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We study the spectral theory for the first-order system J u ′ + q u = w f of differential equations on the real interval (a , b) where J is a constant, invertible, skew-hermitian matrix and q and w are matrices whose entries are distributions of order 0 with q hermitian and w non-negative. Specifically, we construct a generalized Weyl-Titchmarsh m -function with corresponding spectral measure τ and a generalized Fourier transform after imposing certain conditions on J , q , and w. Different conditions are motivated and studied in the later sections. A Fatou-type identity needed for our result is recorded in the appendix. [ABSTRACT FROM AUTHOR]

We present a method to estimate the number of irreducible components of the Fermi varieties of periodic Schrödinger operators on graphs in terms of suitable asymptotics. Our main theorem is an abstract bound for the number of irreducible components of Laurent polynomials in terms of such asymptotics. We then show how the abstract bound implies irreducibility in many lattices of interest, including examples with more than one vertex in the fundamental cell such as the Lieb lattice as well as certain models obtained by the process of graph decoration. [ABSTRACT FROM AUTHOR]

In this short note, we prove positivity of the Lyapunov exponent for 1D continuum Anderson models by leveraging some classical tools from inverse spectral theory. The argument is much simpler than the existing proof due to Damanik–Sims–Stolz, and it covers a wider variety of random models. Along the way we note that a Large Deviation Theorem holds uniformly on compacts. [ABSTRACT FROM AUTHOR]

This is the third and the last paper in a series of papers on spectral and scattering theory for the Schrödinger operator on a manifold possessing an escape function, for example a manifold with asymptotically Euclidean and/or hyperbolic ends. Here we discuss the time-dependent scattering theory. A long-range perturbation is allowed, and scattering by obstacles, possibly non-smooth and/or unbounded in a certain way, is included in the theory. We also resolve a conjecture by Hempel–Post–Weder on cross-ends transmissions between two or more ends, formulated in a time-dependent manner. [ABSTRACT FROM AUTHOR]

We extend some results of M. G. Krein to the class of entire functions which can be represented as ratios of discrete Cauchy transforms in the plane. As an application we obtain new versions of de Branges' Ordering Theorem for nearly invariant subspaces in a class of Hilbert spaces of entire functions. Examples illustrating sharpness of the obtained results are given. We also discuss applications to spectral theory of rank one perturbations of normal operators. [ABSTRACT FROM AUTHOR]

Abstract We provide a complete spectral analysis of all self-adjoint operators acting on ℓ 2 (Z) which are associated with two doubly infinite Jacobi matrices with entries given by q − n + 1 δ m , n − 1 + q − n δ m , n + 1 and δ m , n − 1 + α q − n δ m , n + δ m , n + 1 , respectively, where q ∈ (0 , 1) and α ∈ R. As an application, we derive orthogonality relations for the Ramanujan entire function and the third Jackson q -Bessel function. [ABSTRACT FROM AUTHOR]

The inverse problem of recovering a smooth simply connected multisheet planar domain from its Steklov spectrum is equivalent to the problem of determination, up to a gauge transform, of a smooth positive function a on the unit circle from the spectrum of the operator a Λ, where Λ is the Dirichlet-to-Neumann operator of the unit disk. Zeta-invariants are defined by Z m ( a ) = Tr [ ( a Λ ) 2 m − ( a D ) 2 m ] for every smooth function a . In the case of a positive a , zeta-invariants are determined by the Steklov spectrum. We obtain some estimate from below for Z m ( a ) in the case of a real function a . On using the estimate, we prove the compactness of a Steklov isospectral family of planar domains in the C ∞ -topology. We also describe all real functions a satisfying Z m ( a ) = 0 . [ABSTRACT FROM AUTHOR]

