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.
We study the question of well-posedness of the Cauchy problem for Schrödinger maps from $\rone \times \rtwo$ to the sphere $\stwo$ or to H2, the hyperbolic space. The idea is to choose an appropriate gauge change so that the derivatives of the map will satisfy a certain nonlinear Schrödinger system of equa…
Study long wave limits for Schrodinger maps into Kahler manifolds.
problem Understanding long wave behavior in Schrodinger maps systems.
method Develop KdV type systems on tangent spaces for general Schrodinger maps.
result Obtained KdV type systems for Schrodinger maps into Kahler manifolds.
In this paper, we introduce a new notion named as Schrödinger soliton. So-called Schrödinger solitons are defined as a class of special solutions to the Schrödinger flow equation from a Riemannian manifold or a Lorentzian manifold M into a Kähler manifold N. If the target manifold N admits a Killing potential, th…
In this paper we establish the equivalence of solutions between Schrödinger map into S2 or H2 and their associated gauge invariant Schrödinger equations. We also establish the existence of global weak solutions into H2 in two space dimensions. We extend these ideas for maps into com…
Gauss map of skew mean curvature flow satisfies Schrödinger flow.
problem Understanding the geometry of skew mean curvature flow.
method Exploring the oriented Grassmannian manifold and embedding it into exterior product space.
result Gauss map of SMCF satisfies a Schrödinger flow equation.
BM2 learns Schrödinger bridges using neural networks.
problem Learning dynamic transport maps between two distributions.
method Coupled Bridge Matching (BM2) with neural networks. result Preliminary theoretical analysis and numerical experiments show BM2's effectiveness. 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.
We establish the global well-posedness of the initial value problem for the Schrodinger map flow for maps from the real line into Kahler manifolds and for maps from the circle into Riemann surfaces. This partially resolves a conjecture of W.-Y. Ding.
Global Schrödinger map flows to Kähler manifolds proved for high dimensions with small data.
problem Global existence of Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces.
method Decay estimates of moving frame dependent quantities in caloric gauge setting, combined with a bootstrap-iteration scheme.
result Global existence of Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces for high dimensions.
New algorithm computes Schrödinger Bridge for unpaired data translation.
problem Computing optimal transport maps for unpaired data translation.
method Schrödinger Bridge Flow, a discretization of a flow of path measures.
result Eliminates the need to train multiple DDM-like models.
New algorithm improves on existing methods for solving transport problems.
problem Finding a map to transport one distribution to another.
method Iterative Markovian Fitting (IMF) and Diffusion Schrödinger Bridge Matching (DSBM).
result DSBM significantly improves over previous SB numerics and recovers various transport methods.
Symplectic method solves infinite-dimensional Schrödinger equations.
problem Solving Schrödinger equations on infinite-dimensional Hilbert spaces with unbounded Hamiltonians.
method Analytic vectors, manifolds modelled on normed spaces, symplectic differential geometry, Marsden--Weinstein reduction.
result Mapped t-dependent Schrödinger equations onto projective spaces. The paper constructs TQFTs and Schrödinger representations for Heisenberg group.
problem Constructing TQFTs and Schrödinger representations for Heisenberg group.
method Using Lagrangian correspondences and q-deformation of U(1).
result Normalization of Schrödinger bimodule action reproduces abelian TQFT.
In this article, we prove that the equation of the Schrödinger maps from R2 to the hyperbolic 2-space H2 is SU(1,1)-gauge equivalent to the following 1+2 dimensional nonlinear Schrödinger-type system of unknown three complex functions p,q,r and a real function u: {c} iq_t+q_{z{\bar z}}-2u q+2({\ba…
Study subgroup actions on mapping class groups using Heisenberg representations.
problem Untwisting representations of mapping class groups on Heisenberg subgroups.
method Restrict and analyze twisted representations of mapping class groups to Heisenberg subgroups.
result Untwisting representations on Torelli group for any Heisenberg representation.
Solve-training trains neural nets to map physical solutions efficiently.
problem Representing complex physical solutions with neural networks.
method Variational training using loss functions from physical models.
result Effective neural network representation of solution maps without expensive labels.
