Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/9811176

The uniqueness of the solution of the Schrodinger equation with discontinuous coefficients

Willi Jager·Yoshimi Saito
View original source ↗Read source document ↗Export metadata ↓

Abstract

Consider the Schroeodinger equation: - Du(x) - l(x)u + s(x)u = 0, where D is the Laplacian, l(x) > 0 and s(x) is dominated by l(x). We shall extend the celebrated Kato's result on the asymptotic behavior of the solution to the case where l(x) has unbounded discontinuity. The result will be used to establish the limiting absorption principle for a class of reduced wave operators with discontinuous coefficients.

Explore related subjects

math.SP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Willi Jager, Yoshimi Saito. 1998-11-30. The uniqueness of the solution of the Schrodinger equation with discontinuous coefficients. https://doi.org/10.1142/s0129055x98000318

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The sharp one-dimensional Lieb-Thirring inequality for the sum of eigenvalues

We prove that the best constant in the one-dimensional Lieb-Thirring inequality with exponent one is $4/(3\sqrt{3}π)$, confirming the Lieb-Thirring conjecture in this case. Moreover, we extend the inequality to operator-valued potentials and obtain the bound $L_{1,d}\leq(2/\sqrt3)L_{1,d}^{\mathrm{cl}}$ in every dimension.

math.SP↗
arXiv2026

Singular value decomposition of unbounded operators

The singular value decomposition has been established for matrices, Hilbert--Schmidt operators, trace-class operator, compact operators, and bounded operators, but surprisingly not for unbounded operators. Unfortunately, most interesting operators in applied math are unbounded, as any operators involving some form of derivatives --- gradient, exterior derivatives, Laplacians, Fourier and other transforms of derivatives, Hamiltonians, etc. --- are likely unbounded. In this article, we fill in this last missing piece by establishing the existence of singular value decompositions for unbounded operators in three natural forms: multiplication-operator, direct-integral, and operator-valued-measure. We show it inherits classical properties of finite-dimensional singular value decomposition including approximation results, relationships with fundamental subspaces, and the Moore--Penrose inverse. This discovery opens the door to the singular value decompositions of a myriad of well-known unbounded operators in mathematics, physics, statistics, and finnance --- gradients on Euclidean spaces and manifolds, Petrov--Galerkin method, finite-difference operators, Hilbert--Schmidt operators, Hilbert complexes, supersymmetric quantum mechanics, Sturm--Liouville theory, nonparametric density estimation, and the Black--Scholes equation. The resulting decompositions reveal a number of novel insights, among many others: bosonic and fermionic states in supersymmetric quantum mechanics arise as left and right singular vectors of generalized ladder operators; the Riesz transform appears as the left singular operator of the Euclidean gradient; and the Hodge decomposition follows directly from the singular value decompositions of the exterior derivatives.

math.SP↗
arXiv2026

A quadratic comparison of Neumann eigenvalues on thin convex domains in arbitrary dimenstion

Let $Ω\subset\mathbb R^n$ be a bounded convex domain that is thin around a chosen diameter segment. We compare its Neumann spectrum with the spectrum of that segment weighted by the $(n-1)$-dimensional volumes of its perpendicular sections. We prove an $O(\varepsilon^2)$ comparison of the mean-zero inverse operators and, consequently, an $O(\varepsilon^2)$ eigenvalue comparison for every fixed index in every dimension $n\ge2$. The constants depend only on the dimension and the eigenvalue index. Thin rectangles show that the quadratic exponent is optimal.

math.SP↗