Search

Stability and optimal convergence analysis of a non-uniform implicit-explicit L1 finite element method (IMEX-L1-FEM) is studied for a class of time-fractional linear partial differential/integro-differential equations with non-self-adjoint elliptic part having (space-time) variable coefficients. The proposed scheme is based on a combination of an IMEX-L1 method on graded mesh in the temporal direction and a finite element method in the spatial direction. With the help of a discrete fractional Grönwall inequality, global almost optimal error estimates in L 2 - and H 1 -norms are derived for the problem with initial data u 0 ∈ H 0 1 (Ω) ∩ H 2 (Ω). The novelty of our approach is based on managing the interaction of the L1 approximation of the fractional derivative and the time discrete elliptic operator to derive the optimal estimate in H 1 -norm directly. Furthermore, a super convergence result is established when the elliptic operator is self-adjoint with time and space varying coefficients, and as a consequence, an L ∞ error estimate is obtained for 2D problems that too with the initial condition is in H 0 1 (Ω) ∩ H 2 (Ω). All results proved in this paper are valid uniformly as α → 1 − , where α is the order of the Caputo fractional derivative. Numerical experiments are presented to validate our theoretical findings. [ABSTRACT FROM AUTHOR]

We consider a non-relativistic particle in a one-dimensional box with all possible quantum boundary conditions that make the kinetic-energy operator self-adjoint. We determine the Wigner functions of the corresponding eigenfunctions and analyze in detail their classical limit, governed by their behavior in the high-energy regime. We show that the quantum boundary conditions split into two classes: all local and regular boundary conditions collapse to the same classical boundary condition, while a dependence on singular non-local boundary conditions persists in the classical limit. [ABSTRACT FROM AUTHOR]

Let M be an n-dimensional oriented compact submanifold with constant normalized scalar curvature R ≥ c in the space form Fn+p(c). Denote by H and RicM the mean curvature and the Ricci curvature of M respectively. By applying Cheng-Yau’s self-adjoint operator, we first prove that if M is a hypersurface in a unit sphere, and RicM ≥ (n−2)(1+H2), then M is totally umbilical. Furthermore, we investigate the submanifolds in Fn+p(c) with flat normal bundle satisfying RicM ≥ (n − 2)(c + H2) > 0, and obtain a complete classification theorem. [ABSTRACT FROM AUTHOR]

We study singular Sturm–Liouville operators of the form 1 r j (− d d x p j d d x + q j) , j = 0 , 1 , in L 2 ((a , b) ; r j) with endpoints a and b in the limit point case, where, in contrast to the usual assumptions, the weight functions r j have different signs near a and b. In this situation the associated maximal operators become self-adjoint with respect to indefinite inner products and their spectral properties differ essentially from the Hilbert space situation. We investigate the essential spectra and accumulation properties of nonreal and real discrete eigenvalues; we emphasize that here also perturbations of the indefinite weights r j are allowed. Special attention is paid to Kneser type results in the indefinite setting and to L 1 perturbations of periodic operators. [ABSTRACT FROM AUTHOR]

Quantum graphs were introduced to model free electrons in organic molecules using a self-adjoint Hamiltonian on a network of intervals. A second graph quantization describes wave propagation on a graph by specifying scattering matrices at the vertices. A question that is frequently raised is the extent to which these models are the same or complementary. In particular, are all energy-independent unitary vertex scattering matrices associated with a self-adjoint Hamiltonian? Here we review results related to this issue. In addition, we observe that a self-adjoint Dirac operator with four component spinors produces a secular equation for the graph spectrum that matches the secular equation associated with wave propagation on the graph when the Dirac operator describes particles with zero mass and the vertex conditions do not allow spin rotation at the vertices. [ABSTRACT FROM AUTHOR]

We prove that the spectrum of the two-photon quantum Rabi Hamiltonian consists of two eigenvalue sequences. (Em+)m=0,∞ (Em-)m=0∞ satisfying a three-term asymptotic formula with the remainder estimate 0(m-1 1n m) when m tends to infinity. Our asymptotic formula can be written so that the third term is given by an explicit oscillatory integral and an explicit remainder estimate. [ABSTRACT FROM AUTHOR]

