CyberSec Research

In this paper, we extend the definition of c-entropy to canonical L-systems with non-dissipative state-space operators. We also introduce the concepts of dissipation and accumulation coefficients for such systems. In addition, we examine the coupling of these L-systems and derive closed form expressions for the corresponding c-entropy.

We study the area ranges where the two possible isoperimetric domains on the infinite cylinder $\mathbb{S}^{1}\times \R$, namely, geodesic disks and cylindrical strips of the form $\mathbb{S}^1\times [0,h]$, satisfy P\'{o}lya's conjecture. In the former case, we provide an upper bound on the maximum value of the radius for which the conjecture may hold, while in the latter we fully characterise the values of $h$ for which it does hold for these strips. As a consequence, we determine a necessary and sufficient condition for the isoperimetric domain on $\mathbb{S}^{1}\times \R$ corresponding to a given area to satisfy P\'{o}lya's conjecture. In the case of the cylindrical strip, we also provide a necessary and sufficient condition for the Li-Yau inequalities to hold.

We establish $\frac{1}{2}$-H\"older continuity, or even the Lipschitz property, for the spectral measures of half-line discrete Schr\"odinger operators under suitable boundary conditions and exponentially decaying small potentials. These are the first known examples, apart from the free case, of Schr\"odinger operators with Lipschitz continuous spectral measures, as a direct consequence of the Dirichlet boundary condition. Notably, we show that the asymptotic behavior of the time-averaged quantum return probability, either $\log (t) / t$ or $1 / t$, as in the case of the free Laplacian, remains unchanged in this setting. Furthermore, we prove the persistence of the purely absolutely continuous spectrum and the $\frac{1}{2}$-H\"older continuity of the spectral measure for (Diophantine) quasi-periodic operators under exponentially decaying small perturbations. These results are optimal and hold for all energies.

In this paper, we investigate the spectrum of a class of multidimensional quasi-periodic Schr\"odinger operators that exhibit a Cantor spectrum, which provides a resolution to a question posed by Damanik, Fillman, and Gorodetski \cite{DFG}. Additionally, we prove that for a dense set of irrational frequencies with positive Hausdorff dimension, the Hausdorff (and upper box) dimension of the spectrum of the critical almost Mathieu operator is positive, yet can be made arbitrarily small.

In this paper, we obtain an explicit formula for the heat kernel on the infinite Cayley graph of the modular group $\operatorname{PSL}_2\mathbb{Z}$, given by the presentation $\langle a,b\mid a^2=1, b^3=1\rangle$. Our approach extends the method of Chung--Yau in~\cite{MR1667452} by observing that the Cayley graph strongly and regularly covers a weighted infinite line. We solve the spectral problem on this line to obtain an integral expression for its heat kernel, and then lift this to the Cayley graph using spectral transfer principles for strongly regular coverings. The explicit formula allows us to determine the Laplace spectrum, containing eigenvalues and continuous parts. As a by-product, we suggest a conjecture on the lower bound for the spectral gap of Cayley graphs of $\operatorname{PSL}_2\mathbb{F}_p$ with our generators, inspired by the analogy with Selberg's $1/4$-conjecture. Numerical evidence is provided for small primes.

We consider Dirac-Schr\"odinger operators over odd-dimensional Euclidean space. The conditions for the potential are based on those of C. Callias in his famous paper on the corresponding index problem. However, we treat the case where the potential can take values in unbounded operators of a separable Hilbert space, and crucially, we also do not assume that the potential needs to be invertible outside a compact region. Hence, the Dirac-Schr\"odinger operator is not necessarily Fredholm. In the setup we discuss, it however still admits a related trace formula in terms of the underlying potential. In this paper we express the trace formula for these Callias-type operators in terms of higher order spectral shift functions, leading to a functional equation which generalizes a known functional equation found first by A. Pushnitski. To the knowledge of the author, this paper presents the first multi-dimensional non-Fredholm extension of the Callias index theorem involving higher order spectral shift functions. More precisely, we also show that under a Lebesgue point condition on the higher order spectral shift function associated to the potential, the Callias-type operator admits a regularized index, even in non-Fredholm settings. This corresponds to a known Witten index result in the one-dimensional case shown by A. Carey et al. The regularized index that we introduce is a minor extension of the classical Witten index, and we present an index formula, which generalizes the classical Callias index theorem. As an example, we treat the case of $(d+1)$-massless Dirac-Schr\"odinger operators, for which we calculate the associated higher order spectral shift functions.

We prove that Toeplitz operators associated with a Bernstein-Markov measure on a compact complex manifold endowed with a big line bundle form an algebra under composition. As an application, we derive a Szeg\H{o}-type spectral equidistribution result for this class of operators. A key component of our approach is the off-diagonal asymptotic analysis of the Bergman kernel, also known as the Christoffel-Darboux kernel.

