Infinite Multiplicity Research Articles

Stability property and essential spectrum of linear retarded functional differential equations

In this work, we shall study stability and stabilization of a class of retarded functional differential equations in Banach spaces. We present sufficient (and necessary, sometimes) conditions for weak, asymptotic and exponential stability properties. It is shown that retarded growth bounds are totally determined by retarded non-isolated spectra, isolated eigenvalues with infinite algebraic multiplicity and spectral bounds. Stabilization problem via compact perturbations, a topic which is practically useful in the context of control theory of dynamical systems, is considered as well.

Infinite multiplicity for an inhomogeneous supercritical problem in entire space

Let $K(x)$ be a positive function in $R^N, N \geq 3$ and satisfy$\lim\limits_{|x|\rightarrow \infty} K(x) = K_\infty$ where$K_\infty$ is a positive constant. When $p > \frac{N + 1}{N - 3}, N\geq 4$, we prove the existence of infinitely many positivesolutions to the following supercritical problem: \begin{eqnarray*}\Delta u(x) + K(x)u^p = 0, u>0 \quad in \quad R^N, \lim_{|x|\rightarrow \infty} u(x) = 0.\end{eqnarray*} If in addition we have, for instance, $\lim\limits_{|x| \rightarrow\infty}|x|^\mu (K(x) - K_\infty ) = C_0 \neq 0, 0 \frac{N + 2}{N - 2}$.

Some Remarks on the Eigenvalue Multiplicities of the Laplacian on Infinite Locally Finite Trees

We consider the continuous Laplacian on an infinite uniformly locally finite network under natural transition conditions as continuity at the ramification nodes and the classical Kirchhoff flow condition at all vertices in a L ∞-setting. The characterization of eigenvalues of infinite multiplicity for trees with finitely many boundary vertices (von Below and Lubary, Results Math 47:199–225, 2005, 8.6) is generalized to the case of infinitely many boundary vertices. Moreover, it is shown that on a tree, any eigenspace of infinite dimension contains a subspace isomorphic to $${\ell^\infty({\mathbb N})}$$ . As for the zero eigenvalue, it is shown that a locally finite tree either is a Liouville space or has infinitely many linearly independent bounded harmonic functions if the edge lengths do not shrink to zero anywhere. This alternative is shown to be false on graphs containing circuits.

Optimal transport and dynamics of expanding circle maps acting on measures

AbstractIn this paper we compute the derivative of the action on probability measures of an expanding circle map at its absolutely continuous invariant measure. The derivative is defined using optimal transport: we use the rigorous framework set up by Gigli to endow the space of measures with a kind of differential structure. It turns out that 1 is an eigenvalue of infinite multiplicity of this derivative, and we deduce that the absolutely continuous invariant measure can be deformed in many ways into atomless, nearly invariant measures. We also show that the action of standard self-covering maps on measures has positive metric mean dimension.

Spectral analysis on infinite Sierpiński fractafolds

A fractafold, a space that is locally modeled on a specified fractal, is the fractal equivalent of a manifold. For compact fractafolds based on the Sierpinski gasket, it was shown by the first author how to compute the discrete spectrum of the Laplacian in terms of the spectrum of a finite graph Laplacian. A similar problem was solved by the second author for the case of infinite blowups of a Sierpinski gasket, where spectrum is pure point of infinite multiplicity. Both works used the method of spectral decimations to obtain explicit description of the eigenvalues and eigenfunctions. In this paper we combine the ideas from these earlier works to obtain a description of the spectral resolution of the Laplacian for noncompact fractafolds. Our main abstract results enable us to obtain a completely explicit description of the spectral resolution of the fractafold Laplacian. For some specific examples we turn the spectral resolution into a "Plancherel formula". We also present such a formula for the graph Laplacian on the 3-regular tree, which appears to be a new result of independent interest. In the end we discuss periodic fractafolds and fractal fields.

