proper subset) and note that a complete answer remains open. They therefore formulate the following conjecture/question describing the expected regimes and ask to also describe the structure of the corresponding eigensections.

References

Then the general results about the lowest eigenvalues are expected, so that we put it as the following conjecture/question: Let $(X,J,\Theta)$ be a connected, compact complex Hermitian manifold and let $(L,h_{L})$, $(E,h_E)$ be holomorphic line bundles on $X$ with smooth Hermitian metrics. Assume $h_L$ to be positive. For $f\in\mathscr{C}\infty(X,R)$ (it can take negative values), set $\kappa(f):=\sup_{x\in X}\mathrm{ord}x(f)\in N\cup{+\infty}$ (where $\mathrm{ord}_x(f)$ denotes the vanishing order of $f$ at $x$), show that as $p\rightarrow +\infty$ \begin{equation} \min \mathrm{Spec}*(T2{f,p})=\begin{cases} \simeq p{-\kappa(f)} & ,\text{\, if \,} \kappa(f)<+\infty; \ \simeq e{-c_f \sqrt{p}}& , \text{\, if \,} \kappa(f)=+\infty \text{\,and\,} \supp{f}=X \ \simeq e{-c_f p} &, \text{\, if \,} \kappa(f)=+\infty \text{\,and\,} X\setminus \supp{f} \neq\varnothing \end{cases} \label{eq:5.18Jan} \end{equation} where $c_f>0$ is some constant depending on $f$, and $\mathrm{Spec}*(T2_{f,p}):=\mathrm{Spec}(T2_{f,p})\setminus{0}$, the sign $\simeq$ means up to a multiplication of some nonzero constant. Moreover, a related interesting question is to describe the corresponding eigensections on $X$. Note that in Section \ref{ss5.3Jan}, we give the examples on $\mathbb{CP}1$ where the lowest eigenvalues of $T2_{f,p}$ are the cases listed in eq:5.18Jan. Following the results on the off-diagonal decay of $P_p(x,y)$ given in , it is possible that we need to assume the analyticity on $\Theta$ and $h_L$, $h_E$ to get the three nice cases in eq:5.18Jan.

— Toeplitz operators and zeros of square-integrable random holomorphic sections  (2404.15983 - Drewitz et al., 2024) in Question (Section 5.2: Random zeros and lowest Toeplitz eigenvalues on compact manifolds)