The spectral properties of non-self-adjoint extensions A [ B ] of a symmetric operator in a Hilbert space are studied with the help of ordinary and quasi boundary triples and the corresponding Weyl functions. These extensions are given in terms of abstract boundary conditions involving an (in general non-symmetric) boundary operator B . In the abstract part of this paper, sufficient conditions for sectoriality and m-sectoriality as well as sufficient conditions for A [ B ] to have a non-empty resolvent set are provided in terms of the parameter B and the Weyl function. Special attention is paid to Weyl functions that decay along the negative real line or inside some sector in the complex plane, and spectral enclosures for A [ B ] are proved in this situation. The abstract results are applied to elliptic differential operators with local and non-local Robin boundary conditions on unbounded domains, to Schrödinger operators with δ -potentials of complex strengths supported on unbounded hypersurfaces or infinitely many points on the real line, and to quantum graphs with non-self-adjoint vertex couplings. [ABSTRACT FROM AUTHOR]

For analytic operator functions, we prove accumulation of branches of complex eigenvalues to the essential spectrum. Moreover, we show minimality and completeness of the corresponding system of eigenvectors and associated vectors. These results are used to prove sufficient conditions for eigenvalue accumulation to the poles and to infinity of rational operator functions. Finally, an application of electromagnetic field theory is given. [ABSTRACT FROM AUTHOR]

In this paper, we study a classical two species Lotka–Volterra competition–diffusion–advection system, where the diffusion and advection rates of two competitors are supposed to be proportional. By employing the principal spectral theory, we first establish a key a priori estimate on the co-existence (positive) steady state, which is a powerful tool to link the local and global dynamics. We then further present a complete classification on all possible long-time dynamical behaviors by appealing to the theory of monotone dynamical systems. Lastly, we apply these results to a special situation where two species are competing for the same resources and obtain a sharp criteria in term of certain variable parameters for all kinds of global dynamics. This work gives a positive answer to the conjecture proposed by Lou et al. in [34] by considering a more general model under certain conditions, and also, can be seen as a further development of He and Ni [19] for competition–diffusion system, where we bring new ingredients in the arguments to overcome the difficulty caused by the involvement of advection. [ABSTRACT FROM AUTHOR]

A one-channel operator is a self-adjoint operator on ℓ 2 ( G ) for some countable set G with a rank 1 transition structure along the sets of a quasi-spherical partition of G . Jacobi operators are a very special case. In essence, there is only one channel through which waves can travel across the shells to infinity. This channel can be described with transfer matrices which include scattering terms within the shells and connections to neighboring shells. Not all of the transfer matrices are defined for some countable set of energies. Still, many theorems from the world of Jacobi operators are translated to this setup. The results are then used to show absolutely continuous spectrum for the Anderson model on certain finite dimensional graphs with a one-channel structure. This result generalizes some previously obtained results on antitrees. [ABSTRACT FROM AUTHOR]

We are concerned with the geometric properties of the surface plasmon resonance (SPR). SPR is a non-radiative electromagnetic surface wave that propagates in a direction parallel to the negative permittivity/dielectric material interface. It is known that the SPR oscillation is very sensitive to the material interface. However, we show that the SPR oscillation asymptotically localizes at places with high magnitude of curvature in a certain sense under an assumption equivalent to convexity in the three-dimensional setting. Our work leverages the Heisenberg picture of quantization and quantum ergodicity first derived by Shnirelman, Zelditch, Colin de Verdière and Helffer-Martinez-Robert, as well as certain novel and more general ergodic properties of the Neumann-Poincaré operator to analyze the SPR field, which are of independent interest to the spectral theory and the potential theory. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the behavior of the eigenvalues of a magnetic Aharonov–Bohm operator with half-integer circulation and Dirichlet boundary conditions in a bounded planar domain. We establish a sharp relation between the rate of convergence of the eigenvalues as the singular pole is approaching a boundary point and the number of nodal lines of the eigenfunction of the limiting problem, i.e. of the Dirichlet-Laplacian, ending at that point. The proof relies on the construction of a limit profile depending on the direction along which the pole is moving, and on an Almgren-type monotonicity argument for magnetic operators. [ABSTRACT FROM AUTHOR]

