Weakly resonant tunneling interactions for adiabatic quasi-periodic Schrodinger operators

The periodic magnetic Schrödinger operators: spectral gaps and tunneling effect

A periodic Schrödinger operator on a noncompact Riemannian manifold $M$ such that $H^1(M, \mathbb R)=0$ endowed with a properly discontinuous cocompact isometric action of a discrete group is considered. Under some additional conditions on the magnetic field existence of an arbitrary large number of gaps in the spectrum of such an operator in the semiclassical limit is established. The proofs are based on the study of the tunneling effect in the corresponding quantum system.

Landau-Zener tunneling for dephasing Lindblad evolutions

We derive an analog of the Landau-Zener adiabatic tunneling formula for an open, two-level system coupled to a memoryless, dephasing bath. The derivation rests on a geometric view of the spectral subspaces as adiabatic invariants.

关于薛定谔方程解在未达绝热极限时的稳定性

On the stability of solutions to Schrödinger's equation short of the adiabatic limit

本文证明了一个适用于短于绝热极限时间尺度的绝热定理，通过直接刻画扰动的跨子空间效应，揭示了子空间动力学可主导全局动力学，从而解释了量子退火中比标准绝热定理更快的收敛现象。

Tunneling effects between tori in double wells

We consider tunneling between 2 symmetric potential wells for a 2-d Schrodinger operator, in the case of eigenvalues associated with quasi-modes supported on KAM or Birkhoff tori.

The adiabatic limit of Schrödinger operators on fibre bundles

We consider Schrödinger operators $H=-Δ_{g_\varepsilon} + V$ on a fibre bundle $M\stackrelπ{\to}B$ with compact fibres and a metric $g_\varepsilon$ that blows up directions perpendicular to the fibres by a factor ${\varepsilon^{-1}\gg 1}$. We show that for an eigenvalue $λ$ of the fibre-wise part of $H$, satisfying a local gap condition, and every $N\in \mathbb{N}$ there exists a subspace of $L^2(M)$ that is invariant under $H$ up to errors of order $\varepsilon^{N+1}$. The dynamical and spectral features of $H$ on this subspace can be described by an effective operator on the fibre-wise $λ$-e…

Optimal parametrizations of adiabatic paths

The parametrization of adiabatic paths is optimal when tunneling is minimized. Hamiltonian evolutions do not have unique optimizers. However, dephasing Lindblad evolutions do. The optimizers are simply characterized by an Euler-Lagrange equation and have a constant tunneling rate along the path irrespective of the gap. Application to quantum search algorithms recovers the Grover result for appropriate scaling of the dephasing. Dephasing rates that beat Grover imply hidden resources in Lindblad operators.

半经典隧道效应对于某些具有复值势的1D薛定谔算子

Semiclassical tunneling for some 1D Schrödinger operators with complex-valued potentials

研究半经典薛定谔算子在复值势下的隧道效应，发现谱中存在指数接近的特征值对，且非零α时最小特征值对快速旋转。

The Tunneling Effect for Schrödinger operators on a Vector Bundle

In the semiclassical limit h to 0, we analyze a class of self-adjoint Schrödinger operators H_h = h^2 L + h W + V id_E acting on sections of a vector bundle E over an oriented Riemannian manifold M where L is a Laplace type operator, W is an endomorphism field and the potential energy V has non-degenerate minima at a finite number of points m_1, ... m_r in M, called potential wells. Using quasimodes of WKB-type near m_j for eigenfunctions associated with the low lying eigenvalues of H_h, we analyze the tunneling effect, i.e. the splitting between low lying eigenvalues, which e.g. arises in cer…

On the eigenvalues for slowly varying perturbations of a periodic Schrödinger operator

In this paper, I consider one-dimensional periodic Schr{ö}dinger operators perturbed by a slowly decaying potential. In the adiabatic limit, I give an asymptotic expansion of the eigenvalues in the gaps of the periodic operator. When one slides the perturbation along the periodic potential, these eigenvalues oscillate. I compute the exponentially small amplitude of the oscillations.

Resonant tunneling in a system with correlated pure point spectrum

We consider resonant tunneling between disorder localized states in a potential energy displaying perfect correlations over large distances. The phenomenon described here may be of relevance to models exhibiting many-body localization. Furthermore, in the context of single particle operators, our examples demonstrate that exponential resolvent localization does not imply exponential dynamical localization for random Schrödinger operators with correlated potentials.

对数凹性与隧穿：凸函数（带尖峰）的绝热量子优化

Log-concavity and tunneling: adiabatic quantum optimization for convex functions (with a spike)

本文扩展了绝热量子优化中隧穿效应的分析，建立离散对数凹性作为基态的关键结构性质，并应用于凸势及带局部极小值的势，证明了新的谱隙界，并将尖峰汉明重量问题的微扰分析推广到更广泛的势族。

Inverse tunneling estimates and applications to the study of spectral statistics of random operators on the real line

We present a proof of Minami type estimates for one dimensional random Schrödinger operators valid at all energies in the localization regime provided a Wegner estimate is known to hold. The Minami type estimates are then applied to various models to obtain results on their spectral statistics. The heuristics underlying our proof of Minami type estimates is that close by eigenvalues of a one-dimensional Schrödinger operator correspond either to eigenfunctions that live far away from each other in space or they come from some tunneling phenomena. In the second case, one can undo the tunneling a…

均匀磁场中径向电势阱之间的隧穿效应

Tunneling effect between radial electric wells in a homogeneous magnetic field

本文研究二维均匀磁场中对称径向紧支双势阱 Schrödinger 算子的隧穿现象，给出前两个特征值之差在半经典极限下指数小的渐近公式。

非零边界条件下F=1旋量非线性薛定谔方程的绝热微扰理论

Adiabatic perturbation theory for the $F=1$ spinor nonlinear Schrödinger equation with nonvanishing boundary conditions

本文针对非零边界条件下的F=1旋量非线性薛定谔方程，在黎曼-希尔伯特问题框架内建立绝热微扰理论，推导单孤子参数演化的有限维动力系统，边界条件消失时可简化为已有结果。