This paper introduces stochastic covariance models in Hilbert spaces with stationary affine instantaneous covariance processes. We explore the applications of these models in the context of forward curve dynamics within fixed-income and commodity markets. The affine instantaneous covariance process is defined on positive self-adjoint Hilbert–Schmidt operators, and we prove the existence of a unique limit distribution for subcritical affine processes, provide convergence rates of the transition kernels in the Wasserstein distance of order p ∈ [ 1 , 2 ] , and give explicit formulas for the first two moments of the limit distribution. Our results allow us to introduce affine stochastic covariance models in the stationary covariance regime and to investigate the behaviour of the implied forward volatility for large forward dates in commodity forward markets. [ABSTRACT FROM AUTHOR]

In the reaction–diffusion systems, the Laplacian is usually a self-adjoint operator. Incorporating advection terms generally destroys this and yields a quite complicated decomposition of phase space in the Center Manifold Reduction (CMR) method. Considering that the Multiple Time Scales (MTS) method can be directly applied to bifurcation analysis and normal form derivation based on algebraic operations, and the computational process is relatively universal, we apply the MTS method to analyze the Hopf bifurcation problem of reaction–diffusion–advection systems with time delay. First, we introduce the MTS method for the system with a specific boundary condition, which is used to derive the normal form of Hopf bifurcation. Calculating the normal form using the MTS method can be summarized as determining the bifurcation parameters, conducting Taylor expansion based on the MTS idea, and eliminating secular terms. Then, the key coefficients in the normal form are explicitly given, which are also compared with the results obtained by the CMR method, and both methods lead to the same bifurcation results. Finally, we use the MTS method to calculate the normal form of Hopf bifurcation for a predator–prey model with mixed boundary conditions. We find that the spatially nonhomogeneous periodic solutions appear near the equilibrium in the system when the time delay exceeds the critical value, and the theoretical results are illustrated by numerical simulations. [ABSTRACT FROM AUTHOR]

We revisit the non-perturbative renormalization of a class of simple polaron models with resting fermions. The considered dispersion relations and form factors are allowed to be highly singular, such that infinite self-energies and wave function renormalizations may occur and the diagonalizing dressing transformations might not be implementable on Fock space. Instead of taking cutoffs, we rigorously interpret the self-energies and wave function renormalizations as elements of suitable vector spaces, as well as a field extension of the complex numbers. Moreover, we define two extended state vector spaces (ESSs) over this field extension, which contain a dense subspace of Fock space, but also incorporate non-square-integrable wave functions. Elements of these ESSs can be seen as states of virtual particles, described in a mathematically rigorous way. The Hamiltonian without cutoffs, formally infinite counterterms, and the dressing transformation can then be defined as linear operators between certain subspaces of the two Fock space extensions. This way, we obtain a renormalized Hamiltonian which can be realized as a densely defined self-adjoint operator on Fock space. [ABSTRACT FROM AUTHOR]

The infinite-dimensional hypercube (IDH) is an infinite connected graph with infinite degree at each its vertex, and can be viewed as an infinite-dimensional analog of finite-dimensional hypercubes. In this paper, we investigate a self-adjoint operator determined by the topology of the IDH and a function on the nonnegative integers, which can be interpreted as the Hamiltonian of a free fermion field. We prove that, under some mild conditions, the operator has only pure point spectrum and its spectrum is even a compact interval of the real line. We also obtain some commutation relations concerning the operator, which are of physical interest. [ABSTRACT FROM AUTHOR]

We consider the stabilization of a class of linear evolution systems z ′ = A z + B v under the observation y = C z by means of a finite dimensional control v. The control is based on the design of a Luenberger observer which can be infinite or finite dimensional (of dimension large enough). In the infinite dimensional case, the operator A is supposed to generate an analytical semigroup with compact resolvent and the operators B and C are unbounded operators whereas in the finite dimensional case, A is assumed to be a self-adjoint operator with compact resolvent, B and C are supposed to be bounded operators. In both cases, we show that if (A, B) and (A, C) verify the Fattorini-Hautus Criterion, then we can construct an observer-based control v of finite dimension (greater or equal than largest geometric multiplicity of the unstable eigenvalues of A) such that the evolution problem is exponentially stable. As an application, we study the stabilization of the diffusion system. [ABSTRACT FROM AUTHOR]