New algorithm preserves transport maps for better diffusion model training.
problem Training diffusion models with task-specific optimality structures.
method Generalized Schrödinger Bridge Matching (GSBM), inspired by conditional stochastic optimal control.
result GSBM better preserves transport maps, enabling stable convergence and improved scalability.
On a fixed smooth compact Riemann surface with boundary (M0,g), we show that the Cauchy data space (or Dirichlet-to-Neumann map $\mc{N}$) of the Schrödinger operator Δ+V with V∈C2(M0) determines uniquely the potential V. We also discuss briefly the corresponding consequences for potential scattering at 0 …
It is well-known that the LIE(Locally Induction Equation) admit soliton-type solutions and same soliton solutions arise from different and apparently irrelevant physical models. By comparing the solitons of LIE and Killing magnetic geodesics, we observe that these solitons are essentially decided by two families of iso…
Conservation laws, heirarchies, scattering theory and Bäcklund transformations are known to be the building blocks of integrable partial differential equations. We identify these as facets of a theory of Poisson group actions, and apply the theory to the ZS-AKNS nxn heirarchy (which includes the non-linear Schrödinger …
Generative AI connects to Schrödinger bridge problems with soft constraints for stability.
problem Stability issues in generative AI due to hard terminal constraints.
method Soft-constrained Schrödinger bridge formulation and convergence analysis.
result Existence and convergence of optimal solutions as penalty grows.
Study Heisenberg homology on surface configurations, revealing new representations of mapping class groups.
problem Homology of surface configurations with Heisenberg group representations.
method Analysis of unordered configurations in a surface, using Heisenberg group actions and representations.
result Obtained genuine and projective representations of mapping class groups from Heisenberg group actions.
New uncertainty principle for Schrödinger equations on hyperbolic manifolds.
problem Uncertainty principle for Schrödinger equations on hyperbolic manifolds.
method General strategy of Escauriaza-Kenig-Ponce-Vega, new Carleman estimates, logarithmic convexity, new mollifier and weight function.
result Similar rigidity phenomenon as in Euclidean space persists in hyperbolic geometry.
Unified framework for Schrödinger Bridge solutions between arbitrary densities.
problem Generalizing generative models to arbitrary distributions.
method Unified closed-form framework for SB dynamics.
result Direct inference of SB dynamics from samples.
Study on maps with horizontal α-harmonic properties in 1D and 2D.
problem Characterizing maps with specific harmonic properties in different dimensions.
method Investigates Schrödinger type systems and variational problems for maps tangent to a planes distribution.
result Shows regularity of horizontal α-harmonic maps in 1D and 2D. Characterizes Schrödinger operator boundedness on weighted Riemannian manifolds.
problem Classifying functions V for bounded Schrödinger operator Δ−V. method Investigates weighted L2-boundedness of Hodge projector. result Characterizes function V for Schrödinger operator boundedness. 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.
Geometric framework connects quantum and fluid dynamics.
problem Understanding Newton's equations on diffeomorphism spaces.
method Introducing a geometric framework and Madelung transform.
result Madelung transform is a symplectomorphism and Kähler map.
3MSBM learns smooth trajectories from multiple snapshots.
problem Capturing long-range temporal dependencies in complex systems.
method Lifts dynamics to phase space, generalizes stochastic bridges to multi-marginal conditional problems, learns transport maps preserving intermediate marginals.
result Significantly improves convergence and scalability in capturing complex dynamics.
Theory of covariant Schrödinger semigroups on Riemannian manifolds developed.
problem Developing theory for Schrödinger semigroups on Riemannian manifolds.
method Sobolev spaces, heat kernels, differential operators, Wiener measure, Dynkin and Kato potentials.
result Properties and continuity of covariant Schrödinger semigroups established.
New formulation of Schrödinger connections preserves vector lengths in geometry.
problem Preserving vector lengths in non-Euclidean geometries.
method Coordinate-free formulation, differential geometry, torsion, non-metricity.
result Explicit example of non-static Einstein manifold with torsion.
