Related papers: Resonance index and singular spectral shift functi…

We explore connections between Krein's spectral shift function $\xi(\lambda,H_0,H)$ associated with the pair of self-adjoint operators $(H_0,H)$, $H=H_0+V$ in a Hilbert space $\calH$ and the recently introduced concept of a spectral shift…

We obtain a representation formula for the derivative of the spectral shift function $\xi(\lambda; B, \epsilon)$ related to the operators $H_0(B,\epsilon) = (D_x - By)^2 + D_y^2 + \epsilon x$ and $H(B, \epsilon) = H_0(B, \epsilon) + V(x,y),…

The self-similar functions of zero spectral degree are defined. The singular points of these functions are investigated and full classification of points is given. The connection with spectral problems (Stieltjes string) is pointed out.

The development of electron spin resonance (ESR) combined with scanning tunneling spectroscopy (STM) is undoubtedly one of the main experimental breakthroughs in surface science of the last decade thanks to joining the extraordinarily high…

Let~$H_0$ and~$V$ be self-adjoint operators such that~$V$ admits a factorisation $V = F^*JF$ with bounded self-adjoint $J$ and $|H_0|^{1/2}$-compact~$F.$ Flow of singular spectrum of the path of self-adjoint operators $H_0 + rV,$ $r \in…

We study spectral properties of the Neumann-Poincar\'e operator on planar domains with corners with particular emphasis on existence of continuous spectrum and pure point spectrum. We show that the rate of resonance at continuous spectrum…

In this paper, we derive a simple equality that relates the spectral function $I(k,\omega)$ and the fidelity susceptibility $\chi_F$, i.e. $% \chi_F=\lim_{\eta\rightarrow 0}\frac{\pi}{\eta} I(0, i\eta)$ with $\eta$ being the half-width of…

We study spectra of Schr\"odinger operators on $\RR^d$. First we consider a pair of operators which differ by a compactly supported potential, as well as the corresponding semigroups. We prove almost exponential decay of the singular values…

We theoretically explore the role of mesoscopic fluctuations and noise on the spectral and temporal properties of systems of $\mathcal{PT}$-symmetric coupled gain-loss resonators operating near the exceptional point, where eigenvalues and…

A complex function $f(z)$ is called a Herglotz-Nevanlinna function if it is holomorphic in the upper half-plane ${\mathbb C}_+$ and maps ${\mathbb C}_+$ into itself. By a maximum principle a Herglotz-Nevanlinna function which takes a real…

A notion of the radial index of an isolated singular point of a 1-form on a singular (real or complex) variety is discussed. For the differential of a function it is related to the Euler characteristic of the Milnor fibre of the function. A…

We consider the 3D Schr\"odinger operator $H_0$ with constant magnetic field $B$ of scalar intensity $b>0$, and its perturbations $H_+$ (resp., $H_-$) obtained by imposing Dirichlet (resp., Neumann) conditions on the boundary of the bounded…

This paper shows that an integrable approximation of the spring pendulum, when tuned to be in $1:1:2$ resonance, has monodromy. The stepwise precession angle of the swing plane of the resonant spring pendulum is shown to be a rotation…

Let $f \in C^n(\mathbb{R})$ be such that $\Vert f^{(n)} \Vert_\infty < \infty$. Let $f^{[n]} \in C(\mathbb{R}^{n+1})$ be the $n$th order divided difference. A special case of our main result states that for $1 < p < \infty$ we have \[\Vert…

This paper is concerned with non-Hermitian degeneracy and exceptional points associated with resonances in an acoustic scattering problem with sound-hard obstacles. The aim is to find non-Hermitian degenerate (defective) resonances using…

Stationary points or derivative zero crossings of a regression function correspond to points where a trend reverses, making their estimation scientifically important. Existing approaches to uncertainty quantification for stationary points…

It is an established fact that a finite difference operator approximates a derivative with a fixed algebraic rate of convergence. Nevertheless, we exhibit a new finite difference operator and prove it has spectral accuracy. Its rate of…

In this paper we study the asymptotic expansion of the spectral shift function for the slowly varying perturbations of periodic Schr\"odinger operators. We give a weak and pointwise asymptotics expansions in powers of $h$ of the derivative…

Recent observations suggest that the spectral index of the primordial perturbations is very close to unity, as expected in models of slow roll inflation. It is still possible for such models to produce spectra which are scale dependent. We…

This article investigates the unique determination of a radial refractive index n from spectral data. First, we demonstrate that for piecewise twice continuously differentiable functions, n is not uniquely determined by the special…