In its original form the peak model for rank one supersingular perturbations of class ɦ--4 or higher of a nonnegative self-adjoint operator requires that the Gram matrix of the model should be diagonal. Here we remove the restriction on the Gram matrix. In particular we explain the origin of the Krein Q-function associated with the Gram matrix. [ABSTRACT FROM AUTHOR]

In this paper, the long-time asymptotics of the solution to the Cauchy problem is described by means of the evolution unitary group of the self-adjoint Mehler operator. Spectral analysis of the latter operator is also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we study the notions of Browder operator and Browder S-spectrum of bounded right linear operator defined over the right quaternionic Hilbert space. Some properties of Browder operator and stability of the ascent, descent and Browder S-spectrum have been investigated in the right quaternionic setting. We also characterize the property of invariant Browder operators and study the spectral mapping theorem of Browder S-spectrum for self-adjoint operators in quaternionic setting. This investigation concludes by exploring the Browder S-spectrum of the sum of two bounded linear operators. [ABSTRACT FROM AUTHOR]

The quantum injectivity problem classifies frames which are injective with respect to self-adjoint Hilbert-Schmidt operators. In this paper, we aim to analyze quantum injective frames in terms of the excess of frame elements in R n . Especially, we investigate the connection between the injectivity, full spark and phase retrieval frames. In addition, we detect injective (alternate) dual frames and show that the family of quantum injective dual frames is open and dense in the set of all dual frames. Finally, the stability of quantum injective frames will be addressed. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove the small data global existence of solutions to the semilinear damped wave equation$ \partial_t^2 u + Au + \partial_t u = |u|^{p-1}u $on a measure space $ X $ with a self-adjoint operator $ A $ on $ L^2(X) $. Under a certain decay estimate on the corresponding heat semigroup, we establish the linear estimates which generalize the so-called Matsumura estimates. Our approach is based on a direct spectral analysis analogous to the Fourier analysis. The self-adjoint operators treated in this paper include some important examples such as the Laplace operators on Euclidean spaces, the Dirichlet Laplacian on an arbitrary open set, the Robin Laplacian on an exterior domain, the Schrödinger operator, the elliptic operator, the Laplacian on Sierpinski gasket, and the fractional Laplacian. [ABSTRACT FROM AUTHOR]

In the space of square integrable functions on the positive semi-axis, two positive self-adjoint operators generated by a one-dimensional free Hamiltonian are constructed. These operators are employed to construct a pair of spectrally absolutely continuous bounded self-adjoint operators whose difference is an operator of rank 1. The perturbation determinant is used to find an explicit form of the M. G. Krein spectral shift function for this pair. It is shown that despite the Asmoothness of the perturbation in the sense of Hölder, the point λ = 1, where the spectral shift function has a discontinuity of the first kind, is not an eigenvalue of the perturbed operator. [ABSTRACT FROM AUTHOR]

Inverse spectral problem for a self-adjoint differential operator, which is the sum of the operator of the third derivative on a finite interval and of the operator of multiplication by a real function (potential), is solved. Closed system of integral linear equations is obtained. Via solution to this system, the potential is calculated. It is shown that the main parameters of the obtained system of equations are expressed via spectral data of the initial operator. It is established that the potential is unambiguously defined by the four spectra. [ABSTRACT FROM AUTHOR]

We revise Krein's extension theory of positive symmetric operators. Our approach using factorization through an auxiliary Hilbert space has several advantages: it can be applied to non-densely defined transformations and it works in both real and complex spaces. As an application of the results and the construction we consider positive self-adjoint extensions of the modulus square operator $ T^*T $ T ∗ T of a densely defined linear transformation T and bounded self-adjoint extensions of a symmetric operator. Krein's results on the uniqueness of positive (respectively, norm preserving) self-adjoint extensions are also revised. [ABSTRACT FROM AUTHOR]

The report addresses a complex problem for artificial intelligence (AI) as self-connected signification, design of selfadjoint operators and semiotic reality, issue and perspective of self-connected digital reality of different types of identity, systems of information infrastructure and digital pesrsonalisation. In addition, AI is also presented as a digital issue of strategic reality and guidelines that the European Commission undertakes for safety and security in the use of this instrument in digital society and markets in the EU and globally. [ABSTRACT FROM AUTHOR]