A variety of inverse Sturm-Liouville problems is considered, including the two-spectrum inverse problem, the problem of recovering the potential from the Weyl function, as well as the recovery from the spectral function. In all cases the potential in the Sturm-Liouville equation is assumed to be complex valued. A unified approach for the approximate solution of the inverse Sturm-Liouville problems is developed, based on Neumann series of Bessel functions (NSBF) representations for solutions and their derivatives. Unlike most existing approaches, it allows one to recover not only the complex-valued potential but also the boundary conditions of the Sturm-Liouville problem. Efficient accuracy control is implemented. The numerical method is direct. It involves only solving linear systems of algebraic equations for the coefficients of the NSBF representations, while eventually the knowledge only of the first NSBF coefficients leads to the recovery of the Sturm-Liouville problem. Numerical efficiency is illustrated by several test examples.

We study the growth of the resolvent of a Hardy--Toeplitz operator $T_b$ with a Laurent polynomial symbol (\emph{i.e., } the matrix $T_b$ is banded), at the neighborhood of a point $w_0\in\partial(\sigma(T_b))$ on the boundary of its spectrum. We show that such growth is inverse linear in some non-tangential domains at the vertex $w_0$, provided that $w_0$ does not belong to a certain finite set on the complex plane.

We link the Boundary Control Theory and the Titchmarsh-Weyl Theory. This provides a natural interpretation of the $A-$amplitude due to Simon and yields a new efficient method to evaluate the Titchmarsh-Weyl $m-$function associated with the Schr\"{o}dinger operator $H=-\partial _{x}^{2}+q\left( x\right) $ on $L_{2}\left( 0,\infty \right) $ with Dirichlet boundary condition at $x=0.$

We establish connections between different approaches to inverse spectral problems: the classical Gelfand--Levitan theory, the Krein method, the Simon theory, the approach proposed by Remling and the Boundary Control method. We show that the Boundary Control approach provides simple and physically motivated proofs of the central results of other theories. We demonstrate also the connections between the dynamical and spectral data and derive the local version of the classical Gelfand--Levitan equations.

We show the equivalence of inverse problems for different dynamical systems and corresponding canonical systems. For canonical system with general Hamiltonian we outline the strategy of studying the dynamic inverse problem and procedure of construction of corresponding de Branges space.

On a star graph $G$ with $n = n_+ + n_-$ edges of unit length, we study the operator $-\frac{\mathrm{d}^2}{\mathrm{d} x^2}$ on $n_+$ and $\frac{\mathrm{d}^2}{\mathrm{d} x^2}$ on $n_-$ edges equipped with Dirichlet boundary conditions at the outer vertices and a Kirchhoff condition at the central vertex. We study the spectral properties of the corresponding indefinite Kirchhoff Laplacian on $G$ and we show that it is similar to a selfadjoint operator in the Hilbert space $L^2(G)$ and that its eigenfunctions form a Riesz basis. Furthermore, we give a complete description of the point spectrum.

We establish a spectral correspondence between random Schr\"odinger operators and deterministic convolution operators on wreath products, generalizing previous results that relate Lamplighter groups to Schr\"odinger operators with Bernoulli potentials. Using this correspondence in both directions, we obtain an elementary criterion for the absolute continuity of convolutions on wreath products, Lifschitz tail estimates for Schr\"odinger operators on Cayley graphs of polynomial growth, and an exact formula for the second moment of the Green function, expressed in terms of the wreath product with an Abelian group of lamps.

We extend the notion of generalized boundary triples and their Weyl functions from extension theory of symmetric operators to adjoint pairs of operators, and we provide criteria on the boundary parameters to induce closed operators with a nonempty resolvent set. The abstract results are applied to Schr\"odinger operators with complex $L^p$-potentials on bounded and unbounded Lipschitz domains with compact boundaries.

We study scattering by metamaterials with negative indices of refraction, which are known to support \emph{surface plasmons} -- long-lived states that are highly localized at the boundary of the cavity. This type of states has found uses in a variety of modern technologies. In this article, we study surface plasmons in the setting of non-trapping cavities; i.e. when all billiard trajectories outside the cavity escape to infinity. We characterize the indices of refraction which support surface plasmons, show that the corresponding resonances lie super-polynomially close to the real axis, describe the localization properties of the corresponding resonant states, and give an asymptotic formula for their number.

In this paper we study the approximation of Dirac operators with $\delta$-shell potentials in the norm resolvent sense. In particular, we consider the approximation of Dirac operators with confining electrostatic and Lorentz scalar $\delta$-shell potentials, where the support of the $\delta$-shell potentials is impermeable to particles modelled by such Dirac operators.

In this article the authors continue the discussion in \cite{ALM} about inverse problems for second order elliptic and hyperbolic equations on metric trees from boundary measurements. In the present paper we prove the identifiability of varying densities of a planar tree-like network of strings along with the complete information on the graph, i.e. the lengths of the edges, the edge degrees and the angles between neighbouring edges. The results are achieved using the Titchmarch-Weyl function for the spectral problem and the Steklov-Poincar{\'e} operator for the dynamic wave equation on the tree. The general result is obtained by a peeling argument which reduces the inverse problem layer-by-layer from the leaves to the clamped root of the tree.