Let Φ be a family of functions analytic in some neighborhood of a complex domain Ω, and let T be a Hilbert space operator whose spectrum is contained in Ω ‾ . Our typical result shows that under some extra conditions, if the closed unit disc is complete K ′ -spectral for φ ( T ) for every φ ∈ Φ , then Ω ‾ is complete K -spectral for T for some constant K . In particular, we prove that under a geometric transversality condition, the intersection of finitely many K ′ -spectral sets for T is again K -spectral for some K ≥ K ′ . These theorems generalize and complement results by Mascioni, Stessin, Stampfli, Badea–Beckermann–Crouzeix and others. We also extend to non-convex domains a result by Putinar and Sandberg on the existence of a skew dilation of T to a normal operator with spectrum in ∂Ω. As a key tool, we use the results from our previous paper [11] on traces of analytic uniform algebras. [ABSTRACT FROM AUTHOR]

We study spectral properties of convolution operators L and their perturbations H = L + v ( x ) by compactly supported potentials. Results are applied to determine the front propagation of a population density governed by operator H with a compactly supported initial density provided that H has positive eigenvalues. If there is no positive spectrum, then the stabilization of the population density is proved. [ABSTRACT FROM AUTHOR]

In this work, we prove a bound on the multiplicity of the singular spectrum for a certain class of Anderson Hamiltonians. The operator in consideration is the form H ω = Δ + ∑ n ∈ Z d ω n P n on the Hilbert space ℓ 2 ( Z d ) , where Δ is discrete laplacian, P n are projection onto ℓ 2 ( { x ∈ Z d : n i l i < x i ≤ ( n i + 1 ) l i } ) for some l 1 , ⋯ , l d ∈ N and { ω n } n are i.i.d. real bounded random variables following an absolutely continuous distribution. We prove that the multiplicity of the singular spectrum is bounded above by 2 d − d independent of { l i } i = 1 d . When l i + 1 ∉ 2 N ∪ 3 N for all i and g c d ( l i + 1 , l j + 1 ) = 1 for i ≠ j , the singular spectrum is also simple. [ABSTRACT FROM AUTHOR]

We show that a generic element of a space of limit-periodic CMV operators has zero-measure Cantor spectrum. We also prove a Craig–Simon type theorem for the density of states measure associated with a stochastic family of CMV matrices and use our construction from the first part to prove that the Craig–Simon result is optimal in general. We discuss applications of these results to a quantum walk model where the coins are arranged according to a limit-periodic sequence. The key ingredient in these results is a new formula which may be viewed as a relationship between the density of states measure of a CMV matrix and its Schur function. [ABSTRACT FROM AUTHOR]

Given a finite alphabet Λ, and a not necessarily finite type subshift X ⊆ Λ ∞ , we introduce a partial action of the free group F ( Λ ) on a certain compactification Ω X of X , which we call the spectral partial action. The space Ω X has already appeared in many papers in the subject, arising as the spectrum of a commutative C*-algebra usually denoted by D X . A good understanding of D X is crucial for the study of C*-algebras related to subshifts, and since the descriptions given of Ω X in the literature are often somewhat terse and obscure, one of our main goals is to present a sensible model for it which allows for a detailed study of its structure, as well as of the spectral partial action, from various points of view, including topological freeness and minimality. We then apply our results to study certain C*-algebras associated to X , introduced by Matsumoto and Carlsen. Thus the spectral partial action permits us to endow the Carlsen–Matsumoto C*-algebra O X with a partial crossed product structure. We combine this with our characterization of the dynamical properties of the spectral partial action, in order to treat the problem of simplicity of O X , considered earlier by several authors. As a new advance, we are able to give necessary and sufficient conditions for O X to be simple, without imposing any restriction on X , and this is done in terms of transparent “graphical” properties of X . As a by-product of our partial action approach, we easily recover some known facts on O X , putting them more in line with mainstream techniques used to treat similar C*-algebras. [ABSTRACT FROM AUTHOR]

