Inverse spectral problem for the Hill operator on the graph with a loop

In this paper, we investigate a generalization of the classical a PT-symmetric Hill operator to lasso graph. The definition of the PT-symmetric Hill operator on lasso graph is given and derived its spectral properties. We solved the inverse problem, proved the uniqueness theorem and provided a constructive procedure for the solution of the inverse problem.

The spectrum of the Hamiltonian with a PT-symmetric periodic optical potential

We give a complete description, provided with a mathematical proof, of the shape of the spectrum of the Hill operator with potential [Formula: see text], where [Formula: see text] We prove that the second critical point [Formula: see text], after which the real parts of the first and second bands disappear, is a number between [Formula: see text] and [Formula: see text]. Moreover, we prove that [Formula: see text] is the degeneration point for the first periodic eigenvalue. Besides, we give a scheme by which one can find arbitrary precise value of the second critical point as well as the [Formula: see text]th critical points after which the real parts of the [Formula: see text]th and [Formula: see text]th bands disappear, where [Formula: see text]

Creep and Fatigue of Single Crystal and Directionally Solidified Nickel-Base Blades via a Unified Approach Based on Hill48 Potential Function: Part 2 — Low Cycle Fatigue

In the first part of this work the modeling of the anisotropic plastic and creep response of Single Crystal (SX) and Directionally Solidified (DS) nickel-base blades was discussed. In this second part, the set of tools based on the Hill’s potential function is extended by showing how Hill like operators can be used to describe basic anisotropic features of the low cycle fatigue of SX and DS nickel-base superalloys. To this end a new Hill operator is introduced in order to set empirical life correlations and straightforward calibration procedures were set. The work is accompanied by applications on real heavy duty gas turbine components.