Quantum Geometry on Quantum Spacetime: Distance, Area and Volume Operators

We develop the first steps towards an analysis of geometry on the quantum spacetime proposed in Doplicher et al. (Commun Math Phys 172:187–220, 1995). The homogeneous elements of the universal differential algebra are naturally identified with operators living in tensor powers of Quantum Spacetime; this allows us to compute their spectra. In particular, we consider operators that can be interpreted as distances, areas, 3- and 4-volumes. The Minkowski distance operator between two independent events is shown to have pure Lebesgue spectrum with infinite multiplicity. The Euclidean distance operator is shown to have spectrum bounded below by a constant of the order of the Planck length. The corresponding statement is proved also for both the space-space and space-time area operators, as well as for the Euclidean length of the vector representing the 3-volume operators. However, the space 3-volume operator (the time component of that vector) is shown to have spectrum equal to the whole complex plane. All these operators are normal, while the distance operators are also selfadjoint. The Lorentz invariant spacetime volume operator, representing the 4-volume spanned by five independent events, is shown to be normal. Its spectrum is pure point with a finite distance (of the order of the fourth power of the Planck length) away from the origin. The mathematical formalism apt to these problems is developed and its relation to a general formulation of Gauge Theories on Quantum Spaces is outlined. As a byproduct, a Hodge Duality between the absolute differential and the Hochschild boundary is pointed out.

On the entropy of actions of nilpotent Lie groups and their lattice subgroups

AbstractWe consider a natural class $\mathcal {ULG}$ of connected, simply connected nilpotent Lie groups which contains ℝn, the group $\mathcal {UT}_n(\mathbb {R})$ of all triangular unipotent matrices over ℝ and many of its subgroups, and is closed under direct products. If $G \in \mathcal {ULG}$, then $\Gamma _1 = G\cap \mathcal {UT}_n(\mathbb {Z})$ is a lattice subgroup of G. We prove that if $G \in \mathcal {ULG}$ and Γ is a lattice subgroup of G, then a free ergodic measure-preserving action T of G on a probability space (X,ℬ,μ) has completely positive entropy (CPE) if and only if the restriction TΓ of T to Γ has CPE. We can deduce from this the following version of a well-known conjecture in this case: the action T has CPE if and only if T is uniformly mixing. Moreover, such T has a Lebesgue spectrum with infinite multiplicity. We further consider an ergodic free action T with positive entropy and suppose TΓ is ergodic for any lattice subgroup Γ of G. This holds, in particular, if the spectrum of T does not contain a discrete component. Then we show the Pinsker algebra Π(T) of T exists and coincides with the Pinsker algebras Π(TΓ) of TΓ for any lattice subgroup Γ of G. In this case, T always has Lebesgue spectrum with infinite multiplicity on the space ℒ20(X,μ)⊖ℒ20(Π(T)) , where ℒ20(Π(T)) contains all Π(T) -measurable functions from ℒ20(X,μ) . To prove these results, we use the following formula: h(T)=∣G(Γ)∣−1hK (TΓ) , where h(T) is the Ornstein–Weiss entropy of T, hK (TΓ) is a Kolmogorov–Sinai entropy of TΓ, and the number ∣G(TΓ)∣ is the Haar measure of the compact subset G(Γ) of G. In particular, h(T)=hK (TΓ1) , and hK (TΓ1)=∣G(Γ)∣−1hK (TΓ) . The last relation is an analogue of the Abramov formula for flows.

The current system of natural minerals and prospects for its expansion

The principle of an unlimited number of mineral species (NMS) is stated. Its scope includes unique types of rocks and ores with their structurally and compositionally complex minerals, in particular nanometersized ones. The unlimited NMS is determined not so much by the vast number of possible combinations of chemical elements as by the infinite multiplicity of structural forms of these combinations, and is associated not so much with the macro- as with the micro-world of minerals, which is still in the early stages of investigation. The number of IMA-approved mineral species is currently about 4300. Predictions for the next few decades suggest that their number will increase to 5000 by 2020 and to 10000–11000 by 2050.

Infinite multiplicity for an inhomogeneous supercritical problem in entire space

In this paper, we will prove the existence of infinitely many positive solutions to the following supercritical problem by using the Liapunov-Schmidt reduction method and asymptotic analysis: { Δ u + u p + f ( x ) = 0 , u > 0 in R n , lim | x | → ∞ u ( x ) → 0. \begin{eqnarray*} \left \{ \begin {array}{ll} \Delta u + u^{p}+f(x)=0,\ \ u>0\ \ \mbox {in}\ \mathbb {R}^{n},\\ \lim _{|x|\to \infty }u(x)\to 0. \end{array} \right . \end{eqnarray*}

