Analysis and Mathematical Physics最新文献

Thanks to the topological Hopf algebra of renormalization of Green’s functions in a gauge field theory, we associate a bi-Heyting algebra to each combinatorial Dyson–Schwinger equation. This setting leads us to characterize subsystems generated by the solution spaces of quantum motions. In addition, we apply the (c_{2})-invariant of Feynman diagrams to build a new Heyting algebra of multiplicative groups which encodes a more general class of subsystems of the physical theory.

If U is a unitary operator on a separable complex Hilbert space (mathcal {H}), an application of the spectral theorem says there is a conjugation C on (mathcal {H}) (an antilinear, involutive, isometry on (mathcal {H})) for which ( C U C = U^{*}.) In this paper, we fix a unitary operator U and describe all of the conjugations C which satisfy this property. As a consequence of our results, we show that a subspace is hyperinvariant for U if and only if it is invariant for any conjugation C for which (CUC = U^{*}).

We study the generalized quasiperiodically forced circle map (f:mathbb {T}^{m}times mathbb {T}^{1}rightarrow mathbb {T}^{m}times mathbb {T}^{1}), which is a natural generalization of the quasiperiodically forced circle map. Our main aim is to show that for each (rho ) in the interior of the fibred rotation set, there is a minimal set such that each orbit on the minimal set is (rho )-bounded.

We study the problem originally communicated by E. Meckes on the asymptotics for the eigenvalues of the kernel of the unitary eigenvalue process of a random (n times n) matrix. The eigenvalues (p_{j}) of the kernel are, in turn, associated with the discrete prolate spheroidal wave functions. We consider the eigenvalue counting function (|G(x,n)|:=#{j:p_j>Ce^{-x n}}), ((C>0) here is a fixed constant) and establish the asymptotic behavior of its average over the interval (x in (lambda -varepsilon , lambda +varepsilon )) by relating the function |G(x, n)| to the solution J(q) of the following energy problem on the unit circle (S^{1}), which is of independent interest. Namely, for given (theta ), (0<theta < 2 pi ), and given q, (0<q<1), we determine the function (J(q) =inf {I(mu ): mu in mathcal {P}(S^{1}), mu (A_{theta }) = q}), where (I(mu ):= int !int log frac{1}{|z - zeta |} dmu (z) dmu (zeta )) is the logarithmic energy of a probability measure (mu ) supported on the unit circle and (A_{theta }) is the arc from (e^{-i theta /2}) to (e^{i theta /2}).

We study the nonholonomic motion of a point particle on the Heisenberg group around the fixed “sun” placed at the origin whose potential is given by the fundamental solution of the sub-Laplacian. In contrast with several recent papers that approach this problem as a variational one (hence a control problem) we study the equations of dynamical motion which are non-variational in nonholonomic mechanics. We find three independent first integrals of the system and show that its bounded trajectories are wound up around certain surfaces of the fourth order. We also describe some particular cases of trajectories.

We consider the one-dimensional Schrödinger operator with properly connecting general point interaction at the origin. We derive a trace formula for trace of difference of resolvents of perturbed and unperturbed Schrödinger operators in terms of a Wronskian which results in an explicit expression for perturbation determinant. Using the estimate for large-time real argument on the trace norm of the resolvent difference of the perturbed and unperturbed Schrödinger operators we express the spectral shift function in terms of perturbation determinant. Under certain integrability conditions on the potential function, we calculate low-energy asymptotics for the perturbation determinant and prove an analog of Levinson’s formula

For a unitary operator U on a separable complex Hilbert space ({mathcal {H}}), we describe the set ({mathscr {C}}_{c}(U)) of all conjugations C (antilinear, isometric, and involutive maps) on ({mathcal {H}}) for which (C U C = U). As this set might be empty, we also show that ({mathscr {C}}_{c}(U) not = varnothing ) if and only if U is unitarily equivalent to (U^{*}).

In a recent paper, Hassan and O’Regan (Rocky Mt J Math 50:599–617, 2020) studied the oscillatory behavior of solutions for some higher-order nonlinear dynamic equations on time scales. In this paper, we investigate an open problem (Hassan and O’Regan in Rocky Mt J Math 50:599–617, 2020) and completely solve it by using the generalized Riccati approach and integral averaging technique. Our findings constitute an addendum to the above paper.

In this paper, we mainly focus on finding transcendental entire solutions of binomial and trinomial equations generated by q-shift operator in ({mathbb {C}}^{2}), to extend the results of Li and Xu (Axioms 126(10):1-19, 2021) and Zheng and Xu (Anal Math 48(1):199-226, 2022) completely in a different direction in terms of their q-shift counterpart. We have observed notable differences in the solutions derived from q-shift equations, including two different variants of q-difference equations, compared to the solutions obtained from the corresponding c-shift equations in ({mathbb {C}}^{2}). Our findings have been supported with several examples that illustrate these differences. Additionally, the introduction of two new lemmas not explored in existing literature further adds depth to the mathematical tools available for addressing such problems.

In this paper, the half inverse problem and interior inverse problems for Dirac operators with discontinuity inside the interval (0, T) is considered. It is shown that (i) if the potential is given on (Big (0,frac{(1+alpha )T}{4}Big )), then one spectrum can uniquely determine the potential on the whole interval; (ii) one spectrum and a set of values of eigenfunctions at some internal points can also uniquely determine the potential on the whole interval.