Let X be a space of homogeneous type. Assume that L is a non negative, selfadjoint operator on L 2 (X) satisfying the sub-Gaussian upper bounds. In this paper, we prove that ‖ (I + L) - n / 2 e i τ L f ‖ 1 , ∞ ≤ C (1 + | τ |) n / 2 ‖ f ‖ 1 , ∀ τ ∈ R. By interpolation, we obtain ‖ (I + L) - n | 1 / p - 1 / 2 | e i τ L f ‖ p ≤ C (1 + | τ |) n | 1 / p - 1 / 2 | ‖ f ‖ p , ∀ τ ∈ R , 1 < p < ∞. [ABSTRACT FROM AUTHOR]

This article is a revised version of the lectures given by the author at KROMSH-2019. These lectures are devoted to describing a few different ways of constructing a model representation for self-adjoint and unitary operators acting in Pontryagin spaces, and a comparison between them. Two of these models are based on the regularized integral Krein–Langer representation of a numerical sequence generated by the powers of a self-adjoint (in the sense of Pontryagin spaces) operator. The steps to deduce both this representation and the spectral function of the corresponding operator are given. In both models (first of which belongs to the author of this paper), the operator is realized as an operator of multiplication by an independent variable, but the space of functions in which it acts is different for each of the models. The third model, introduced by Shulman, is based on his own concept of a quasivector. [ABSTRACT FROM AUTHOR]

On a uniform grid on the real axis, we study the Yanenko–Stechkin–Subbotin problem of extremal function interpolation in the mean in the space , , of two-way real sequences with the least value of the norm of a linear formally self-adjoint differential operator of order with constant real coefficients. In case of even , the value of the least norm in the space , , of the extremal interpolant is calculated exactly if the grid step and the averaging step are related by the inequality . [ABSTRACT FROM AUTHOR]

We consider an abstract sequence { A n } n = 1 ∞ of closed symmetric operators on a separable Hilbert space H . It is assumed that all A n 's have equal deficiency indices (k, k) and thus self-adjoint extensions { B n } n = 1 ∞ exist and are parametrized by partial isometries { U n } n = 1 ∞ on H according to von Neumann's extension theory. Under two different convergence assumptions on the A n 's we give the precise connection between strong resolvent convergence of the B n 's and strong convergence of the U n 's. [ABSTRACT FROM AUTHOR]

We obtain formulas for Petz–Rényi and Umegaki relative entropy from the idea of distribution of a positive self-adjoint operator. Classical results on Rényi and Kullback–Leibler divergences are applied to obtain new results and new proofs for some known results about Petz–Rényi and Umegaki relative entropy. Most important among these, is a necessary and sufficient condition for the finiteness of the Petz–Rényi α -relative entropy. All of the results presented here are valid in both finite and infinite dimensions. In particular, these results are valid for states in Fock spaces and thus are applicable to continuous variable quantum information theory. [ABSTRACT FROM AUTHOR]

Let ßA (ℍ) denote the algebra of bounded linear operators on a complex Hilbert space ℍ that admit A-adjoint operators, where A is a non-zero positive semi-definite operator on ℍ. A commuting operator tuple T = (T1 ,..., Td) ∈ ßA(ℍ)d is called jointly A-normaloid if rA(T) = ∥T∥A, where rA(T) and ∥T∥A represent the joint A-spectral radius and the joint operator A-seminorm of T, respectively. This paper aims to investigate this new class of operators and provides several examples. Furthermore, a characterization of A-normaloidity is established. Additionally, the joint Euclidean A-seminorm of a d-tuple of A-bounded operators T, denoted by , is examined. Specifically, for all positive integers n, we prove that the following equivalence holds for any commuting operator tuple. Here. Finally, several related questions are explored. [ABSTRACT FROM AUTHOR]

Let $H$ be a Hilbert space. In this paper we show among others that, if the selfadjoint operators $A$ and $B$ satisfy the condition $0$ $