The construction of the free Banach lattice generated by a real Banach space is extended to the complex setting. It is shown that for every complex Banach space E there is a complex Banach lattice FBL C [ E ] containing a linear isometric copy of E and satisfying the following universal property: for every complex Banach lattice X C , every operator T : E → X C admits a unique lattice homomorphic extension T ˆ : FBL C [ E ] → X C with ‖ T ˆ ‖ = ‖ T ‖. The free complex Banach lattice FBL C [ E ] is shown to have analogous properties to those of its real counterpart. However, examples of non-isomorphic complex Banach spaces E and F can be given so that FBL C [ E ] and FBL C [ F ] are lattice isometric. The spectral theory of induced lattice homomorphisms on FBL C [ E ] is also explored. [ABSTRACT FROM AUTHOR]

We show that the C ⁎ -algebra generated by the field operators associated to a symplectic space Ξ is graded by the semilattice of all finite dimensional subspaces of Ξ. If Ξ is finite dimensional we give a simple intrinsic description of the components of the grading, we show that the self-adjoint operators affiliated to the algebra have a many channel structure similar to that of N-body Hamiltonians, in particular their essential spectrum is described by a kind of HVZ theorem, and we point out a large class of operators affiliated to the algebra. [ABSTRACT FROM AUTHOR]

In this paper we study k -equivariant wave maps from the hyperbolic plane into the 2-sphere as well as the energy critical equivariant S U ( 2 ) Yang–Mills problem on 4-dimensional hyperbolic space. The latter problem bears many similarities to a 2-equivariant wave map into a surface of revolution. As in the case of 1-equivariant wave maps considered in [9] , both problems admit a family of stationary solutions indexed by a parameter that determines how far the image of the map wraps around the target manifold. Here we show that if the image of a stationary solution is contained in a geodesically convex subset of the target, then it is asymptotically stable in the energy space. However, for a stationary solution that covers a large enough portion of the target, we prove that the Schrödinger operator obtained by linearizing about such a harmonic map admits a simple positive eigenvalue in the spectral gap. As there is no a priori nonlinear obstruction to asymptotic stability, this gives evidence for the existence of metastable states (i.e., solutions with anomalously slow decay rates) in these simple geometric models. [ABSTRACT FROM AUTHOR]

In this paper we extend the H ∞ functional calculus to quaternionic operators and to n -tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated functional calculus, called S -functional calculus. The S -functional calculus has two versions: one for quaternionic-valued functions and one for Clifford algebra-valued functions and can be considered the Riesz–Dunford functional calculus based on slice hyperholomorphicity, because it shares with it the most important properties. The S -functional calculus is based on the notion of S -spectrum which, in the case of quaternionic normal operators on a Hilbert space, is also the notion of spectrum that appears in the quaternionic spectral theorem. The main purpose of this paper is to construct the H ∞ functional calculus based on the notion of S -spectrum for both quaternionic operators and for n -tuples of noncommuting operators. We remark that the H ∞ functional calculus for ( n + 1 ) -tuples of operators applies, in particular, to the Dirac operator. [ABSTRACT FROM AUTHOR]

We study the point spectrum of the nonlinear Dirac equation in any spatial dimension, linearized at one of the solitary wave solutions. We prove that, in any dimension, the linearized equation has no embedded eigenvalues in the part of the essential spectrum beyond the embedded thresholds. We then prove that the birth of point eigenvalues with nonzero real part (the ones which lead to linear instability) from the essential spectrum is only possible from the embedded eigenvalues or thresholds, and therefore can not take place beyond the embedded thresholds. We also prove that “in the nonrelativistic limit” ω → m , the point eigenvalues can only accumulate to 0 and ± 2 m i . [ABSTRACT FROM AUTHOR]

