Reviews in Mathematical Physics

We consider a model for a massive uncharged non-relativistic particle interacting with a massless bosonic field, widely referred to as the Nelson model. It is well known that an ultraviolet renormalized Hamilton operator exists in this case. Further, due to translation-invariance, it decomposes into fiber operators. In this paper, we treat the renormalized fiber operators. We give a description of the operator and form domains and prove that the fiber operators do not have a ground state. Our results hold for any non-zero coupling constant and arbitrary total momentum. Our proof for the absence of ground states is a new generalization of methods recently applied to related models. A major enhancement we provide is that the method can be applied to models with degenerate ground state eigenspaces.

Given a spectral triple on a unital $C^{*}$-algebra $A$ and an equicontinuous action of a discrete group $G$ on $A$, a spectral triple on the reduced crossed product $C^{*}$-algebra $A{⋊}_{r}G$ was constructed by Hawkins, Skalski, White and Zacharias in [On spectral triples on crossed products arising from equicontinuous actions, Math. Scand. 113(2) (2013) 262–291], extending the construction by Belissard, Marcolli and Reihani in [Dynamical systems on spectral metric spaces, preprint (2010), arXiv:1008.4617], by using the Kasparov product to make an ansatz for the Dirac operator. Supposing that the triple on $A$ is equivariant for an action of $G$, we show that the triple on $A{⋊}_{r}G$ is equivariant for the dual coaction of $G$. If moreover an equivariant real structure $J$ is given for the triple on $A$, we give constructions for two inequivalent real structures on the triple $A{⋊}_{r}G$. We compute the KO-dimension with respect to each real structure in terms of the KO-dimension of $J$ and show that the first- and the second-order conditions are preserved. Lastly, we characterize an equivariant orientation cycle on the triple on $A{⋊}_{r}G$ coming from an equivariant orientation cycle on the triple on $A$. We show, along the paper, that our constructions generalize the respective constructions of the equivariant spectral triple on the noncommutative $2$-torus.

A theory of graded manifolds can be viewed as a generalization of differential geometry of smooth manifolds. It allows one to work with functions which locally depend not only on ordinary real variables, but also on $\mathbb{Z}$-graded variables which can either commute or anticommute, according to their degree. To obtain a consistent global description of graded manifolds, one resorts to sheaves of graded commutative associative algebras on second countable Hausdorff topological spaces, locally isomorphic to a suitable “model space”. This paper aims to build robust mathematical foundations of geometry of graded manifolds. Some known issues in their definition are resolved, especially the case where positively and negatively graded coordinates appear together. The focus is on a detailed exposition of standard geometrical constructions rather than on applications. Necessary excerpts from graded algebra and graded sheaf theory are included.

We consider Frobenius objects in the category Span, where the objects are sets and the morphisms are isomorphism classes of spans of sets. We show that such structures are in correspondence with data that can be characterized in terms of simplicial sets. An interesting class of examples comes from groupoids.

Our primary motivation is that Span can be viewed as a set-theoretic model for the symplectic category, and thus Frobenius objects in Span provide set-theoretic models for classical topological field theories. The paper includes an explanation of this relationship.

Given a finite commutative Frobenius object in Span, one can obtain invariants of closed surfaces with values in the natural numbers. We explicitly compute these invariants in several examples, including examples arising from abelian groups.

We extend in this work the Jitomirskaya–Last inequality [Power-law subordinacy and singular spectra i. Half-line operators, Acta Math. 183 (1999) 171–189] and Last and Simon [Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (1999) 329–367] criterion for the absolutely continuous spectral component of a half-line Schrödinger operator to the special class of matrix-valued Jacobi operators $H:l^2(\mathbb{Z},{ℂ}^l)\to l^2(\mathbb{Z},{ℂ}^l)$ given by the law ${[Hu]}_n:=D_{n-1}{u}_{n-1}+D_n{u}_{n+1}+V_n{u}_n$, where ${(D_n)}_n{(V_n)}_n$ are bilateral sequences of $l\times l$ self-adjoint matrices such that $0<{inf}_{n\in \mathbb{Z}}{s}_l[D_n]\le {sup}_{n\in \mathbb{Z}}{s}_1[D_n]<\infty$ (here, $s_k[A]$ stands for the $k$th singular value of $A$). Moreover, we also show that the absolutely continuous components of even multiplicity of minimal dynamically defined matrix-valued Jacobi operators are constant, extending another result from Last and Simon [Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (1999) 329–367] originally proven for scalar Schrödinger operators.