Search results for "Eigenvalue"

Improved embedded molecular cluster model

2002

We demonstrate that boundary effects (i.e., displacements of the cluster boundary atoms from their lattice sites and the difference between effective charges of the perfect crystal atoms and those of the cluster atoms in the case of a cluster with no point defect in it) in an embedded molecular cluster (EMC) model can be radically reduced. A new embedding scheme is proposed. It includes search for the structural elements (SE) of which perfect crystal is composed, use of corresponding to these SE expression for the total energy, and application of the degree of localization of equations consistent with the wave functions of the cluster. To get equations for the cluster wave functions, the pr…

Musical pitch quantization as an eigenvalue problem

2020

How can discrete pitches and chords emerge from the continuum of sound? Using a quantum cognition model of tonal music, we prove that the associated Schrödinger equation in Fourier space is invariant under continuous pitch transpositions. However, this symmetry is broken in the case of transpositions of chords, entailing a discrete cyclic group as transposition symmetry. Our research relates quantum mechanics with music and is consistent with music theory and seminal insights by Hermann von Helmholtz.

A simple tool to forecast the natural frequencies of thin-walled cylinders

2023

Abstract. This paper presents an approximate method to predict the natural frequencies of thin-walled cylinders. By taking inspiration from a previous work of one of the authors, the starting point of the proposed approach is a proper construction of reasonable eigenfunctions. However, a new simple tool based on the principle of virtual work has been developed to estimate the natural frequencies and the amplitude of vibration without complex numerical resolution. Moreover, the applicability of the model is extended to all the most common constraint conditions. The identification of the natural frequencies of a continuous cylinder is reduced to an eigenvalue problem based on a matrix whose e…

PT Symmetry and Weyl Asymptotics

2012

For a class of PT-symmetric operators with small random perturbations, the eigenvalues obey Weyl asymptotics with probability close to 1. Consequently, when the principal symbol is nonreal, there are many nonreal eigenvalues.

Complex powers and non-compact manifolds

2002

We study the complex powers $A^{z}$ of an elliptic, strictly positive pseudodifferential operator $A$ using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras, ``extended Weyl algebras,'' whose definition was inspired by Guillemin's paper on the subject. An extended Weyl algebra can be thought of as an algebra of ``abstract pseudodifferential operators.'' Many algebras of pseudodifferential operators are extended Weyl algebras. Several results typical for algebras of pseudodifferential operators (asymptotic completeness, construction of Sobolev spaces, boundedness between apropriate Sobolev spaces, >...) generalize to…

On the zeros of Jacobi polynomials

1994

Corrigendum to “Imaginary time propagation code for large-scale two-dimensional eigenvalue problems in magnetic fields” [Comput. Phys. Comm. 184(3) (…

2016

Zur Existenz von Lösungen gewisser Randwertaufgaben

1971

With the aid of some known results about integral equations of the Hammerstein type there is proofed an existence theorem for the following class of boundary value problems−y″−l 2 y′=f(x,y),y(a)=y(b)=0,l 2>0 mit|f(x, y)|=0,l 3 (x)>0. The existence range is determined by the greatest eigenvalue of some linear problem.

Weak commutation relations of unbounded operators and applications

2011

Four possible definitions of the commutation relation $[S,T]=\Id$ of two closable unbounded operators $S,T$ are compared. The {\em weak} sense of this commutator is given in terms of the inner product of the Hilbert space $\H$ where the operators act. Some consequences on the existence of eigenvectors of two number-like operators are derived and the partial O*-algebra generated by $S,T$ is studied. Some applications are also considered.

On the Fučík spectrum of the p-Laplacian with no-flux boundary condition

2023

In this paper, we study the quasilinear elliptic problem \begin{align*} \begin{aligned} -\Delta_{p} u&= a\l(u^+\r)^{p-1}-b\l(u^-\r)^{p-1} \quad && \text{in } \Omega,\\ u & = \text{constant} &&\text{on } \partial\Omega,\\ 0&=\int_{\partial \Omega}\left|\nabla u\right|^{p-2}\nabla u\cdot \nu \,\diff \sigma,&& \end{aligned} \end{align*} where the operator is the $p$-Laplacian and the boundary condition is of type no-flux. In particular, we consider the Fu\v{c}\'{\i}k spectrum of the $p$-Laplacian with no-flux boundary condition which is defined as the set $\fucik$ of all pairs $(a,b)\in\R^2$ such that the problem above has a nontrivial solution. It turns out…