Related papers: On the trace formula for higher-order ODO
The main goal of this work is to provide a description of the {trace per unit volume} in terms of the {Dixmier trace} (regularized by the resolvent of the harmonic oscillator) for a large class of two-dimensional \emph{magnetic operators}…
We present an exact first-order perturbation theory for the eigenmodes in systems with interfaces causing material discontinuities. We show that when interfaces deform, higher-order terms of the perturbation series can contribute to the…
Scientific studies often require the precise calculation of derivatives. In many cases an analytical calculation is not feasible and one resorts to evaluating derivatives numerically. These are error-prone, especially for higher-order…
An explicit formula to find symmetry recursion operators for partial differential equations (PDEs) is obtained from new results connecting variational integrating factors and non-variational symmetries. The formula is special case of a…
We study the semi-classical trace formula at a critical energy level for a $h$-pseudo-differential operator on $\mathbb{R}^{n}$ whose principal symbol has a totally degenerate critical point for that energy. We compute the contribution to…
Expressions for the derivatives with respect to order of modified Bessel functions evaluated at integer orders and certain integral representations of associated Legendre functions with modulus argument greater than unity are used to…
Higher-order tensors appear in various areas of mechanics as well as physics, medicine or earth sciences. As these tensors are highly complex, most are not well understood. Thus, the analysis and the visualization process form a highly…
This paper is devoted to a new first order Taylor-like formula where the corresponding remainder is strongly reduced in comparison with the usual one which appears in the classical Taylor's formula. To derive this new formula, we introduce…
We study the spectral functional tr f(D+A) for a suitable function f, a self-adjoint operator D having compact resolvent, and a certain class of bounded self-adjoint operators A. Such functionals were introduce by Chamseddine and Connes in…
In this paper, we extend several time reversible numerical integrators to solve the Lorentz force equations from second order accuracy to higher order accuracy for relativistic charged particle tracking in electromagnetic fields. A fourth…
Let $A$ be a selfadjoint operator in a separable Hilbert space, $K$ a selfadjoint Hilbert-Schmidt operator, and $f\in C^n(\mathbb{R})$. We establish that $\varphi(t)=f(A+tK)-f(A)$ is $n$-times continuously differentiable on $\mathbb{R}$ in…
A new totally algebraic formalism based on general, abstract ladder operators has been proposed. This approach heavily grounds in the superoperator formalism of Primas. However it is necessary to introduce many improvements in his…
We study the optimal recovery problem for isotropic functions defined by second-order differential operators using both function and gradient values. We derive the upper bound for n-th optimal error with an explicit constant, which is…
We consider a Schr\"odinger operator with complex-valued potentials on the line. The operator has essential spectrum on the half-line plus eigenvalues (counted with algebraic multiplicity) in the complex plane without the positive…
The trace formulas commonly used in discussing spectral properties of quantum or wave systems are derived simply and directly from the Bogomolny transfer operator. Special cases are the Gutzwiller formula, the Berry-Tabor formula, and the…
We derive inequalities for sums of eigenvalues of Schr\"{o}dinger operators on finite intervals and tori. In the first of these cases, the inequalities converge to the classical trace formulae in the limit as the number of eigenvalues…
Linear temporal logic (LTL) is a specification language for finite sequences (called traces) widely used in program verification, motion planning in robotics, process mining, and many other areas. We consider the problem of learning LTL…
We study the numerical differentiation formulae for functions given in grids with arbitrary number of nodes. We investigate the case of the infinite number of points in the formulae for the calculation of the first and the second…
In this paper we construct high order numerical methods for solving third and fourth orders nonlinear functional differential equations (FDE). They are based on the discretization of iterative methods on continuous level with the use of the…
Two trace formulas for the spectra of arbitrary Hermitian matrices are derived by transforming the given Hermitian matrix $H$ to a unitary analogue. In the first type the unitary matrix is $e^{i(\lambda\II - H)}$ where $\lambda$ is the…