We present the representation of solution sets of the operator equation XAX = BX on Hilbert spaces by taking advantage of the invariant subspace of B, including solution sets of nonzero solutions, normal solutions and self-adjoint solutions. Moreover, necessary conditions and sufficient conditions are established for existence of nonzero solutions of the operator equation XAX = BX which commute with B. [ABSTRACT FROM AUTHOR]

We investigate the link between inverse problems and final state observability for a general class of parabolic systems. We generalize a stability result for initial data known for the case of self-adjoint dissipative operators. More precisely, we consider a system governed by an analytic semigroup. Under the assumption of final state observability, we prove a logarithmic stability estimate depending on the analyticity angle of the semigroup. This is done by using a general logarithmic convexity result. The abstract result is illustrated by considering the Ornstein–Uhlenbeck equation. [ABSTRACT FROM AUTHOR]

This study investigates the inverse problem of determining the right-hand side of a telegraph equation given in a Hilbert space. The main equation under consideration has the form (D t ρ) 2 u (t) + 2 α D t ρ u (t) + A u (t) = p (t) q + f (t) , where 0 < t ≤ T , 0 < ρ < 1 and D t ρ is the Caputo derivative. The equation contains a self-adjoint positive operator A and a time-varying multiplier p(t) in the source function, which, like the solution of the equation, is unknown. To solve the inverse problem, an additional condition B [ u (t) ] = ψ (t) is imposed, where B is an arbitrary bounded linear functional. The existence and uniqueness of a solution to the problem are established and stability inequalities are derived. It should be noted that, as far as we know, such an inverse problem for the telegraph equation is considered for the first time. Examples of the operator A and the functional B are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we study the small‐λ$\lambda$ spectral asymptotics of an integral operator K$\mathcal {K}$ defined on two multi‐intervals J$J$ and E$E$, when the multi‐intervals touch each other (but their interiors are disjoint). The operator K$\mathcal {K}$ is closely related to the multi‐interval finite Hilbert transform (FHT). This case can be viewed as a singular limit of self‐adjoint Hilbert–Schmidt integral operators with so‐called integrable kernels, where the limiting operator is still bounded, but has a continuous spectral component. The regular case when dist(J,E)>0$\text{dist}(J,E)>0$, and K$\mathcal {K}$ is of the Hilbert–Schmidt class, was studied in an earlier paper by the authors. The main assumption in this paper is that U=J∪E$U=J\cup E$ is a single interval (although part of our analysis is valid in a more general situation). We show that the eigenvalues of K$\mathcal {K}$, if they exist, do not accumulate at λ=0$\lambda =0$. Combined with the results in an earlier paper by the authors, this implies that Hp$H_p$, the subspace of discontinuity (the span of all eigenfunctions) of K$\mathcal {K}$, is finite dimensional and consists of functions that are smooth in the interiors of J$J$ and E$E$. We also obtain an approximation to the kernel of the unitary transformation that diagonalizes K$\mathcal {K}$, and obtain a precise estimate of the exponential instability of inverting K$\mathcal {K}$. Our work is based on the method of Riemann–Hilbert problem and the nonlinear steepest descent method of Deift and Zhou. [ABSTRACT FROM AUTHOR]

Based on the recent construction of a self-adjoint momentum operator for a particle confined in a one-dimensional interval, we extend the construction to arbitrarily shaped regions in any number of dimensions. Different components of the momentum vector do not commute with each other unless very special conditions are met. As such, momentum measurements should be considered one direction at a time. We also extend other results, such as the Ehrenfest theorem and the interpretation of the Heisenberg uncertainty relation to higher dimensions. [ABSTRACT FROM AUTHOR]

In this note, we present some results on the convergence of Nekrasov partition functions as power series in the instanton counting parameter. We focus on U(N) N = 2 gauge theories in four dimensions with matter in the adjoint and in the fundamental representations of the gauge group, respectively, and find rigorous lower bounds for the convergence radius in the two cases: if the theory is conformal, then the series has at least a finite radius of convergence, while if it is asymptotically free it has infinite radius of convergence. Via AGT correspondence, this implies that the related irregular conformal blocks of W N algebrae admit a power expansion in the modulus converging in the whole plane. By specifying to the SU(2) case, we apply our results to analyze the convergence properties of the corresponding Painlevé τ -functions. [ABSTRACT FROM AUTHOR]

