Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1007.4605

Symmetrized Perturbation Determinants and Applications to Boundary Data Maps and Krein-Type Resolvent Formulas

Fritz Gesztesy·Maxim Zinchenko
View original source ↗Read source document ↗Export metadata ↓

Abstract

The aim of this paper is twofold: On one hand we discuss an abstract approach to symmetrized Fredholm perturbation determinants and an associated trace formula for a pair of operators of positive-type, extending a classical trace formula. On the other hand, we continue a recent systematic study of boundary data maps, that is, 2 \times 2 matrix-valued Dirichlet-to-Neumann and more generally, Robin-to-Robin maps, associated with one-dimensional Schrödinger operators on a compact interval [0,R] with separated boundary conditions at 0 and R. One of the principal new results in this paper reduces an appropriately symmetrized (Fredholm) perturbation determinant to the 2\times 2 determinant of the underlying boundary data map. In addition, as a concrete application of the abstract approach in the first part of this paper, we establish the trace formula for resolvent differences of self-adjoint Schrödinger operators corresponding to different (separated) boundary conditions in terms of boundary data maps.

Explore related subjects

math.SPmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Fritz Gesztesy, Maxim Zinchenko. 2010-07-27. Symmetrized Perturbation Determinants and Applications to Boundary Data Maps and Krein-Type Resolvent Formulas. https://doi.org/10.1112/plms%2Fpdr024

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↗