This paper refines the Gaussian Sinkhorn algorithm for general multivariate models.
problem Finite-dimensional solutions for general Gaussian multivariate models.
method Recursive formulation of the Sinkhorn algorithm for Gaussian models, including closed form expressions of entropic transport maps and Schrödinger bridges.
result Refined convergence analysis of Gaussian Sinkhorn algorithms.
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.
Schrödinger and Onsager's ideas linked in nonequilibrium thermodynamics.
problem Linking Schrödinger's variational problem with Onsager's nonequilibrium statistical mechanics.
method Analyzing the historical context and comparing the two approaches.
result Schrödinger's ideas have not yet been fully integrated into the classical context of Onsager's work.
Local solutions found for nonautonomous Schrödinger flows on Kähler manifolds.
problem Existence and uniqueness of solutions for nonautonomous Schrödinger flows.
method Proved the existence of local solutions under certain conditions.
result Existence and uniqueness of solutions with higher regularity.
Estimates the first eigenvalue of a Schrödinger operator on closed surfaces.
problem Estimating the first eigenvalue of a Schrödinger operator on closed surfaces.
method Based on Schoen-Yau's work, derives an estimate.
result Derives an estimate of the first eigenvalue.
Geometrically solves Schrödinger flow on sphere.
problem Solving periodic Cauchy problem for Schrödinger flow on sphere.
method Explicit geometric algorithm for construction of solutions.
result Explicit geometric algorithm for solving Schrödinger flow on sphere.
Study finds optimal martingale coupling between two distributions with minimal entropy.
problem Finding the optimal martingale coupling between two distributions with minimal relative entropy.
method Solving a dual problem to find the log-density of the optimal coupling, which represents the marginal and martingale constraints.
result The log-density of the optimal coupling is given by a triplet of real functions representing the marginal and martingale constraints.
Paper proves uniqueness of Schrödinger flow on specific manifolds.
problem Proving uniqueness of Schrödinger flow on manifolds.
method Intrinsic proof using distance functions and gauge language.
result Uniqueness of Schrödinger flow from a general complete Riemannian manifold to a complete Kähler manifold.
Extends martingale Schrödinger bridge to arbitrary dimensions and characterizes it.
problem Tackles the martingale Schrödinger bridge in arbitrary dimensions.
method Identifies continuous-time counterpart and relates to variational problems.
result Continuous martingale Schrödinger bridge coincides with Föllmer martingale in irreducible case.
Study proposes a new early-warning framework for high-dimensional complex systems.
problem Predicting critical transitions in complex systems like epileptic seizures.
method Integrates manifold learning with stochastic dynamical system modeling, using Schrödinger bridge theory.
result Demonstrates higher sensitivity and robustness in epilepsy prediction.
Study on Schrödinger operators on Zoll manifolds, focusing on pseudo-spectra.
problem Analyzing the pseudo-spectra of Schrödinger operators on Zoll manifolds.
method Asymptotic analysis of pseudo-spectra and numerical range of non-self-adjoint Schrödinger operators.
result Obtained asymptotic results on the pseudo-spectra of 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.
This work studies the contraction coefficients of Schrödinger bridge problems in linear systems.
problem Optimally controlling the evolution of a system's state density over time.
method Analyzes and improves the convergence rates of dynamic Schrödinger systems via geometric and control-theoretic interpretations.
result New insights into improving computation of worst-case contraction coefficients by preconditioning.
We solve a Schrödinger bridge with a quadratic state cost, finding a closed-form solution.
problem Optimizing diffusion processes between given distributions.
method Regularized Schrödinger bridge with a quadratic state cost.
result Closed-form solution for the Markov kernel of the regularized Schrödinger bridge.
Paper solves local well-posedness for Schrödinger flow into sphere with natural boundary conditions.
problem Local well-posedness of Schrödinger flow into S2 with natural boundary conditions. method Developed a new approximation scheme to solve the problem.
result Solved the local well-posedness problem for the Schrödinger flow into S2 with natural boundary conditions. Study of Schrödinger flows on S6 using octonions.
problem Schrödinger flows on S6 and related geometric properties. method Using G2-structure on O, study of G2-binormal motion of curves in R7. result Equivalence of G2-binormal motion to Schrödinger flows and nonlinear Schrödinger-type system.