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.
Characterizes Schrödinger operator boundedness on weighted Riemannian manifolds.
problem Classifying functions V V V for bounded Schrödinger operator Δ − V Δ-V Δ − V . method Investigates weighted L 2 L^2 L 2 -boundedness of Hodge projector. result Characterizes function V V V for Schrödinger operator boundedness. Abstract notes on generative modeling techniques.
problem Improving generative modeling techniques.
method Connections between optimal transport and Schrödinger bridge, flow matching.
result Showed connections between mathematical principles and generative modeling techniques.
Paper introduces a new generative learning model using Schrödinger bridge diffusion in latent space.
problem Learning distributions from divergent data distributions.
method Pre-training with large-scale models, Schrödinger bridge diffusion model in latent space.
result Effective control of second-order Wasserstein distance between generated and target distributions.
Unified framework for robust, stable, and efficient density ratio estimation.
problem Density-chasm and support-chasm problems in density ratio estimation.
method Dequantified diffusion-Schrödinger bridge (D3RE) framework with DDBI and DSBI.
result Offers uniform approximation and bounded time scores in theory and empirical performance.
We give a new lower bound for the first gap λ 2 − λ 1 λ_2 - λ_1 λ 2 − λ 1 of the Dirichlet eigenvalues of the Schr{ö}dinger operator on a bounded convex domain Ω Ω Ω in R n ^n n or S n ^n n and greatly sharpens the previous estimates. The new bound is explicit and computable.
In this paper, we partially settle down the long standing open problem of the finite time blow-up property about the nonlinear Schr o ¨ \ddot{o} o ¨ dinger equations on some Riemannian manifolds like the standard 2-sphere S 2 S^2 S 2 and the hyperbolic 2-space H 2 ( − 1 ) H^{2}(-1) H 2 ( − 1 ) . Using the similar idea, we establish such blow-up results on…
The goal of this article is twofold: in a first part, we prove Gaussian estimates for the heat kernel of Schr{ö}dinger operators delta + V whose potential V is "small at infinity" in an integral sense. In a second part, we prove sharp boundedness result for the associated Riesz transform with potential d(delta+V) --1/2…
For the spherical Laplacian on the sphere and for the Dirichlet Laplacian in the square}, Antonie Stern claimed in her PhD thesis (1924) the existence of an infinite sequence of eigenvalues whose corresponding eigenspaces contain an eigenfunction with exactly two nodal domains. These results were given complete proofs …
We solved the Schr{ö}dinger equation for a particle in a uniform magnetic field in the n-dimensional torus. We obtained a complete set of solutions for a broad class of problems; the torus T^n = R^n / Λ is defined as a quotient of the Euclidean space R^n by an arbitrary n-dimensional lattice Λ. The lattice is not neces…
Study shows observability for Schrödinger equations on product manifolds with specific conditions.
problem Observability of Schrödinger equations on product manifolds with product metrics.
method Proof of observability in finite time on open subsets satisfying Vertical Geometric Control Condition, under gap condition on spectrum of F(g).
result Observability on ω for the Schrödinger equation is strictly weaker than Geometric Control Condition on product of spheres.
Suppose that G = ( V , E ) G=(V, E) G = ( V , E ) is a finite graph with the vertex set V V V and the edge set E E E . Let Δ Δ Δ be the usual graph Laplacian. Consider the following nonlinear Schr o ¨ \ddot{o} o ¨ dinger type equation of the form { − Δ u − α u = f ( x , u ) , u ∈ W 1 , 2 ( V ) , \left \{ \begin{array}{lcr} -Δu-αu=f(x,u),\\ u\in W^{1,2}(V),\\ \end{array} \right. { − Δ u − α u = f ( x , u ) , u ∈ W 1 , 2 ( V ) , on graph G G G , where $f(x…
New inequalities for spectral zeta kernels on spheres and manifolds.
problem Establishing new inequalities for spectral zeta functions.
method Applying Kato's inequalities and majorisation techniques.
result Generalized Kato's comparison inequalities to higher dimensions.
Generative model for time series using Schrödinger bridge.
problem Creating synthetic time series data with temporal dynamics.
method Schrödinger bridge approach for entropic interpolation via optimal transport.
result The method generates synthetic time series that respect temporal dynamics.
CMCD sampler connects transport and variational inference for efficient sampling.
problem Efficient sampling and generative modeling in Bayesian computation.
method Developed a principled framework using divergences on path space, CMCD sampler with adaptive dynamics.
result CMCD sampler outperforms competing approaches across various experiments.
