The aim of this review paper is to revisit the geometric framework of the Teukolsky equation in a form that is suitable for analysts working on this equation. We introduce spinor bundles, the Newman–Penrose formalism and the Geroch–Held–Penrose (GHP) formalism. In particular, we develop the case of Kerr spacetimes, for which we provide detailed computations.

The study of spectral properties of natural geometric elliptic partial differential operators acting on smooth sections of vector bundles over Riemannian manifolds is a central theme in global analysis, differential geometry and mathematical physics. Instead of studying the spectrum of a differential operator [Formula: see text] directly one usually studies its spectral functions, that is, spectral traces of some functions of the operator, such as the spectral zeta function [Formula: see text] and the heat trace [Formula: see text]. The kernel [Formula: see text] of the heat semigroup [Formula: see text], called the heat kernel, plays a major role in quantum field theory and quantum gravity, index theorems, non-commutative geometry, integrable systems and financial mathematics. We review some recent progress in the study of spectral asymptotics. We study more general spectral functions, such as [Formula: see text], that we call quantum heat traces. Also, we define new invariants of differential operators that depend not only on the eigenvalues but also on the eigenfunctions, and, therefore, contain much more information about the geometry of the manifold. Furthermore, we study some new invariants, such as [Formula: see text], that contain relative spectral information of two differential operators. Finally, we show how the convolution of the semigroups of two different operators can be computed by using purely algebraic methods.

In this paper, we develop a systematic approach to treat Dirac operators [Formula: see text] with singular electrostatic, Lorentz scalar, and anomalous magnetic interactions of strengths [Formula: see text], respectively, supported on points in [Formula: see text], curves in [Formula: see text], and surfaces in [Formula: see text] that is based on boundary triples and their associated Weyl functions. First, we discuss the one-dimensional case which also serves as a motivation for the multidimensional setting. Afterwards, in the two- and three-dimensional situation we construct quasi, generalized, and ordinary boundary triples and their Weyl functions, and provide a detailed characterization of the associated Sobolev spaces, trace theorems, and the mapping properties of integral operators which play an important role in the analysis of [Formula: see text]. We make a substantial step towards more rough interaction supports [Formula: see text] and consider general compact Lipschitz hypersurfaces. We derive conditions for the interaction strengths such that the operators [Formula: see text] are self-adjoint, obtain a Krein-type resolvent formula, and characterize the essential and discrete spectrum. These conditions include purely Lorentz scalar and purely non-critical anomalous magnetic interactions as well as the confinement case, the latter having an important application in the mathematical description of graphene. Using a certain ordinary boundary triple, we also show the self-adjointness of [Formula: see text] for arbitrary critical combinations of the interaction strengths under the condition that [Formula: see text] is [Formula: see text]-smooth and derive its spectral properties. In particular, in the critical case, a loss of Sobolev regularity in the operator domain and a possible additional point of the essential spectrum are observed.

The Jensen, Jensen–Mercer, and Hermite–Hadamard inequalities have been key in advancing convex inequalities for scalars and operators, with several applications in fields like mathematical physics, operator theory and mathematical inequalities. This paper presents a new argument leading to interesting refinements of these celebrated inequalities. Applications that include entropy and operator inequalities will be presented.

We use the method of eigenvalue level spacing developed by Dietlein and Elgart [Level spacing and Poisson statistics for continuum random Schrödinger operators, J. Eur. Math. Soc. (JEMS) 23(4) (2021) 1257–1293] to prove that the local eigenvalue statistics (LES) for the Anderson model on [Formula: see text], with uniform higher-rank [Formula: see text], single-site perturbations, is given by a Poisson point process with intensity measure [Formula: see text], where [Formula: see text] is the density of states at energy [Formula: see text] in the region of localization near the spectral band edges. This improves the result of Hislop and Krishna [Eigenvalue statistics for random Schrödinger operators with non-rank one perturbations, Comm. Math. Phys. 340(1) (2015) 125–143], who proved that the LES is a compound Poisson process with Lévy measure supported on the set [Formula: see text]. Our proofs are an application of the ideas of Dietlein and Elgart to these higher-rank lattice models with two spectral band edges, and illustrate, in a simpler setting, the key steps of the proof of Dietlein and Elgart.