We consider some basic problems associated with quantum mechanics of systems having a time-dependent Hilbert space. We provide a consistent treatment of these systems and address the possibility of describing them in terms of a time-independent Hilbert space. We show that in general the Hamiltonian operator does not represent an observable of the system even if it is a self-adjoint operator. This is related to a hidden geometric aspect of quantum mechanics arising from the presence of an operator-valued gauge potential. We also offer a careful treatment of quantum systems whose Hilbert space is obtained by endowing a time-independent vector space with a time-dependent inner product. [ABSTRACT FROM AUTHOR]

In this paper we study the approximate controllability of fractional partial differential equations involving the higher order Hilfer type time-fractional derivative associated with a non-negative self-adjoint operator $ A $ with compact resolvent on $ L^2(\Omega) $, where $ \Omega\subset{{\mathbb{R}}}^N $ ($ N\geq 1 $) is an open set. More precisely, we show that if $ 0\le\nu\le 1 $, $ 1<\mu\le 2 $ and $ \Omega\subset{{\mathbb{R}}}^N $ is an open set, then there is a control function $ f $ such that the system$ \begin{equation*} \begin{cases} {{\mathbb{D}}}^{\mu,\nu}_tu+Au = f\chi_{\omega}\;\;&\mbox{ in }\;\Omega\times(0,T),\\ (I_t^{(1-\nu)(2-\mu)}u)(\cdot,0) = u_0 &\mbox{ in }\;\Omega,\\ (\partial_tI_t^{(1-\nu)(2-\mu)}u)(\cdot,0) = u_1 &\mbox{ in }\;\Omega, \end{cases} \end{equation*} $is memory approximately controllable for any $ T>0 $, $ u_0\in D(A^{1/\mu}) $, $ u_1\in L^2(\Omega) $ and any non-empty open set $ \omega\subset\Omega $. The same result holds for $ u_0\in D(A^{1/2}) $ and $ u_1\in L^2(\Omega) $. We also provide a method to construct such a control function $ f $ which is very useful in numerical treatment and other applications. [ABSTRACT FROM AUTHOR]

In this paper several tensorial norm inequalities for continuous functions of selfadjoint operators in Hilbert spaces have been obtained. Multiple inequalities are obtained with variations due to the convexity properties of the mapping f || 1/6 [f(A) x 1 + 4f - A - 1 + 1 B 2 - + 1 f(B) - - Z 1 0 f((1 - k)A < 1 + k1 B)dk || 5\36 1 B-A 1 f'I,+8. [ABSTRACT FROM AUTHOR]

The dynamics of Markovian open quantum systems are described by Lindblad master equations, generating a quantum dynamical semigroup. An important concept for such systems is (Davies) irreducibility, i.e. the question whether there exist non-trivial invariant subspaces. Steady states of irreducible systems are unique and faithful, i.e. they have full rank. In the 1970s, Frigerio showed that a system is irreducible if the Lindblad operators span a self-adjoint set with trivial commutant. We discuss a more general and powerful algebraic criterion, showing that a system is irreducible if and only if the multiplicative algebra generated by the Lindblad operators La and the operator i H + ∑ a L a † L a , involving the Hamiltonian H, is the entire operator space. Examples for two-level systems, show that a change of Hamiltonian terms as well as the addition or removal of dissipators can render a reducible system irreducible and vice versa. Examples for many-body systems show that a large class of spin chains can be rendered irreducible by dissipators on just one or two sites. Additionally, we discuss the decisive differences between (Davies) reducibility and Evans reducibility for quantum channels and dynamical semigroups which has lead to some confusion in the recent physics literature, especially, in the context of boundary-driven systems. We give a criterion for quantum reducibility in terms of associated classical Markov processes and, lastly, discuss the relation of the main result to the stabilization of pure states and argue that systems with local Lindblad operators cannot stabilize pure Fermi-sea states. [ABSTRACT FROM AUTHOR]

We consider a self-adjoint operator T on a separable Hilbert space, with pure-point and simple spectrum with accumulations at finite points. Explicit conditions are stated on the eigenvalues of T and on the bounded perturbation V ensuring the global stability of the spectral nature of T + ɛV, ε ∈ R. [ABSTRACT FROM AUTHOR]

