In this article we develop new techniques for studying concentration of Laplace eigenfunctions $\phi_\lambda$ as their frequency, $\lambda$, grows. The method consists controlling $\phi_\lambda(x)$ by decomposing into a superposition geodesic beams that run through the point $x$. Each beam is localized in phase-space on tube centered around whose radius shrinks slightly slower than $\lambda^{-\...

Direct simulation methods of Electron Paramagnetic Resonance (EPR) spectra are an alternative to standard eigenfunction-based approaches. Both methods can be employed to find numerical solutions of the Stochastic Liouville Equation (SLE) in order to compute spectral lineshapes. It can be shown rigorously that the domain of validity of eigenfunction expansion solutions of the SLE covers only a l...

we investigate the boundary-value problem generated by the sturm-liouville equation withdiscontinuous coefficients, eigenparameter dependent boundary conditions and transmission conditionsat the point of discontinuity. with a different approach we introduce an adequate hilbert spaceformulation, investigate some properties of eigenvalues, green’s function and resolvent operator, andfind simple c...

The article describes the eigenvalue and eigenfunction problems. Basic properties, some applications and examples in system analysis are provided.

The boundary-value problem for the linear horizontally-forced sloshing problem can be solved using two different classes of eigenfunction expansions. The first will be referred to as the ”cosine” expansion since the organizing centre is a cosine series in the x−direction (the horizontal direction), and the second is called the “vertical eigenfunction expansion” since the organizing centre is a ...

Abstract Inspired by the squared eigenfunction symmetry constraint, we introduce a new τk-flow by “extending” a specific tn-flow of discrete KP hierarchy (DKPH). We construct extended discrete KPH (exDKPH), which consists of tnflow, τk-flow and tn evolution of eigenfunction and adjoint eigenfunctions, and its Lax representation. The exDKPH contains two types of discrete KP equation with self-co...

The comparison and matching based on partial description of the 3D model is the current focus of study in the shape analysis. The partial description based on the eigenfunction of the Laplace-Beltrami operator is an important way. A large number of eigenfunction values of any point on the surface of the model form a eigenvector; based on this vector, K-means clustering method will be used to qu...