We consider a class of Schrödinger operators with complex decaying potentials on the lattice. Using some classical results from complex analysis we obtain some trace formulae and use them to estimate zeros of the Fredholm determinant in terms of the potential.

We review the work of Tosio Kato on the mathematics of non-relativistic quantum mechanics and some of the research that was motivated by this. Topics in this second part include absence of embedded eigenvalues, trace class scattering, Kato smoothness, the quantum adiabatic theorem and Kato’s ultimate Trotter Product Formula.

We review the work of Tosio Kato on the mathematics of non-relativistic quantum mechanics and some of the research that was motivated by this. Topics in this first part include analytic and asymptotic eigenvalue perturbation theory, Temple–Kato inequality, self-adjointness results, and quadratic forms including monotone convergence theorems.

We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori \(\dot{W}^1_q\)-estimates for any \(q\in [2,\infty )\) when the coefficients are merely measurable functions in one fixed direction. For the system on bounded Lipschitz domains with a small Lipschitz...

We study spectral properties of a class of global infinite order pseudo-differential operators and obtain the asymptotic behaviour of the spectral counting functions of such operators. Unlike their finite order counterparts, their spectral asymptotics are not of power-log-type but of log-type. The ultradistributional setting of such operators of infinite order makes the theory...

We study the bilinear Weyl product acting on quasi-Banach modulation spaces. We find sufficient conditions for continuity of the Weyl product and we derive necessary conditions. The results extend known results for Banach modulation spaces.

We extend the resolvent estimate on the sphere to exponents off the line \(\frac{1}{r}-\frac{1}{s}=\frac{2}{n}\). Since the condition \(\frac{1}{r}-\frac{1}{s}=\frac{2}{n}\) on the exponents is necessary for a uniform bound, one cannot expect estimates off this line to be uniform still. The essential ingredient in our proof is an \((L^{r}, L^{s})\) norm estimate on the operator...

This is a survey on Nichols algebras of diagonal type with finite dimension, or more generally with arithmetic root system. The knowledge of these algebras is the cornerstone of the classification program of pointed Hopf algebras with finite dimension, or finite Gelfand–Kirillov dimension; and their structure should be indispensable for the understanding of the representation...

Let \(\pi \) be a set of primes. According to H. Wielandt, a subgroup H of a finite group X is called a \(\pi \)-submaximal subgroup if there is a monomorphism \(\phi :X\rightarrow Y\) into a finite group Y such that \(X^\phi \) is subnormal in Y and \(H^\phi =K\cap X^\phi \) for a \(\pi \)-maximal subgroup K of Y. In his talk at the celebrated conference on finite groups in...

We define a distance function on the bordered punctured disk \(0<|z|\le 1/e\) in the complex plane, which is comparable with the hyperbolic distance of the punctured unit disk \(0<|z|<1.\) As an application, we will construct a distance function on an n-times punctured sphere which is comparable with the hyperbolic distance. We also propose a comparable quantity which is not...

Decompositions of linear ordinary differential equations (ode’s) into components of lower order have successfully been employed for determining their solutions. Here this approach is generalized to nonlinear ode’s. It is not based on the existence of Lie symmetries, in that it is a genuine extension of the usual solution algorithms. If an equation allows a Lie symmetry, the...

In this article we analyze the notions of amenability and paradoxical decomposition from an algebraic perspective. We consider this dichotomy for locally finite extended metric spaces and for general algebras over fields. In the context of algebras we also study the relation of amenability with proper infiniteness. We apply our general analysis to two important classes of...

For \(n\in \mathbb {N}\) the nth alternating harmonic number $$\begin{aligned} H_n^*:=\sum _{k=1}^n(-1)^{k-1}\frac{1}{k} \end{aligned}$$is given in the form $$\begin{aligned} H_n^*=\ln 2 +\frac{(-1)^{n+1}}{4\left\lfloor \frac{n+1}{2}\right\rfloor } +\sum _{i=1}^{q-1}\frac{(4^i-1)B_{2i}}{(2i)\left( 2\left\lfloor \frac{n+1}{2}\right\rfloor \right) ^{2i}}+r_q(n) \end{aligned}$$where...

We study the Nemytskii operators \(u\mapsto |u|\) and \(u\mapsto u^{\pm }\) in fractional Sobolev spaces \(H^s({\mathbb {R}}^n)\), \(s>1\).

In this paper, we study semi-slant submanifolds and their warped products in Kenmotsu manifolds. The existence of such warped products in Kenmotsu manifolds is shown by an example and a characterization. A sharp relation is obtained as a lower bound of the squared norm of second fundamental form in terms of the warping function and the slant angle. The equality case is also...

The principal aim of this paper is to derive an abstract form of the third Green identity associated with a proper extension T of a symmetric operator S in a Hilbert space \(\mathfrak {H}\), employing the technique of quasi boundary triples for T. The general results are illustrated with couplings of Schrödinger operators on Lipschitz domains on smooth, boundaryless, compact...

At the turn of this century Durand, and Lagarias and Pleasants established that key features of minimal subshifts (and their higher-dimensional analogues) to be studied are linearly repetitive, repulsive and power free. Since then, generalisations and extensions of these features, namely \(\alpha \)-repetitive, \(\alpha \)-repulsive and \(\alpha \)-finite (\(\alpha \ge 1\)), have...

Weighted Sobolev spaces play a main role in the study of Sobolev orthogonal polynomials. The aim of this paper is to prove several important properties of weighted Sobolev spaces: separability, reflexivity, uniform convexity, duality and Markov-type inequalities.

We construct the first examples of algorithmically complex finitely presented residually finite groups and the first examples of finitely presented residually finite groups with arbitrarily large (recursive) Dehn functions, and arbitrarily large depth functions. The groups are solvable of class 3.

This is an expository paper on the theory of gradient flows, and in particular of those PDEs which can be interpreted as gradient flows for the Wasserstein metric on the space of probability measures (a distance induced by optimal transport). The starting point is the Euclidean theory, and then its generalization to metric spaces, according to the work of Ambrosio, Gigli and...

We provide an introduction to mathematical theory of scattering resonances and survey some recent results.

We study the inverse problems for the second order hyperbolic equations of general form with time-dependent coefficients assuming that the boundary data are given on a part of the boundary. The main result of this paper is the determination of the time-dependent Lorentzian metric by the boundary measurements. This is achieved by the adaptation of a variant of the boundary control...

In this paper, we establish a new multiplicative Sobolev inequality. As applications, we refine and extend the results in Kukavica and Ziane (J Math Phys 48:065203, 2007) and Cao (Discrete Contin Dyn Syst 26:1141–1151, 2010) simultaneously.

We derive trace formulas of the Buslaev–Faddeev type for quantum star graphs. One of the new ingredients is high energy asymptotics of the perturbation determinant.

We use the type theory for rings of operators due to Kaplansky to describe the structure of modules that are invariant under automorphisms of their injective envelopes. Also, we highlight the importance of Boolean rings in the study of such modules. As a consequence of this approach, we are able to further the study initiated by Dickson and Fuller regarding when a module...