We completely determine the spectra of composition operators induced by linear fractional self-maps of the unit disc acting on weighted Dirichlet spaces; extending earlier results by Higdon [8] and answering the open questions in this context. [ABSTRACT FROM AUTHOR]

For a class of Sierpinski-type self-affine measures on R 2 , a characterization for all maximal orthogonal sets of exponentials is given. By this characterization, a sufficient condition for a maximal orthogonal set to be a basis of L 2 ( μ ) is given. [ABSTRACT FROM AUTHOR]

We propose a construction for spectral triple on algebras associated with subshifts. One-dimensional subshifts provide concrete examples of Z -actions on Cantor sets. The C ⁎ -algebra of this dynamical system is generated by functions in C ( X ) and a unitary element u implementing the action. Building on ideas of Christensen and Ivan, we give a construction of a family of spectral triples on the commutative algebra C ( X ) . There is a canonical choice of eigenvalues for the Dirac operator D which ensures that [ D , u ] is bounded, so that it extends to a spectral triple on the crossed product. We study the summability of this spectral triple, and provide examples for which the Connes distance associated with it on the commutative algebra is unbounded, and some for which it is bounded. We conjecture that our results on the Connes distance extend to the spectral triple defined on the noncommutative algebra. [ABSTRACT FROM AUTHOR]

We establish an algorithm for a criterion of the diagonalisability of a matrix over a local field by a unitary matrix. For this sake, we define the notion of normality of a p -adic operator, and give several criteria for the normality. We study the relation between the normality and the reduction. In the finite dimensional case, the normality of an operator is equivalent to the diagonalisability of a matrix by a unitary matrix. Therefore we also study the relation between the diagonalisability and the reduction. For example, we show that the diagonalisation of the reduction gives a partition of unity corresponding to the reduction of the spectrum, which gives a functorial lift of the eigenspace decomposition of the reduction. [ABSTRACT FROM AUTHOR]

On R + d , endowed with the Laguerre probability measure μ α , we define a Hodge–Laguerre operator L α = δ δ ⁎ + δ ⁎ δ acting on differential forms. Here δ is the Laguerre exterior differentiation operator, defined as the classical exterior differential, except that the partial derivatives ∂ x i are replaced by the ‘‘Laguerre derivatives’’ x i ∂ x i , and δ ⁎ is the adjoint of δ with respect to inner product on forms defined by the Euclidean structure and the Laguerre measure μ α . We prove dimension-free bounds on L p , 1 < p < ∞ , for the Riesz transforms δ L α − 1 / 2 and δ ⁎ L α − 1 / 2 . As applications we prove the strong Hodge–de Rahm–Kodaira decomposition for forms in L p and deduce existence and regularity results for the solutions of the Hodge and de Rham equations in L p . We also prove that for suitable functions m the operator m ( L α ) is bounded on L p , 1 < p < ∞ . [ABSTRACT FROM AUTHOR]