In this paper, we establish relations between special values of Dirichlet L -functions and those of spectral zeta functions or L -functions of cycle graphs. In fact, they determine each other in a natural way. These two kinds of special values are bridged together by a combinatorial derivative formula obtained from studying spectral zeta functions of the first order self-adjoint differential operators on the unit circle. [ABSTRACT FROM AUTHOR]

Let H be a Hilbert space. In this paper we show among others that, if the selfadjoint operators A and B satisfy the condition 0 < m ≤ A, B ≤ M, for some constants m, M, then... for all v ∈ [0;1]: We also have the inequalities for Hadamard product... for all v ∈ [0,1]. [ABSTRACT FROM AUTHOR]

We estimate the lowest eigenvalue in the gap of the essential spectrum of a Dirac operator with mass in terms of a Lebesgue norm of the potential. Such a bound is the counterpart for Dirac operators of the Keller estimates for the Schrödinger operator, which are equivalent to some Gagliardo-Nirenberg-Sobolev interpolation inequalities. Domain, self-adjointness, optimality and critical values of the norms are addressed, while the optimal potential is given by a Dirac equation with a Kerr nonlinearity. A new critical bound appears, which is the smallest value of the norm of the potential for which eigenvalues may reach the bottom of the gap in the essential spectrum. The Keller estimate is then extended to a Lieb-Thirring inequality for the eigenvalues in the gap. Most of our result are established in the Birman-Schwinger reformulation. [ABSTRACT FROM AUTHOR]

Let $H$ be a Hilbert space. In this paper we show among others that, if $f,$ $g$ are synchronous and continuous on $I$ and $A,$ $B$ are selfadjoint with spectra ${Sp}\left( A\right) ,$ ${Sp}\left( B\right) \subset I,$ then% \begin{equation*} \left( f\left( A\right) g\left( A\right) \right) \otimes 1+1\otimes \left( f\left( B\right) g\left( B\right) \right) \geq f\left( A\right) \otimes g\left( B\right) +g\left( A\right) \otimes f\left( B\right) \end{equation*}% and the inequality for Hadamard product% \begin{equation*} \left( f\left( A\right) g\left( A\right) +f\left( B\right) g\left( B\right) \right) \circ 1\geq f\left( A\right) \circ g\left( B\right) +f\left( B\right) \circ g\left( A\right) . \end{equation*}% Let either $p,q\in \left( 0,\infty \right) $ or $p,q\in \left( -\infty ,0\right) $. If $A,$ $B>0,$ then \begin{equation*} A^{p+q}\otimes 1+1\otimes B^{p+q}\geq A^{p}\otimes B^{q}+A^{q}\otimes B^{p}, \end{equation*}% and% \begin{equation*} \left( A^{p+q}+B^{p+q}\right) \circ 1\geq A^{p}\circ B^{q}+A^{q}\circ B^{p}. \end{equation*}

Let L be a non negative, selfadjoint operator on L 2 (X) , where X is a metric space endowed with a doubling measure. Consider the Schrödinger group for fractional powers of L. If the heat flow e − t L satisfies suitable conditions of Davies–Gaffney type, we obtain the following estimate in Hardy spaces associated to L : ‖ (I + L) − β / 2 e i τ L γ / 2 f ‖ H L p (X) ≤ C (1 + | τ |) n s p ‖ f ‖ H L p (X) where p ∈ (0 , 1 ] , γ ∈ (0 , 1 ] , β / γ = n | 1 2 − 1 p | = n s p and τ ∈ R. If in addition e − t L satisfies a localized L p 0 → L 2 polynomial estimate for some p 0 ∈ [ 1 , 2) , we obtain ‖ (I + L) − β / 2 e i τ L γ / 2 f ‖ p 0 , ∞ ≤ C (1 + | τ |) n s p 0 ‖ f ‖ p 0 , ∀ τ ∈ R , provided 0 < γ ≠ 1 , β / γ = n | 1 2 − 1 p | = n s p and τ ∈ R. By interpolation, the second estimate implies also, for all p ∈ (p 0 , p 0 ′) , the strong (p , p) type estimate ‖ (I + L) − β / 2 e i τ L γ / 2 f ‖ p ≤ C (1 + | τ |) n s p ‖ f ‖ p. The applications of our theory span a diverse spectrum, ranging from the Schrödinger operator with an inverse square potential to the Dirichlet Laplacian on open domains. It showcases the effectiveness of our theory across various settings. [ABSTRACT FROM AUTHOR]

