No Arabic abstract
We consider operators of the form H+V where H is the one-dimensional harmonic oscillator and V is a zero-order pseudo-differential operator which is quasi-periodic in an appropriate sense (one can take V to be multiplication by a periodic function for example). It is shown that the eigenvalues of H+V have asymptotics of the form lambda_n(H+V)=lambda_n(H)+W(sqrt n)n^{-1/4}+O(n^{-1/2}ln(n)) as nto+infty, where W is a quasi-periodic function which can be defined explicitly in terms of V.
We show the asymptotic behavior of the eigenvalues of the non-linear integral system related to the (p,q)-Laplacian.
The Neumann-Poincare operator is a boundary-integral operator associated with harmonic layer potentials. This article proves the existence of eigenvalues within the essential spectrum for the Neumann-Poincare operator for certain Lipschitz curves in the plane with reflectional symmetry, when considered in the functional space in which it is self-adjoint. The proof combines the compactness of the Neumann-Poincare operator for curves of class $C^{2,alpha}$ with the essential spectrum generated by a corner. Eigenvalues corresponding to even (odd) eigenfunctions are proved to lie within the essential spectrum of the odd (even) component of the operator when a $C^{2,alpha}$ curve is perturbed by inserting a small corner.
In this note we consider a one-dimensional quantum mechanical particle constrained by a parabolic well perturbed by a Gaussian potential. As the related Birman-Schwinger operator is trace class, the Fredholm determinant can be exploited in order to compute the modified eigenenergies which differ from those of the harmonic oscillator due to the presence of the Gaussian perturbation. By taking advantage of Wangs results on scalar products of four eigenfunctions of the harmonic oscillator, it is possible to evaluate quite accurately the two lowest-lying eigenvalues as functions of the coupling constant $lambda$.
We consider the one-dimensional quantum harmonic oscillator perturbed by a linear operator which is a polynomial of degree $2$ in $(x,-{rm i}partial_x)$, with coefficients quasi-periodically depending on time. By establishing the reducibility results, we describe the growth of Sobolev norms. In particular, the $t^{2s}-$polynomial growth of ${mathcal H}^s-$norm is observed in this model if the original time quasi-periodic equation is reduced to a constant Stark Hamiltonian.
We consider the twisted waveguide $Omega_theta$, i.e. the domain obtained by the rotation of the bounded cross section $omega subset {mathbb R}^{2}$ of the straight tube $Omega : = omega times {mathbb R}$ at angle $theta$ which depends on the variable along the axis of $Omega$. We study the spectral properties of the Dirichlet Laplacian in $Omega_theta$, unitarily equivalent under the diffeomorphism $Omega_theta to Omega$ to the operator $H_{theta}$, self-adjoint in ${rm L}^2(Omega)$. We assume that $theta = beta - epsilon$ where $beta$ is a $2pi$-periodic function, and $epsilon$ decays at infinity. Then in the spectrum $sigma(H_beta)$ of the unperturbed operator $H_beta$ there is a semi-bounded gap $(-infty, {mathcal E}_0^+)$, and, possibly, a number of bounded open gaps $({mathcal E}_j^-, {mathcal E}_j^+)$. Since $epsilon$ decays at infinity, the essential spectra of $H_beta$ and $H_{beta - epsilon}$ coincide. We investigate the asymptotic behaviour of the discrete spectrum of $H_{beta - epsilon}$ near an arbitrary fixed spectral edge ${mathcal E}_j^pm$. We establish necessary and quite close sufficient conditions which guarantee the finiteness of $sigma_{rm disc}(H_{beta-epsilon})$ in a neighbourhood of ${mathcal E}_j^pm$. In the case where the necessary conditions are violated, we obtain the main asymptotic term of the corresponding eigenvalue counting function. The effective Hamiltonian which governs the the asymptotics of $sigma_{rm disc}(H_{beta-epsilon})$ near ${mathcal E}_j^pm$ could be represented as a finite orthogonal sum of operators of the form $-mufrac{d^2}{dx^2} - eta epsilon$, self-adjoint in ${rm L}^2({mathbb R})$; here, $mu > 0$ is a constant related to the so-called effective mass, while $eta$ is $2pi$-periodic function depending on $beta$ and $omega$.