We study the canonical heat flow (H t) t ≥ 0 on the cotangent module L 2 (T ⁎ M) over an RCD (K , ∞) space (M , d , m) , K ∈ R. We show Hess–Schrader–Uhlenbrock's inequality and, if (M , d , m) is also an RCD ⁎ (K , N) space, N ∈ (1 , ∞) , Bakry–Ledoux's inequality for (H t) t ≥ 0 w.r.t. the heat flow (P t) t ≥ 0 on L 2 (M). Variable versions of these estimates are discussed as well. In conjunction with a study of logarithmic Sobolev inequalities for 1-forms, the previous inequalities yield various L p -properties of (H t) t ≥ 0 , p ∈ [ 1 , ∞ ]. Then we establish explicit inclusions between the spectrum of its generator, the Hodge Laplacian Δ → , of the negative functional Laplacian −Δ, and of the Schrödinger operator − Δ + K. In the RCD ⁎ (K , N) case, we prove compactness of Δ → − 1 if M is compact, and the independence of the L p -spectrum of Δ → on p ∈ [ 1 , ∞ ] under a volume growth condition. We terminate by giving an appropriate interpretation of a heat kernel for (H t) t ≥ 0. We show its existence in full generality without any local compactness or doubling, and derive fundamental estimates and properties of it. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following problem involving fractional Laplacian operator: (1) ( − Δ ) α u = | u | 2 α ⁎ − 2 − ε u + λ u in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain in R N , ε ∈ [ 0 , 2 α ⁎ − 2 ) , 0 < α < 1 , 2 α ⁎ = 2 N N − 2 α , and ( − Δ ) α is either the spectral fractional Laplacian or the restricted fractional Laplacian. We show for problem (1) with the spectral fractional Laplacian that for any sequence of solutions u n of (1) corresponding to ε n ∈ [ 0 , 2 α ⁎ − 2 ) , satisfying ‖ u n ‖ H ≤ C in the Sobolev space H defined in (1.2) , u n converges strongly in H provided that N > 6 α and λ > 0 . The same argument can also be used to obtain the same result for the restricted fractional Laplacian. An application of this compactness result is that problem (1) possesses infinitely many solutions under the same assumptions. [ABSTRACT FROM AUTHOR]

We prove three results giving sufficient and/or necessary conditions for discreteness of the spectrum of Schrödinger operators with non-negative matrix-valued potentials, i.e., operators acting on ψ ∈ L 2 ( R n , C d ) by the formula H V ψ : = − Δ ψ + V ψ , where the potential V takes values in the set of non-negative Hermitian d × d matrices. The first theorem provides a characterization of discreteness of the spectrum when the potential V is in a matrix-valued A ∞ class, thus extending a known result in the scalar case ( d = 1 ). We also discuss a subtlety in the definition of the appropriate matrix-valued A ∞ class. The second result is a sufficient condition for discreteness of the spectrum, which allows certain degenerate potentials, i.e., such that det ⁡ ( V ) ≡ 0 . To formulate the condition, we introduce a notion of oscillation for subspace-valued mappings. Our third and last result shows that if V is a 2 × 2 real polynomial potential, then − Δ + V has discrete spectrum if and only if the scalar operator − Δ + λ has discrete spectrum, where λ ( x ) is the minimal eigenvalue of V ( x ) . [ABSTRACT FROM AUTHOR]

In this paper, we consider spectral properties of Riesz product measures supported on homogeneous Cantor sets and we show the existence of spectral measures with arbitrary Hausdorff dimensions, including non-atomic zero-dimensional spectral measures and one-dimensional singular spectral measures. [ABSTRACT FROM AUTHOR]

Let A be a regular and semisimple commutative Banach algebra with structure space Δ ( A ) . Continuing the investigations of [7] and [9] , we establish various results on weak spectral sets in Δ ( A ) . To each closed subset of Δ ( A ) we associate a descending sequence of subsets of Δ ( A ) which proves to be a powerful tool in the study of weak spectral synthesis. Applications concern injection type properties, unions of weak spectral sets and projective tensor products. A number of interesting examples is discussed: algebras of m -times continuously differentiable functions and of Lipschitz functions, and L 1 ( R N ) . [ABSTRACT FROM AUTHOR]

The isospectral renormalization group is a powerful method to analyze the spectrum of operators in quantum field theory. It was introduced in 1995 (see [2,4] ) and since then it has been used to prove several results for non-relativistic quantum electrodynamics. After the introduction of the method there have been many works in which extensions, simplifications or clarifications are presented (see [7,11,13] ). In this paper we present a new approach in which we construct a flow of operators parametrized by a continuous variable in the positive real axis. While this is in contrast to the discrete iteration used before, this is more in spirit of the original formulation of the renormalization group introduced in theoretical physics in 1974 [22] . The renormalization flow that we construct can be expressed in a simple way: it can be viewed as a single application of the Feshbach–Schur map with a clever selection of the spectral parameter. Another advantage of the method is that there exists a flow function for which the renormalization group that we present is the orbit under this flow of an initial Hamiltonian. This opens the possibility to study the problem using different techniques coming from the theory of evolution equations. [ABSTRACT FROM AUTHOR]

