By a recent observation, the Laplacians on the Riemannian manifolds the author used for isospectrality constructions are nothing but the Zeeman-Hamilton operators of free charged particles. These manifolds can be considered as prototypes of the so called Zeeman manifolds. This observation allows to develop a spectral theory both on open Z-manifolds and their compact submanifolds. The theory on open manifolds leads to a new nonperturbative approach to the infinities of QED. This idea exploits that the quantum Hilbert space decomposes into subspaces (Zeeman zones) which are invariant under the actions both of this Zeeman-Laplace operator and the natural complex Heisenberg group representation. Thus a well defined particle theory and zonal geometry can be developed on each zone separately. The most surprising result is that quantities divergent on the global setting are finite on the zonal setting. Even the zonal Feynman integral is well defined. The results include explicit computations of objects such as the zonal spectra, the waves defining the zonal point-spreads, the zonal Wiener-Kac resp. Dirac-Feynman flows, and the corresponding partition functions.
The observation adds new view-point also to the problem of finding intertwining operators by which isospectral pairs of metrics with different local geometries on compact submanifolds can be constructed. Among the examples the author constructed the most surprising are the isospectrality families containing both homogeneous and locally inhomogeneous metrics. The observation provides even quantum physical interpretation to the isospectrality.
The problem of infinities (divergent integrals) appearing in quantum field theory or elementary particle physics are treated by renormalization in the current theories. By this perturbative tool the desired finite quantities are produced by differences of infinities. This paper gives a new non-perturbative approach to this problem. Namely, the Hilbert space on which the quantum Hamilton operator $H$ is acting, is decomposed into subspaces, called Zeeman zones, which are invariant under the actions both of $H$ and the natural complex Heisenberg group representation. Thus well defined particle t…
The purpose of this note is to extend to any space dimension the bilinear estimate for eigenfunctions of the Laplace operator on a compact manifold (without boundary) obtained in a previous work in dimension 2. We also give some related trilinear estimates.
We give an overview over the application of functional equations, namely the classical Poincaré and renewal equations, to the study of the spectrum of Laplace operators on self-similar fractals. We compare the techniques used to those used in the euclidean situation. Furthermore, we use the obtained information on the spectral zeta function to define the Casimir energy of fractals. We give numerical values for this energy for the Sierpiński gasket.
We give a formula relating the $L^2$-isoperimetric profile to the spectral distribution of the Laplace operator associated to a finitely generated group $Γ$ or a Riemannian manifold with a cocompact, isometric $Γ$-action. As a consequence, we can apply techniques from geometric group theory to estimate the spectral distribution of the Laplace operator in terms of the growth and the Følner's function of the group, generalizing previous estimates by Gromov and Shubin. This leads, in particular, to sharp estimates of the spectral distributions for several classes of solvable groups. Furthermore, …
We prove sharp upper bounds for sums of eigenvalues (and other spectral functionals) of Laplace-like operators, including bi-Laplacian and fractional Laplacian. We show that among linear images of a highly symmetric domain, our spectral functionals are maximal on the original domain. We exploit the symmetries of the domain, and the operator, avoiding necessity of finding good test functions for variational problems. This is especially important for fractional Laplacian, since exact solutions are not even known on intervals, making it hard to find good test functions. To achieve our goals we ge…
We study the eigenvalue problem for the Dirichlet Laplacian in bounded simply connected plane domains $Ω\subset\mathbb{C}$ using conformal transformations of the original problem to the weighted eigenvalue problem for the Dirichlet Laplacian in the unit disc $\mathbb{D}$. This allows us to estimate the variation of the eigenvalues of the Dirichlet Laplacian upon domain perturbation via energy type integrals for a large class of "conformal regular" domains which includes all quasidiscs, i.e. images of the unit disc under quasiconformal homeomorphisms of the plane onto itself. Boundaries of such…