Friction-induced reverse chatter in rigid-body mechanisms with impacts

The focus of this paper is on the possibility of formulating a consistent and unambiguous forward simulation model of planar rigid-body mechanical systems with isolated points of intermittent or sustained contact with rigid constraining surfaces in the presence of dry friction. In particular, the analysis considers paradoxical ambiguities associated with the coexistence of sustained contact and one or several alternative forward trajectories that include phases of free-flight motion. Special attention is paid to the so-called Painleve paradoxes where sustained contact is possible even if the contact-independent contribution to the normal acceleration would cause contact to cease. Here, through taking the infinite-stiffness limit of a compliant contact model, the ambiguity in the case of a condition of sustained stick is resolved in favour of sustained contact, whereas the ambiguity in the case of a condition of sustained slip is resolved by eliminating the possibility of reaching such a condition from an open set of initial conditions. A more significant challenge to the goal of an unambiguous forward simulation model is afforded by the discovery of open sets of initial conditions and parameter values associated with the possibility of a left accumulation point of impacts or reverse chatter—a transition to free flight through an infinite sequence of impacts with impact times accumulating from the right on a limit point and with impact velocities diverging exponentially away from the limit point, even where the contact-independent normal acceleration supports sustained contact. In this case, the infinite-stiffness limit of the compliant formulation establishes that, under a specific set of open conditions, the possibility of reverse chatter in the rigid-contact model is an irresolvable ambiguity in the forward dynamics based at the terminal point of a phase of sustained slip. Indeed, as the existence of a left accumulation point of impacts is associated with a one-parameter family of possible forward trajectories, the ambiguity is of infinite multiplicity. The conclusions of the theoretical analysis are illustrated and validated through numerical analysis of an example single-rigid-body mechanical model.

Bernays, Dooyeweerd and Gödel – the remarkable convergence in their reflections on the foundations of mathematics

In spite of differences the thought of Bernays, Dooyeweerd and Gödel evinces a remarkable convergence. This is particularly the case in respect of the acknowledgment of the difference between the discrete and the continuous, the foundational position of number and the fact that the idea of continuity is derived from space (geometry–Bernays). What is furthermore similar is the recognition of what is primitive (and indefinable) as well as the account of the coherence of what is unique, such as when Gödel observes something quasi-spatial in the character of sets. It is shown that Dooye-weerd’s theory of modal aspects provides a philosophical framework that exceeds his own restrictive understanding of infinity (to the potential infinite) and at the same time makes it possible to account for key insights found in the thought of Bernays and Gödel. When Laugwitz says that discreteness rules within the sphere of the numerical, he says nothing more than what Dooyeweerd had in mind with his idea that discrete quantity, as the meaning-nucleus of the arithmetical aspect, qualifies every element within the structure of the quantitative aspect. And when Bernays says that analysis expresses the idea of the continuum in arithmetical language his mode of speech is equivalent to saying that mathematical analysis could seen as being founded upon the spatial anticipation within the modal structure of the arithmetical aspect. The view of the actual infinite (the at once infinite) in terms of an “as if” approach (Bernays), that is, as appreciated as a regulative hypothesis through which every successively infinite multiplicity of numbers could be envisaged as being giving all at once as an infinite totality, provides a sound understanding of the at once infinite and makes it plain why every form of arithmeticism fails. Such attempts have to call upon Cantor’s proof of the non-denumerability of the real numbers–and this proof pre-supposes the use of the at once infinite which, in turn, pre-supposes the (irreducibile) spatial order of simultaneity and the spatial whole-parts relation.

A magnetic Schrödinger operator on a periodic graph

This paper looks at a magnetic Shrödinger operator on a graph of special form in . It is called an armchair graph because graphs of this form with operators on them are used as a possible model for the so-called armchair nanotube in the homogeneous magnetic field which has amplitude and is parallel to the axis of the nanotube. The spectrum of the operator in question consists of an absolutely continuous part (spectral bands, separated by gaps) and finitely many eigenvalues of infinite multiplicity. The asymptotic behaviour of gaps for fixed and high energies is described; it is proved that for all values of , apart from a discrete set containing , there exists an infinite system of nondegenerate gaps with length as . The dependence of the spectrum on the magnetic field is investigated and the existence of gaps independent of is proved for certain special potentials. The asymptotic behaviour of gaps as is described.Bibliography: 32 titles.

Operators That Commute with Slant Toeplitz Operators

Let H be a separable Hilbert space and {en : n ∈ Z} be an orthonormal basis in H. A bounded operator T is called the slant Toeplitz operator if 〈T ej, ei〉 = c2i− j, where cn is the nth Fourier coefficient of a bounded Lebesgue measurable function φ on the unit circle T = {z ∈ C : |z| = 1}. It has been shown [9], with some assumption on the smoothness and the zeros of φ, that T ∗ is similar to either the constant multiple of a shift or to the constant multiple of the direct sum of a shift and a rank one unitary, with infinite multiplicity. These results, together with the theory of shifts (e.g., in [11]), allows us to identify all bounded operators on H commuting with such T .