The study examines conditions for scalar curvature and Dirac operators on singular spaces.
problem Existence of scalar curvature measures and Dirac operators on singular spaces.
method Investigation of smooth manifolds with singular Riemannian metrics.
result Sufficient conditions for the existence of scalar curvature measures and Dirac operators.
The paper examines scalar fourth-order linear differential operators and their invariants.
problem Equivalence problem of scalar fourth-order linear differential operators.
method Investigation of differential invariants.
result Application of differential invariants to the equivalence problem.
Develops connections between operator K-theory and positive scalar curvature.
problem Positive scalar curvature on closed spin manifolds and Gromov's band width conjecture.
method Quantitative index theory and related techniques.
result The propagation of the index of the Dirac operator is inversely related to the curvature lower bound.
New findings extend rigidity results to broader classes of manifolds.
problem Extending rigidity results to non-warped product spaces.
method Establishing rigidity theorems for manifolds conformal to those with nonnegative curvature.
result New families of manifolds exhibit scalar-mean rigidity.
Using holographic renormalization coupled with the Caffarelli/Silvestre\cite{caffarelli} extension theorem, we calculate the precise form of the boundary operator dual to a bulk scalar field rather than just its average value. We show that even in the presence of interactions in the bulk, the boundary operator dual to …
New operators and curvatures derived from embedded manifolds.
problem Finding obstructions and coupling extrinsic operators.
method Explicit computation of extrinsic Paneitz operator and its applications.
result New extrinsically-coupled fourth and sixth order operators.
Estimates mean curvature, scalar curvature, shape operator in warped products.
problem Estimating geometric properties in warped product spaces.
method Local and global upper estimates for curvature and shape operator.
result Results on pseudo-hyperbolic spaces and space forms.
The paper establishes distance estimates for manifolds with lower scalar curvature bounds.
problem Distance estimates on manifolds with lower scalar curvature bounds.
method Introduced a definition of relative index via a deformed Dirac operator trick and proved index coincidence with Callias operators.
result Proved short neck inequality and quantitative shielding result with positive scalar curvature.
Dirac operator invertibility proven for specific manifolds.
problem Invertibility of twisted Dirac operator on manifolds.
method Closed connected spin manifold with non-negative scalar curvature, flat Hilbert module bundle.
result Dirac operator is invertible under given conditions.
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.
Rigidity theorem for scalar curvature on odd-dimensional singular manifolds.
problem Understanding scalar curvature on manifolds with cone-like singularities.
method Analysis of abstract cone operators, spectral flow argument, and twisted Dirac operators.
result Lipschitz rigidity for scalar curvature on Riemannian spin manifolds with cone-like singularities in odd dimensions.
Abstract cone operators prove scalar curvature comparisons on singular manifolds.
problem Proving scalar curvature inequalities on manifolds with cone singularities.
method Using index theory for twisted Dirac operators on Lipschitz bundles.
result Lipschitz rigidity for scalar curvature on odd-dimensional manifolds.
In this paper, the Dirac, twistor and Killing equations on Weyl manifolds with CSpin structures are investigated. A conformal Schr"odinger-Lichnerowicz formula is presented and used to show integrability conditions for these equations. By introducing the Killing equation for spinors of arbitrary weight, the result of A…
The study proves curvature rigidity for convex polytopes.
problem Proving curvature rigidity for convex polytopes.
method Using Fredholm theory for Dirac operators and a theorem of Fefferman and Phong.
result Scalar curvature rigidity theorem for convex polytopes proved.
The study constructs Yamabe operators on OC manifolds and proves their properties.
problem Investigating Yamabe operators on OC manifolds and their invariants.
method Construction and analysis of OC Yamabe operators, transformation formula proof, Green function construction.
result Yamabe operators on OC manifolds have specific scalar positivity properties.
For a scalar evolution equation u t = K ( t , x , u , u x , … , u n ) , n ≥ 2 u_t=K(t,x,u,u_x,\ldots, u_n), n\geq 2 u t = K ( t , x , u , u x , … , u n ) , n ≥ 2 the cohomology spaces H 1 , s ( R ∞ ) H^{1,s}({\mathcal R}^\infty) H 1 , s ( R ∞ ) vanishes for s ≥ 3 s\geq 3 s ≥ 3 while the space H 1 , 2 ( R ∞ ) H^{1,2}({\mathcal R}^\infty) H 1 , 2 ( R ∞ ) is isomorphic to the space of variational operators. The cohomology space H 1 , 2 ( R ∞ ) H^{1,2}({\mathcal R}^\infty) H 1 , 2 ( R ∞ ) is also shown to be …
Adapts Bartnik method to Hilbert manifold structure for vacuum constraint equations.
problem Vacuum constraint equations on compact manifolds of any dimension ≥ 3.
method Adapts Bartnik method to provide Hilbert manifold structure.
result Fibers of scalar curvature and constraint operator are Hilbert submanifolds.