We prove thermodynamic and ultraviolet stable stability bounds for lattice scalar QCD quantum models, with multiflavor real or complex scalar Bose matter fields and a compact, connected gauge Lie group [Formula: see text], [Formula: see text] with Lie algebra dimension [Formula: see text]. Our models are defined on a finite hypercubic lattice [Formula: see text], [Formula: see text], [Formula: see text], with [Formula: see text], even, sites on a side, [Formula: see text] sites, and with free boundary conditions. The models action is a sum of a minimally coupled Bose-gauge part and a Wilson pure-gauge plaquette action. We use local, scaled scalar multiflavor Bose fields. The scaling is global, [Formula: see text]-dependent and noncanonical, and corresponds to an a priori renormalization. The Wilson action is a sum over positive plaquette actions times a factor [Formula: see text], with the gauge coupling [Formula: see text] in [Formula: see text], [Formula: see text]. By local gauge invariance, to eliminate the excess of gauge variables, sometimes we use an enhanced temporal gauge, leaving only [Formula: see text] for [Formula: see text], retained bonds. Fixing this gauge does not alter the value of the partition function. Considering the original physical, unscaled partition function [Formula: see text], where [Formula: see text] is the unscaled (bare) hopping parameter and [Formula: see text] are the boson fields bare masses, and letting [Formula: see text] and [Formula: see text], we show that the scaled partition function [Formula: see text] satisfies the thermodynamic and ultraviolet stable stability bounds [Formula: see text], with finite constants [Formula: see text], independent of the lattice size [Formula: see text] of [Formula: see text] and the spacing [Formula: see text]. For the normalized finite-lattice free energy [Formula: see text], a finite thermodynamic limit ([Formula: see text]) for [Formula: see text], and then the continuum limit [Formula: see text], both exist in the sense of subsequences. They give the model normalized free energies [Formula: see text]. The finiteness of [Formula: see text] is the only question addressed here! The use of the Weyl integration formula is essential in showing these bounds. It allows us to replace the gauge integral over [Formula: see text] gauge bond matrix elements by the integration over its [Formula: see text] eigenvalues. A new global upper bound on the Wilson plaquette action is obtained, which is quadratic in the gluon fields. Our method bypasses the use of diamagnetic inequality and can be extended to treat more general lattices and Lie gauge groups.

In this paper we are interested in generalizing Keller-type eigenvalue estimates for the non-selfadjoint Schrödinger operator to the Dirac operator, imposing some suitable rigidity conditions on the matricial structure of the potential, without necessarily requiring the smallness of its norm.

This paper is a continuation of work by the third author, which studied quantum walks with special long-range perturbations of the coin operator. In this paper, we consider general long-range perturbations of the coin operator and prove the non-existence of a singular continuous spectrum and embedded eigenvalues. The proof relies on the gauge transformation and construction of generalized eigenfunctions (Jost solutions) which was studied in the short-range case.

In [Spectral asymmetry and Riemannian geometry. III, Math. Proc. Cambridge Philos. Soc. 79 (1976) 71–99] Atiyah, Patodi and Singer introduced spectral flow for elliptic operators on odd-dimensional compact manifolds. They argued that it could be computed from the Fredholm index of an elliptic operator on a manifold of one higher dimension. A general proof of this fact was produced by Robbin–Salamon [The spectral flow and the Maslov index, Bull. London Math. Soc. 27(1) (1995) 1–33, MR 1331677]. In [F. Gesztesy, Y. Latushkin, K. Makarov, F. Sukochev and Y. Tomilov, The index formula and the spectral shift function for relatively trace class perturbations, Adv. Math. 227(1) (2011) 319–420, MR 2782197], a start was made on extending these ideas to operators with some essential spectrum as occurs on non-compact manifolds. The new ingredient introduced there was to exploit scattering theory following the fundamental paper [A. Pushnitski, The spectral flow, the Fredholm index, and the spectral shift function, in Spectral Theory of Differential Operators, American Mathematical Society Translations: Series 2, Vol. 225 (American Mathematical Society, Providence, RI, 2008), pp. 141–155, MR 2509781]. These results do not apply to differential operators directly, only to pseudo-differential operators on manifolds, due to the restrictive assumption that spectral flow is considered between an operator and its perturbation by a relatively trace-class operator. In this paper, we extend the main results of these earlier papers to spectral flow between an operator and a perturbation satisfying a higher pth Schatten class condition for [Formula: see text], thus allowing differential operators on manifolds of any dimension [Formula: see text]. In fact our main result does not assume any ellipticity or Fredholm properties at all and proves an operator theoretic trace formula motivated by [M.-T. Benameur, A. Carey, J. Phillips, A. Rennie, F. Sukochev and K. Wojciechowski, An analytic approach to spectral flow in von Neumann algebras, in Analysis, Geometry and Topology of Elliptic Operators (World Scientific Publisher, Hackensack, NJ, 2006), pp. 297–352, MR 2246773; A. Carey, H. Grosse and J. Kaad, On a spectral flow formula for the homological index, Adv. Math. 289 (2016) 1106–1156, MR 3439708]. We illustrate our results using Dirac type operators on [Formula: see text] for arbitrary [Formula: see text] (see Sec. 8). In this setting Theorem 6.4 substantially extends Theorem 3.5 of [A. Carey, F. Gesztesy, H. Grosse, G. Levitina, D. Potapov, F. Sukochev and D. Zanin, Trace formulas for a class of non-Fredholm operators: a review, Rev. Math. Phys. 28(10) (2016) 1630002, MR 3572626], where the case d = 1 was treated.