Zigzag nanoribbons in external electric fields

We consider the Schrodinger operator on nanoribbons (tight-binding models) in an external electric potential V on the plane. The corresponding electric field is perpendicular to the axis of the nanoribbon. If V = 0, then the spectrum of the Schrodinger operator consists of two non-flat bands and one flat band (an eigenvalue with infinite multiplicity) between them. If we switch on a weak electric potential V → 0, then there are two cases: (1) this eigenvalue splits into the small spectral band. We determine the asymptotics of the spectral bands for small fields. (2) the unperturbed eigenvalue remains the flat band. We describe all potentials when the unperturbed eigenvalue remains the flat band and when one splits into the small band of the continuous spectrum.

Zigzag nanoribbons in external electric and magnetic fields

We consider the Schrodinger operators on zigzag nanoribbons (tight-binding models) in external magnetic and electric fields. If these fields are absent, then the spectrum of the Schrodinger operator consists of two non-flat bands and one flat band (an eigenvalue with infinite multiplicity) between them. We describe all magnetic and electric fields for which the unperturbed flat band remains the flat band and when one splits into the small band of the continuous spectrum. Also we determine spectral asymptotics for small fields and solve inverse spectral problem.

Resonances and spectral shift function for a magnetic Schrödinger operator

We consider the three-dimensional Schrödinger operator H0 with a constant magnetic field and subject to an electric potential v0 depending only on the variable along the magnetic field x3. The operator H0 has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb H0 by smooth scalar potentials V=O(⟨(x1,x2)⟩−δ⊥⟨x3⟩−δ∥), δ⊥>2, δ∥>1. We assume also that V and v0 have an analytic continuation, in the magnetic field direction, in a complex sector outside a compact set. We define the resonances of H=H0+V as the eigenvalues of the nonself-adjoint operator obtained from H by analytic distortions of Rx3. We study their distribution near any fixed real eigenvalue of H0, 2bq+λ for q∊N. In a ring centered at 2bq+λ with radii (r,2r), we establish an upper bound, as r tends to 0, of the number of resonances. This upper bound depends on the decay of V at infinity only in the directions (x1,x2). Finally, we deduce a representation of the derivative of the spectral shift function for the operator pair (H0,H) in terms of resonances. This representation justifies the Breit–Wigner approximation and implies a local trace formula.

Effective Masses for Zigzag Nanotubes in Magnetic Fields

We consider the Schr\"odinger operator with a periodic potential on quasi-1D models of zigzag single-wall carbon nanotubes in magnetic field. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We obtain identities and a priori estimates in terms of effective masses and gap lengths.

Schrödinger Operators on Zigzag Nanotubes

We consider the Schrodinger operator with a periodic potential on quasi-1D models of zigzag single-wall carbon nanotubes. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe all compactly supported eigenfunctions with the same eigenvalue. We define a Lyapunov function, which is analytic on some Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove that all resonances are real. We determine the asymptotics of the periodic and antiperiodic spectrum and of the resonances at high energy. We show that there exist two types of gaps: i) stable gaps, where the endpoints are periodic and anti-periodic eigenvalues, ii) unstable (resonance) gaps, where the endpoints are resonances (i.e., real branch points of the Lyapunov function). We describe all finite gap potentials. We show that the mapping: potential \(\rightarrow\) all eigenvalues is a real analytic isomorphism for some class of potentials.

REVERSED WAVELET FUNCTIONS AND SUBSPACES

Let the operators D and T be the dilation-by-2 and translation-by-1 on [Formula: see text], which are both bilateral shifts of infinite multiplicity. If ψ(·) in [Formula: see text] is a wavelet, then {DmTnψ(·)}(m,n)∈ℤ2is an orthonormal basis for the Hilbert space [Formula: see text] but the reversed set {TnDmψ(·)}(n,m)∈ℤ2is not. In this paper we investigate the role of the reversed functions TnDmψ(·) in wavelet theory. As a consequence, we exhibit an orthogonal decomposition of [Formula: see text] into T-reducing subspaces upon which part of the bilateral shift T consists of a countably infinite direct sum of bilateral shifts of multiplicity one, which mirrors a well-known decomposition of the bilateral shift D.

Spectrum of the $\overline {\partial }$-Neumann Laplacian on polydiscs

The spectrum of the ∂-Neumann Laplacian on a polydisc in C is explicitly computed. The calculation exhibits that the spectrum consists of eigenvalues, some of which, in particular the smallest ones, are of infinite multiplicity.