We establish in this paper some weighted Hardy and Rellich type inequalities on the half line in the framework of equalities, extending recent results proved by Machihara-Ozawa-Wadade and Bez-Machihara-Ozawa. In particular, the one-dimensional classical Rellich inequality is proved in the framework of equality.

Perturbation theory of eigenvalue problems

Rellich type inequalities on the half line in the framework of equalities

This dissertation studies the extension of sparse optimization techniques to the numerical solution of partial differential equations for applications in scientific computing, in particular many-particle systems that are governed by a differential equation. Sparse optimization techniques have attracted much attention due to their substantial computational efficiency and feasibility for large-scale problems such as image processing, compressed sensing, and machine learning. In this dissertation, $\ell^1$-minimization scheme has been studied for the solutions of elliptic and parabolic differential equations. Theoretical considerations for the effectiveness of the scheme, such as the sparsity properties, completeness, consistency, and the asymptotic behavior of the solutions are analyzed.

background

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A Sparse Optimization Framework for the Numerical Solution of PDEs

/pdf/spectral-analysis-of-non-self-adjoint-elliptic-operators-1eq1gme29s.pdf

Spectral analysis of non-self-adjoint elliptic operators

In this review paper, we discuss the relation between recent advances in the theory of partial differential equations and their applications to quantum field theory on curved spacetimes. In particular, we focus on hyperbolic propagators and the role they play in the construction of physically admissible quantum states—the so-called Hadamard states—on globally hyperbolic spacetimes. We will review the notion of a propagator and discuss how it can be constructed in an explicit and invariant fashion, first on a Riemannian manifold and then on a Lorentzian spacetime. Finally, we will recall the notion of Hadamard state and relate the latter to hyperbolic propagators via the wavefront set, a subset of the cotangent bundle capturing the information about the singularities of a distribution.

Partial Differential Equations and Quantum States in Curved Spacetimes

In this paper, we study the multidimensional lattice Schr\"odinger operators with $C^2$-cosine like quasi-periodic (QP) potential. We establish quantitative Green's function estimates, the arithmetic version of Anderson (and dynamical) localization, and the finite volume version of $(\frac 12-)$-H\"older continuity of the integrated density of states (IDS) for such QP Schr\"odinger operators. Our proof is based on an extension of the fundamental multi-scale analysis (MSA) type method of Fr\"ohlich-Spencer-Wittwer [\textit{Comm. Math. Phys.} 132 (1990): 5--25] to the higher lattice dimensions. We resolve the level crossing issue on eigenvalues parameterizations in the case of both higher lattice dimension and $C^2$ regular potential.

Localization and Regularity of the Integrated Density of States for Schr\"odinger Operators on $\mathbb{Z}^d$ with $C^2$-cosine Like Quasi-periodic Potential

Perturbation theory of eigenvalue problems

Citations

Rellich type inequalities on the half line in the framework of equalities

A Sparse Optimization Framework for the Numerical Solution of PDEs

Spectral analysis of non-self-adjoint elliptic operators

Partial Differential Equations and Quantum States in Curved Spacetimes

Localization and Regularity of the Integrated Density of States for Schr\"odinger Operators on $\mathbb{Z}^d$ with $C^2$-cosine Like Quasi-periodic Potential

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