Semiclassical approximations for quantum time correlation functions are presented for both electronically adiabatic and nonadiabatic dynamics along with discussions of the operator ordering and the classical limit. With the combined use of the initial-value representation of the semiclassical propagator, a discrete algorithm to evaluate the Jacobi matrices, semiclassical operator ordering rules...

A family Aα of differential operators depending on a real parameter α is considered. The problem can be formulated in the language of perturbation theory of quadratic forms. The perturbation is only relatively bounded but not relatively compact with respect to the unperturbed form. The spectral properties of the operator Aα strongly depend on α. In particular, for α < √ 2 the spectrum of Aα bel...

In this paper we prove an analogue of the Ramanujan’s master theorem in setting Sturm Liouville operator $$\begin{aligned} \mathcal L=\frac{d^2}{dt^2} + \frac{A'(t)}{A(t)} \frac{d}{dt}, \end{aligned}$$on \((0,\infty )\), where \(A(t)=(\sinh t)^{2\alpha +1}(\cosh t)^{2\beta +1}B(t); \alpha ,\beta > -\frac{1}{2}\) with suitable conditions on B. When \(B\equiv 1\) get back Master for Jacobi operat...

If a Jacobi matrix J is reflectionless on (−2, 2) and has a single an0 equal to 1, then J is the free Jacobi matrix an ≡ 1, bn ≡ 0. I’ll discuss this result and its generalization to arbitrary sets and present several applications, including the following: if a Jacobi matrix has some portion of its an’s close to 1, then one assumption in the Denisov-Rakhmanov Theorem can be dropped. 1. Statemen...

Perhaps the most common theme in Fritz Gesztesy’s broad opus is the study of problems with periodic or almost periodic finite gap differential and difference equations, especially those connected to integrable systems. The present paper reviews recent progress in the understanding of finite gap Jacobi matrices and their perturbations. We’d like to acknowledge our debt to Fritz as a collaborator...

In this paper we present a method of studying convolution operator under the Sonin conditions imposed on kernel. The particular case kernel is fractional integral Riemman–Liouville operator, other various types kernels are Bessel-type function, functions with power-logarithmic singularities at origin e.t.c. We pay special attention to study close power type functions. main our aim Sonin–Abel eq...

This is the second of three lectures on introduction to vertex operator algebras. In this lecture, we shall continue Professor Dong’s lecture to present more fundamental properties of vertex operator algebras. From the mathematical point of view, a vertex operator algebra formally resembles a Lie algebra because the Jacobi identity is used as one of the main axioms. For the Lie algebra aspect o...

We study a reaction-diffusion problem formulated with higher-order operator, non-linear advection, and Fisher-KPP reaction term depending on the spatial variable. The operator induces solutions to oscillate in proximity of an equilibrium condition. Given this oscillatory character, are studied set bounded domains. introduce new extension that allows us open domain RN, but departing from sequenc...