Study magnetic Schrödinger operators in Euclidean space.
problem Semiclassical spectral analysis of magnetic Schrödinger operators.
method Spectral problems for Bochner-Schrödinger operator on manifolds.
result Survey and describe ideas of proofs for magnetic Schrödinger operators.
Estimates small eigenvalues for geometrically finite manifolds.
problem Estimating small eigenvalues of Schrödinger operators.
method Geometrically finite manifolds, Riemannian vector bundles.
result Estimates the number of small eigenvalues.
Sharp Gaussian bounds derived for Schrödinger kernel on Ricci solitons.
problem Analyzing Schrödinger heat kernel on gradient shrinking Ricci solitons.
method Deriving sharp Gaussian upper bounds for the Schrödinger heat kernel.
result Sharp upper and lower bounds for eigenvalues of the Schrödinger operator.
Paper shows how scattering maps of Schrödinger equations relate to metrics.
problem Relating scattering maps of time-dependent Schrödinger equations to metrics.
method Analyzes scattering maps for specific classes of metrics and diffeomorphisms.
result Scattering maps differ by a compact operator if and only if metrics are related by diffeomorphism.
Study shows stability of Schrödinger operator spectral data on a manifold.
problem Determining a manifold and potential function from spectral data.
method Approximation of spectral data on a subset to determine manifold and potential.
result Quantitative stability estimate for Schrödinger operator inverse problem.
The paper proves positivity preservation and self-adjointness for Schrödinger operators on incomplete Riemannian manifolds.
problem Positivity preservation and self-adjointness for Schrödinger-type operators on incomplete Riemannian manifolds.
method Control of potential behavior near the Cauchy boundary, essential self-adjointness proof, core of smooth compactly supported functions.
result Positivity preservation and essential self-adjointness of Schrödinger operators on Lp functions on incomplete Riemannian manifolds. Estimates the first eigenvalue of a Schrödinger operator on minimal submanifolds.
problem Estimating the first eigenvalue of a Schrödinger operator on minimal submanifolds.
method Analyzes the Schrödinger operator L:=−Δ−σ on minimal submanifolds Mn in the unit sphere Sn+m. result Provides an estimate for the first eigenvalue of the Schrödinger operator.
Eigenvalue bounds for Schrödinger operators on Ricci shrinkers and related manifolds.
problem Estimating eigenvalues of Schrödinger operators on Ricci shrinkers and related manifolds.
method Using Ricci shrinkers and Perelman's μ-functional, the paper derives lower bounds for the lowest eigenvalues of Schrödinger operators.
result Lower bounds for the lowest eigenvalues of Schrödinger operators on Ricci shrinkers and related manifolds, with equality conditions characterized.
Novel heat flow estimates on ALE manifolds for Schrödinger operators.
problem Estimating heat flows on ALE manifolds with non-trivial L2-kernel. method Combining Fredholm theory for Dirac type operators and heat kernel advances.
result Established Lp−Lq decay estimates for heat flows. Based on the work of Schoen-Yau, we derive an estimate of the first eigenvalue of a Schrödinger Operator (the Jaocbi operator of minimal surfaces in flat 3-spaces) on surfaces.
We consider non-self-adjoint Schrödinger operators Δ+V where Δ is the Laplace-Beltrami operator on a Zoll manifold X and V∈C∞(X,C). We obtain asymptotic results on the pseudo-spectrum and numerical range of such operators.
Exponential localization of eigensections for Bochner-Schrödinger operator.
problem Understanding spectral properties of Bochner-Schrödinger operator on high tensor powers of Hermitian line bundles.
method Approximation of operator by model Schrödinger operator with constant magnetic field, analysis of spectrum.
result Spectrum of Bochner-Schrödinger operator in gaps is discrete and eigensections decay exponentially.
We prove a lower bound for the number of negative eigenvalues for a Schrödinger operator on a Riemannian manifold via the integral of the potential.
The paper studies asymptotic expansions of operators related to Bochner-Schrödinger on Riemannian manifolds.
problem Asymptotic expansions of operators related to Bochner-Schrödinger on Riemannian manifolds.
method Analyzes the Bochner-Schrödinger operator Hp and its function φ(Hp) in L2(X,Lp⊗E), providing an asymptotic expansion of its smooth Schwartz kernel. result The trace of the operator φ(Hp) admits a complete asymptotic expansion in powers of p−1/2 as po∞. The lowest eigenvalue of the Schrödinger operator −Δ+V on a compact Riemannian manifold without boundary is studied. We focus on the particularly subtle case of a sign changing potential with positive average.
The main contribution of our paper is to give a partial classification of the quasi-exactly solvable Lie algebras of first order differential operators in three variables, and to show how this can be applied to the construction of new quasi-exactly solvable Schrödinger operators in three dimensions.