For approximation numbers a n ( C φ ) of composition operators C φ on weighted analytic Hilbert spaces, including the Hardy, Bergman and Dirichlet cases, with symbol φ of uniform norm <1, we prove that lim n → ∞ ⁡ [ a n ( C φ ) ] 1 / n = e − 1 / Cap [ φ ( D ) ] , where Cap [ φ ( D ) ] is the Green capacity of φ ( D ) in D . This formula holds also for H p with 1 ≤ p < ∞ . [ABSTRACT FROM AUTHOR]

We study a neutral atom with a non-vanishing electric dipole moment coupled to the quantized electromagnetic field. For a sufficiently small dipole moment and small momentum, p → , the one-particle (self-)energy of an atom is proven to be a real-analytic function of its momentum. The main ingredient of our proof is a suitable form of the Feshbach–Schur spectral renormalization group. [ABSTRACT FROM AUTHOR]

A new technique for approximating eigenvalues and eigenvectors of a self-adjoint operator is presented. The method does not incur spectral pollution, uses trial spaces from the form domain, has a self-adjoint algorithm, and exhibits superconvergence. [ABSTRACT FROM AUTHOR]

Given a Banach space X and an operator T ∈ L ( X ) , an almost-invariant half-space (AIHS) is a closed subspace Y satisfying T Y ⊆ Y + F for some finite-dimensional subspace F and dim ⁡ ( Y ) = dim ⁡ ( X / Y ) = ∞ . In this paper we study the connection between AIHS and noncompactness of some subsets of the unit ball, chains of T -invariant subspaces, and different parts of spectrum of T and T ⁎ . We describe a simple template for such subspaces and show that if T is quasinilpotent or weakly compact then it admits an AIHS Y . [ABSTRACT FROM AUTHOR]

Abstract: The spectra of parallel flows (that is, flows governed by first-order differential operators parallel to one direction) are investigated, on both spaces and weighted- spaces. As a consequence, an example of a flow admitting a purely singular continuous spectrum is provided. For flows admitting more regular spectra the density of states is analyzed, and spaces on which it is uniformly bounded are identified. As an application, an ergodic theorem with uniform convergence is proved. [Copyright &y& Elsevier]

Abstract: We construct a family of spectral triples for the Sierpinski gasket K. For suitable values of the parameters, we determine the dimensional spectrum and recover the Hausdorff measure of K in terms of the residue of the volume functional at its abscissa of convergence , which coincides with the Hausdorff dimension of the fractal. We determine the associated Connes' distance showing that it is bi-Lipschitz equivalent to the distance on K induced by the Euclidean metric of the plane, and show that the pairing of the associated Fredholm module with (odd) K-theory is non-trivial. When the parameters belong to a suitable range, the abscissa of convergence of the energy functional takes the value , which we call energy dimension, and the corresponding residue gives the standard Dirichlet form on K. [Copyright &y& Elsevier]

Abstract: We present a study of what may be called an intrinsic metric for a general regular Dirichlet form. For such forms we then prove a Rademacher type theorem. For strongly local forms we show existence of a maximal intrinsic metric (under a weak continuity condition) and for Dirichlet forms with an absolutely continuous jump kernel we characterize intrinsic metrics by bounds on certain integrals. We then turn to applications on spectral theory and provide for (measure perturbation of) general regular Dirichlet forms an Allegretto–Piepenbrink type theorem, which is based on a ground state transform, and a Shnol' type theorem. Our setting includes Laplacian on manifolds, on graphs and α-stable processes. [Copyright &y& Elsevier]