We analyze the spectrum of the operator $\Delta{-1} [\nabla \cdot (K\nabla u)]$ subject to homogeneous Dirichlet or Neumann boundary conditions, where $\Delta$ denotes the Laplacian and $K=K(x,y)$ is a symmetric tensor. Our main result shows that this spectrum can be derived from the spectral decomposition $K=Q \Lambda QT$, where $Q=Q(x,y)$ is an orthogonal matrix and $\Lambda=\Lambda(x,y)$ is a diagonal matrix. More precisely, provided that $K$ is continuous, the spectrum equals the convex hull of the ranges of the diagonal function entries of $\Lambda$. The domain involved is assumed to be bounded and Lipschitz. In addition to studying operators defined on infinite-dimensional Sobolev spaces, we also report on recent results concerning their discretized finite-dimensional counterparts. More specifically, even though $\Delta{-1} [\nabla \cdot (K\nabla u)]$ is not compact, it turns out that every point in the spectrum of this operator can, to an arbitrary accuracy, be approximated by eigenvalues of the associated generalized algebraic eigenvalue problems arising from discretizations. Our theoretical investigations are illuminated by numerical experiments. The results presented in this paper extend previous analyses which have addressed elliptic differential operators with scalar coefficient functions. Our investigation is motivated by both preconditioning issues (efficient numerical computations) and the need to further develop the spectral theory of second order PDEs (core analysis). [ABSTRACT FROM AUTHOR]

We prove that the spectrum of the asymmetric quantum Rabi model consists of two eigenvalue sequences (E m +) m = 0 ∞ , (E m −) m = 0 ∞ , satisfying a two-term asymptotic formula with error estimate of the form O (m − 1 / 4) , when m tends to infinity. [ABSTRACT FROM AUTHOR]

This article, published in Mathematical Notes, discusses lacunary recurrence relations with gaps of length 8 for the Bernoulli and Euler polynomials. The authors use standard notations for these polynomials and numbers and provide formulas and expansions for them. They also mention the history of lacunary recurrence relations for these numbers and polynomials and highlight their own contributions in finding such relations. The authors describe the method of proof for their main result, Theorem 1, and provide corollaries that extend the relations to the Bernoulli and Euler numbers. The article concludes by acknowledging funding support and declaring no conflicts of interest. [Extracted from the article]

On a uniform grid on the real axis , we study the Yanenko–Stechkin–Subbotin problem of extremal function interpolation in the mean in the space of two-way real sequences with the least value of the norm of a linear formally s This problem is considered for the class of sequences for which the generalized finite differences of order corresponding to the operator are bounded in the space . In this paper, the least value of the norm is calculated exactly if the grid step and the averaging step of the function to be interpolated in the mean are related by the inequalities . The paper is a continuation of the research by Yu. N. Subbotin and the author in this problem, initiated by Yu. N. Subbotin in 1965. The result obtained is new, in particular, for the -times differentiation operator . [ABSTRACT FROM AUTHOR]

The properties of the unique nontrivial analytic solution, defined implicitly by a functional equation, are pointed out. This work provides local estimations and global inequalities for the involved solution. The corresponding operatorial equation is studied as well. The second part of the paper is devoted to the full classical moment problem, which is an inverse problem. Two constraints are imposed on the solution. One of them requires the solution to be dominated by a concrete convex operator defined on the positive cone of the domain space. A one-dimensional operator is valued, and a multidimensional scalar moment problem is solved. In both cases, the existence and the uniqueness of the solution are proved. The general idea of the paper is to provide detailed information on solutions which are not expressible in terms of elementary functions. [ABSTRACT FROM AUTHOR]

In this paper several tensorial norm inequalities for continuous functions of selfadjoint operators in Hilbert spaces have been obtained. Multiple inequalities of the form.. are obtained with variations due to the convexity properties of the mapping f. [ABSTRACT FROM AUTHOR]