Study the Bochner-Schrödinger operator's trace in semiclassical limit.
problem Trace formula for Bochner-Schrödinger operator on tensor powers of line and vector bundles.
method Semiclassical analysis of the Bochner-Schrödinger operator Hp on tensor powers of a Hermitian line bundle and vector bundle. result Complete asymptotic expansion of the trace of φ(Hp) in the semiclassical limit po∞. Study the Bochner-Schrödinger operator on symplectic manifolds, proving gap existence and asymptotic kernel behavior.
problem Analyzing the spectrum and asymptotic behavior of the Bochner-Schrödinger operator on symplectic manifolds.
method Rough asymptotic description, existence proof, off-diagonal exponential estimate, complete asymptotic expansion.
result Existence of gaps in the spectrum and asymptotic kernel behavior.
Estimates for Schrödinger operators on manifolds with bounded Ricci curvature.
problem Quantifying unique continuation for Schrödinger operators on manifolds with specific curvature conditions.
method Proving quantitative unique continuation estimates for Schrödinger operators on manifolds with Ricci curvature bounded below.
result Upper bound for energy range and constant in terms of Ricci curvature and parameters of relatively dense set.
Researchers prove Weyl laws for Schrödinger operators on noncompact manifolds.
problem Proving Weyl laws for Schrödinger operators on noncompact manifolds.
method Heat kernel asymptotics, Karamata-Hardy-Littlewood Tauberian theorem, and semiclassical analysis.
result Established both classical and semiclassical Weyl laws for Schrödinger operators on noncompact manifolds.
We describe two-dimensional potential Schrodinger and Dirac operators which are finite-gap at one energy level and have singular spectral curves. It appears that the singularities can be rather complicated. Such Dirac operators appear as the spectral curves of tori immersed into the three-space.
Kernel method approximates dynamical operators from data.
problem Estimating eigenfunctions of dynamical operators from data.
method Kernel-based approach in reproducing kernel Hilbert spaces.
result Eigenfunctions estimated via matrix eigenvalue problems.
New method preserves unitarity for Schrödinger equation learning, reducing errors and improving time generalization.
problem Learning the evolution operator for time-dependent Schrödinger equation with varying Hamiltonians.
method Linear estimator preserving weak unitarity, with theoretical error bounds and time generalization.
result Achieves up to two orders of magnitude smaller relative errors than existing methods.
The paper studies eigenvalues in gaps of the essential spectrum of a Bochner-Schrödinger operator.
problem Eigenvalue distribution in gaps of the essential spectrum of the Bochner-Schrödinger operator.
method Trace asymptotics formula and Weyl type asymptotic formula for eigenvalue counting function.
result The spectrum of Hp in the gap is discrete. We review recent probabilistic results on covariant Schrödinger operators on vector bundles over (possibly locally infinite) weighted graphs, and explain applications like semiclassical limits. We also clarify the relationship between these results and their formal analogues on smooth (possibly noncompact) Riemannian m…
We give a lower estimate of the gap of the first two eigenvalues of the Schrodinger operator with a nonconvex potential in terms of a distance associated with the potential. The results here can be applied to the double well potential.
We establish various Lp estimates for the Schrödinger operator −Δ+V on Riemannian manifolds satisfying the doubling property and a Poincaré inequality, where Δ is the Laplace-Beltrami operator and V belongs to a reverse Hölder class. At the end of this paper we apply our result on Lie groups with polynomial …
A new diffusion method approximates Schrödinger bridge with improved convergence.
problem Approximating Schrödinger bridge with Langevin diffusion.
method Leveraging Langevin diffusion to approximate Schrödinger bridge.
result The difference between the two approximations is proportional to the score function.
This note is devoted to optimal spectral estimates for Schrödinger operators on compact connected Riemannian manifolds without boundary. These estimates are based on the use of appropriate interpolation inequalities and on some recent rigidity results for nonlinear elliptic equations on those manifolds.
Study bottom of spectra on orbifolds via coverings.
problem Behavior of bottom of spectra under orbifold coverings.
method Analysis of scalar Schrödinger operators on orbifolds.
result Results apply to geometrically finite and conformally compact orbifolds.
Study eta invariant on non-compact manifolds with positive scalar curvature.
problem Proving geometric formulas and index theorems for uniformly positive scalar curvature metrics.
method Using Dirac-Schrödinger operators and relative eta invariant.
result New geometric formula for spectral flow and index formula for uniformly positive scalar curvature metrics.