Abstract: Let be a probability space, and let P be a Markov operator on with 1 a simple eigenvalue such that (i.e. μ is an invariant probability measure of P). Then has a spectral gap, i.e. 1 is isolated in the spectrum of , if and only if This strengthens a conjecture of Simon and Høegh-Krohn on the spectral gap for hyperbounded operators solved recently by L. Miclo in [10]. Consequently, for a symmetric, conservative, irreducible Dirichlet form on , a Poincaré/log-Sobolev type inequality holds if and only if so does the corresponding defective inequality. Extensions to sub-Markov operators and non-conservative Dirichlet forms are also presented. [Copyright &y& Elsevier]

Abstract: Inspired by the insightful article [4], we revisit the Nualart–Peccati criterion [13] (now known as the Fourth Moment Theorem) from the point of view of spectral theory of general Markov diffusion generators. We are not only able to drastically simplify all of its previous proofs, but also to provide new settings of diffusive generators (Laguerre, Jacobi) where such a criterion holds. Convergence towards Gamma and Beta distributions under moment conditions is also discussed. [Copyright &y& Elsevier]

Abstract: Following the pioneering work of Duistermaat and Grünbaum, we call a family of polynomials bispectral, if the polynomials are simultaneously eigenfunctions of two commutative algebras of operators: one consisting of difference operators acting on the degree index n, and another one of operators acting on the variable x. The goal of the present paper is to construct and parametrize bispectral extensions of the Askey–Wilson polynomials, where the second algebra consists of q-difference operators. In particular, we describe explicitly measures on the real line for which the corresponding orthogonal polynomials satisfy (higher-order) q-difference equations extending all known families of orthogonal polynomials satisfying q-difference, difference or differential equations in x. [Copyright &y& Elsevier]

Abstract: We study the liberation process for projections: where is a free unitary Brownian motion freely independent from . Its action on the operator-valued angle between the projections induces a flow on the corresponding spectral measures ; we prove that the Cauchy transform of the measure satisfies a holomorphic PDE. We develop a theory of subordination for the boundary values of this PDE, and use it to show that the spectral measure possesses a piecewise analytic density for any and any initial projections of trace . We us this to prove the Unification Conjecture for free entropy and information in this trace setting. [Copyright &y& Elsevier]

Abstract: We study the spectral disjointness of the powers of a rank-one transformation. For a large class of rank-one constructions, including those for which the cutting and stacking parameters are bounded, and other examples such as rigid generalized Chaconʼs maps and Katokʼs map, we prove that different positive powers of the transformation are pairwise spectrally disjoint on the continuous part of the spectrum. Our proof involves the existence, in the weak closure of , of “sufficiently many” analytic functions of the operator . Then we apply these disjointness results to prove Sarnakʼs conjecture for the (possibly non-uniquely ergodic) symbolic models associated to these rank-one constructions: All sequences realized in these models are orthogonal to the Möbius function. [Copyright &y& Elsevier]

Abstract: Let L be a non-negative self-adjoint operator acting on where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup which satisfies generalized m-th order Gaussian estimates. In this article, we study singular and dyadically supported spectral multipliers for abstract self-adjoint operators. We show that in this setting sharp spectral multiplier results follow from Plancherel or Stein–Tomas type estimates. These results are applicable to spectral multipliers for a large class of operators including m-th order elliptic differential operators with constant coefficients, biharmonic operators with rough potentials and Laplace type operators acting on fractals. [Copyright &y& Elsevier]

Abstract: Let be a sequence of digit sets in and let be a sequence of integer numbers bigger than 1. We call the family a Moran iterated function system (IFS), which is a natural generalization of an IFS. We prove, under certain conditions in terms of , that the associated Moran measure μ is a spectral measure, i.e., there exists a countable set such that is an orthonormal basis for . [Copyright &y& Elsevier]