We identify a class of potentials for which the semiclassical estimate $N^{\text{(semi)}}=\frac{1}π\int_0^\infty dr\sqrt{-V(r)θ[-V(r)]}$ of the number $N$ of (S-wave) bound states provides a (rigorous) lower limit: $N\ge {N^{\text{(semi)}}}$, where the double braces denote the integer part. Higher partial waves can be included via the standard replacement of the potential $V(r)$ with the effective $\ell$-wave potential $V_\ell^{\text{(eff)}}(r)=V(r)+\frac{\ell(\ell+1)}{r^2}$. An analogous upper limit is also provided for a different class of potentials, which is however quite severely restricted.
Pattern formation in terms of semiclassically limited distribution on lower-dimensional manifolds for nonlocal Fisher--Kolmogorov--Petrovskii--Piskunov equation
We define mixed states associated with submanifolds with probability densities in quantizable closed K{ä}hler manifolds. Then, we address the problem of comparing two such states via their fidelity. Firstly, we estimate the sub-fidelity and super-fidelity of two such states, giving lower and upper bounds for their fidelity, when the underlying submanifolds are two Lagrangian submanifolds intersecting transversally at a finite number of points, in the semiclassical limit. Secondly, we investigate a family of examples on the sphere, for which we manage to obtain a better upper bound for the fide…
Finite-gap potentials as a semiclassical limit of the thermodynamic Bethe Ansatz
KTH Royal Institute of Technology（皇家理工学院）；University of Chicago（芝加哥大学）；Nordita, KTH Royal Institute of Technology and Stockholm University（诺迪塔，皇家理工学院和斯德哥尔摩大学）；Copenhagen University（哥本哈根大学）
Consider the action of a connected complex reductive group on a finite-dimensional vector space. A fundamental result in invariant theory states that the orbit closure of a vector v is separated from the origin if and only if some homogeneous invariant polynomial is nonzero on v, i.e. v is not in the null cone. Thus, efficiently finding the minimum distance between the orbit closure and the origin can lead to deterministic algorithms for null cone membership, an important polynomial identity testing problem including the non-commutative Edmonds problem. This connection to optimization has rece…
Numerical computation of the density of states of aperiodic multiscale Schrödinger operators
École des Ponts（巴黎路桥学校）；Institut Polytechnique de Paris（巴黎理工学院）；Inria（法国国家数字科学研究所）；New Jersey Institute of Technology（新泽西理工学院）；Zhejiang University（浙江大学）；Université Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CEA Grenoble（法国原子能和替代能源委员会格勒诺布尔中心）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
研究非周期多尺度薛定谔算子态密度的数值计算方法，比较动量空间分解和半经典展开两种方法，通过一维玩具模型检验，发现小 $\epsilon$ 时二者吻合好，不太小时有差异，差异源于范霍夫奇点半经典类似处的 DoS 振荡。
In directed last passage site percolation with i.i.d.~random weights with finite support over a $n\times\lfloor n^α\rfloor$ grid, we prove that for $n$ large enough, the order of the $r$-th central moment, $1\le r<+\infty$, of the last passage time is lower bounded by $n^{r(1-α)/2}$, $0<α<1/3$.
We study the cut-off resolvent of semiclassical Schr{ö}dinger operators on $\mathbb{R}^d$ with bounded compactly supported potentials $V$. We prove that for real energies $λ^2$ in a compact interval in $\mathbb{R}_+$ and for any smooth cut-off function $χ$ supported in a ball near the support of the potential $V$, for some constant $C>0$, one has \begin{equation*} \| χ(-h^2Δ+ V-λ^2)^{-1} χ\|_{L^2\to H^1} \leq C \,\mathrm{e}^{Ch^{-4/3}\log \frac{1}{h} }. \end{equation*} This bound shows in particular an upper bound on the imaginary parts of the resonances $λ$, defined as a pole of the meromorph…