We extend the Feynman-Kac formula for Schrödinger type operators on vector bundles over noncompact Riemannian manifolds to possibly very singular potentials that appear in hydrogen like quantum mechanical problems and that need not be bounded from below or locally square integrable. This path integral formula is then u…
We give a sharp upper bound on the vanishing order of solutions to Schrödinger equation, in the case that the potential is of class C1 on a smooth compact manifold.
New method stabilizes quantum ergodicity for mixed quantization and partial hyperbolicity.
problem Stabilizing quantum ergodicity for complex systems.
method Combines mixed quantization techniques with stable ergodicity results for partially hyperbolic systems.
result Establishes stable quantum ergodicity for spin Hamiltonians.
This monograph develops the theory of covariant Schrödinger semigroups acting on sections of vector bundles over noncompact Riemannian manifolds from scratch. Contents: I. Sobolev spaces on vector bundles II. Smooth heat kernels on vector bundles III. Basis differential operators in Riemannian manifolds IV. Some specif…
This paper deals with eigenvalue optimization problems for a family of natural Schrödinger operators arising in some geometrical or physical contexts. These operators, whose potentials are quadratic in curvature, are considered on closed surfaces immersed in space forms and we look for geometries that maximize the eige…
We show that for a Schrödinger operator with bounded potential on a manifold with cylindrical ends the space of solutions which grows at most exponentially at infinity is finite dimensional and, for a dense set of potentials (or, equivalently for a surface, for a fixed potential and a dense set of metrics), the constan…
We consider a magnetic Schrödinger operator Hh, depending on a semiclassical parameter h>0, on a compact Riemannian manifold. We assume that there is no electric field. We suppose that the minimal value b0 of the intensity of the magnetic field b is strictly positive. We give a survey of the results on asympt…
Analyzes tunneling effects for Schrödinger operators on vector bundles.
problem Tunneling effects in quantum systems with multiple potential wells.
method Quasimodes and WKB analysis near potential wells, interaction matrix for coupling between wells.
result Polynomial prefactor for exponentially small eigenvalue splitting determined by dimension of minimal geodesics.
We propose a twistor construction of surfaces in Lie sphere geometry based on the linear system which copies equations of Wilczynski's projective frame. In the particular case of Lie-applicable surfaces this linear system describes joint eigenfunctions of a pair of commuting Schrödinger operators with magnetic fields.
New theorem shows gaps in magnetic Schrödinger operator spectra for large coupling.
problem Understanding gaps in spectra of magnetic Schrödinger operators.
method Analyzes spectral properties of non-periodic magnetic Schrödinger operators.
result Spectral projections of large coupling operators vanish in K-theory.
The study establishes bounds for Schrödinger operators on Riemannian manifolds.
problem Bounding Schrödinger operators on Riemannian manifolds.
method Utilizes weighted manifolds and Faber-Krahn inequalities to derive bounds.
result Establishes conditions for Schrödinger operators to be positive and for their spectra.
New connections found between curvature and Euler characteristic using Schrödinger operators.
problem Establishing relationships between curvature and topological invariants of Riemannian manifolds.
method Using twisted Dirac operators and scaling of potentials to analyze the kernel of these operators.
result Found conditions under which the Euler characteristic of a manifold can be zero or non-zero.
Study regularity of Schrödinger eigenfunctions with Coulomb-type potentials.
problem Regularity of eigenfunctions for Schrödinger operators with singular potentials.
method Blow-ups of manifolds with corners and Lie manifolds.
result Proves regularity estimates in weighted Sobolev spaces for eigenfunctions.
Study invariant operators and vanishing theorems in CR geometry.
problem Analyzing invariant operators and vanishing theorems in CR geometry.
method Investigates Kohn-Dirac operators and derives CR invariant twistor operators.
result Proves vanishing theorems for harmonic spinors and Kohn-Rossi groups.
Study on catenoid stability using asymmetric potentials.
problem Stability of catenoids and related potentials.
method Exact solvable one-dimensional Schrödinger operator with asymmetric Darboux-Pöschl-Teller potential.
result Explicit construction of the unstable mode of the inner catenoid.
Study non-integer power-law potentials for Schrödinger operators using Lie-Rinehart algebras.
problem Analyzing Schrödinger operators with non-integer power-law potentials.
method Using Lie-Rinehart algebras and microlocal analysis.
result Microlocal analysis can be applied to Schrödinger operators with non-integer power-law potentials.
The study examines determinantal point processes linked to a specific operator on Riemannian manifolds.
problem Understanding the spectral properties and associated point processes of the Bochner-Schrödinger operator.
method Analysis of the Bochner-Schrödinger operator on tensor powers of Hermitian line bundles, focusing on large p asymptotics. result The asymptotic behavior of determinantal point processes associated with the operator's spectral projection is computed, leading to the law of large numbers and central limit theorem.