Proves curvature comparison theorem for manifolds with conical singularities.
problem Comparing scalar mean curvature of manifolds with conical singularities.
method Uses Dirac operator and index theory to prove curvature comparison theorem.
result Proves curvature comparison theorem without knowing the index of the twisted Dirac operator.
Paper sharpens inequality linking curvature and spectrum on manifolds.
problem Linking scalar curvature and the bottom spectrum on complete manifolds.
method Using deformed Dirac operators and relative A ^ \widehat{A} A -cowaist. result Established a sharp inequality between scalar curvature and the bottom spectrum.
In this paper, we use localization algebras to study higher rho invariants of closed spin manifolds with positive scalar curvature metrics. The higher rho invariant is a secondary invariant and is closely related to positive scalar curvature problems. The main result of the paper connects the higher index of the Dirac …
Constructs a solution operator for hyperbolic gluing in higher dimensions.
problem Solving the linearized constant scalar curvature equation at hyperbolic space.
method Constructs a solution operator with good support propagation properties.
result Extends the Mao-Oh-Tao gluing method to the hyperbolic setting, gaining two derivatives.
Maps are shown to be Riemannian products with Ricci-flat fibers.
problem Understanding maps between manifolds and their geometric properties.
method Spin geometry and representation theory of curvature operators.
result Scalar-rigid maps are essentially Riemannian products of base and Ricci-flat fibers.
In this paper we extend to the difference case the notion of Poisson-Lichnerowicz cohomology, an object encapsulating the building blocks for the theory of deformations of Hamiltonian operators. A local scalar difference Hamiltonian operator is a polynomial in the shift operator and its inverse, with coefficients in th…
Teaches Dirac operators for geometry and topology.
problem Interactions between geometry and topology.
method Families of Dirac operators and index theorems.
result Applications to metrics of positive scalar curvature and the three-dimensional Weinstein conjecture.
Investigates second best Einstein manifolds in low dimensions.
problem Finding second best Einstein manifolds in dimensions below 12.
method Algebraic and geometric analysis of curvature operators.
result Shows existence of a new angle smaller than previously known, leading to isometry to the round sphere.
The paper explores index theory for Dirac operators to understand scalar curvature properties.
problem Understanding scalar curvature properties in Riemannian bands and manifolds.
method Index theory for Dirac operators, focusing on scalar curvature and width.
result A quantitative negative upper bound on the infimum of scalar curvature for complete metrics.
A Dirac-type operator on a complete Riemannian manifold is of Callias-type if its square is a Schrödinger-type operator with a potential uniformly positive outside of a compact set. We develop the theory of Callias-type operators twisted with Hilbert C ∗ C^\ast C ∗ -module bundles and prove an index theorem for such operators…
Proves upper bound on filling radius for manifolds with positive scalar curvature.
problem Bounding the filling radius of manifolds with positive scalar curvature.
method Quantitative operator K-theory and index theory.
result Proves a quantitative upper bound on the filling radius.
Researchers solve Yamabe problems for specific operators, finding both uniqueness and nonuniqueness.
problem Prescribing scalar, Q-, or σ₂-curvatures in conformal classes.
method Formally self-adjoint, conformally covariant, polydifferential operators.
result Uniqueness results on the sphere, nonuniqueness in general.
Study spectral flow for Callias operators to prove obstructions to positive scalar curvature.
problem Prove obstructions to positive scalar curvature on manifolds.
method Use spectral flow and odd K-cowaist to derive obstructions.
result Infinite odd K-cowaist is an obstruction to the existence of PSC metrics.
This paper explores conditions for positive scalar curvature on spin^c manifolds.
problem Conditions for the existence of metrics with positive scalar curvature on spin^c manifolds.
method Introduces a new scalar-valued function and uses it to study positivity conditions.
result Equivalence between positivity of the new scalar function and the old twisted scalar curvature.
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.
New rigidity results for warped product domains.
problem Scalar curvature rigidity of domains in warped products.
method Developed a new connection on a twisted spinor bundle and associated Dirac operator.
result Obtained Llarull type scalar curvature rigidity for a general class of domains in a warped product.