Solution of some boundary value problems and initial problems in unique ball leads to the convergence and sumability problems of Fourier series of given function by eigenfunctions of Laplace operator on a sphere - spherical harmonics. Such a series are called as Fourier-Laplace series on sphere. There are a number of works devoted investigation of these expansions in different topologies and for the functions from the various functional spaces. In the present work we consider only localization problems in both usual and generalized (almost everywhere localization) senses in the classes of summ…
Let $P$ be a non-negative self-adjoint Laplace type operator acting on sections of a hermitian vector bundle over a closed Riemannian manifold. In this paper we review the close relations between various $P$-related coefficients such as the mollified spectral counting coefficients, the heat trace coefficients, the resolvent trace coefficients, the residues of the spectral zeta function as well as certain Wodzicki residues. We then use the Wodzicki residue to obtain results about the local counting function of operators of Dirac and Laplace type. In particular, we express the second term of the…
We consider self-adjoint extensions of differential operators of the type $ (-\frac{d^2}{dr^2} + \frac{l(l+1)}{r^2})^3 $ on the real semi-axis for l=1,2 with two kinds of boundary conditions: first that nullify the value of a function and its first derivative and second that nullify the 4th (l=1) or the 3rd (l=2) derivative. We calculate the expressions for the correponding resolvents and derive spectral decompositions. These types of boundary conditions are interesting from the physical point of view, especially the second ones, which give an example of emergence of long-range action in excha…
We generalize the normalized combinatorial Laplace operator for graphs by defining two Laplace operators for hypergraphs that can be useful in the study of chemical reaction networks. We also investigate some properties of their spectra.
Several new spectral properties of the normalized Laplacian defined for oriented hypergraphs are shown. The eigenvalue $1$ and the case of duplicate vertices are discussed; two Courant nodal domain theorems are established; new quantities that bound the eigenvalues are introduced. In particular, the Cheeger constant is generalized and it is shown that the classical Cheeger bounds can be generalized for some classes of hypergraphs; it is shown that a geometric quantity used to study zonotopes bounds the largest eigenvalue from below, and that the notion of coloring number can be generalized and…
Let H be the discrete 3-dimensional Heisenberg group with the standard generators x, y, z. The element Delta of the group algebra for H of the form Delta= (x+x^{-1}+y+y^{-1})/4 is called the Laplace operator. This operator can also be defined as transition operator for random walk on the group. The spectrum of Delta in the regular representation of H is the interval [-1,1]. Let E(A), where A is a subset of [-1,1], be a family of spectral projectors for Delta and m(A)=(E(A)e, e) be the corresponding spectral measure. Here e is the characteristic function of the unit element of the group H. We e…
This paper extends Yosida's mean ergodic theorem in order to compute projections onto non-unitary eigenspaces for spectral operators of scalar-type on locally convex linear topological spaces. For spectral operators with dominating point spectrum, the projections take the form of Laplace averages, which are a generalization of the Fourier averages used when the spectrum is unitary. Inverse iteration and Laplace averages project onto eigenspaces of spectral operators with minimal point spectrum. Two classes of dynamical systems --- attracting fixed points in $\mathbb{C}^{d}$ and attracting limi…
In this article we study the top of the spectrum of the normalized Laplace operator on infinite graphs. We introduce the dual Cheeger constant and show that it controls the top of the spectrum from above and below in a similar way as the Cheeger constant controls the bottom of the spectrum. Moreover, we show that the dual Cheeger constant at infinity can be used to characterize that the essential spectrum of the normalized Laplace operator shrinks to one point.
We consider a generalized Dirac operator on a compact stratified space with an iterated cone-edge metric. Assuming a spectral Witt condition, we prove its essential self-adjointness and identify its domain and the domain of its square with weighted edge Sobolev spaces. This sharpens previous results where the minimal domain is shown only to be a subset of an intersection of weighted edge Sobolev spaces. Our argument does not rely on microlocal techniques and is very explicit. The novelty of our approach is the use of an abstract functional analytic notion of interpolation scales. Our results h…
