Related papers: Asymptotics of eigenvalues of the Schroedinger ope…

The almost Mathieu operator is the discrete Schr\"odinger operator $H_{\alpha,\beta,\theta}$ on $\ell^2(\mathbb{Z})$ defined via $(H_{\alpha,\beta,\theta}f)(k) = f(k + 1) + f(k - 1) + \beta \cos(2\pi \alpha k + \theta) f(k)$. We derive…

We study the family of compact operators $B_{\alpha} = V A_{\alpha} V$, $\alpha>0$ in $L^2(\mathbb R^d)$, $d\ge 1$, where $A_{\alpha}$ is the pseudo-differential operator with symbol $a_{\alpha}(\boldsymbol\xi) = a(\alpha\boldsymbol\xi)$,…

Consider a quasi-periodic Schr\"odinger operator $H_{\alpha,\theta}$ with analytic potential and irrational frequency $\alpha$. Given any rational approximating $\alpha$, let $S_+$ and $S_-$ denote the union, respectively, the intersection…

We consider Schr\"odinger operators on sparse graphs. The geometric definition of sparseness turn out to be equivalent to a functional inequality for the Laplacian. In consequence, sparseness has in turn strong spectral and functional…

Spectral properties of the Schroedinger operator $A_{\lambda} = -\Delta +\lambda V$ on regular metric trees are studied. It is shown that as $\lambda$ goes to zero the behavior of the negative eigenvalues of $A_{\lambda}$ depends on the…

In this paper following the same methods in [M. Kadakal, O. Sh. Mukhtarov, Sturm-Liouville problems with discontinuities at two points, Comput. Math. Appl., 54 (2007) 1367-1379] we investigate discontinuous two-point boundary value problems…

We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schr\"odinger operator with an attractive $\delta'$-interaction of a fixed strength, the support of which is a $C^2$-smooth contour. Under…

We consider Schr\"odinger operator in dimension $d\ge 2$ with a singular interaction supported by an infinite family of concentric spheres, analogous to a system studied by Hempel and coauthors for regular potentials. The essential spectrum…

In this note, we study the asymptotics of a spherical integral that is a multiplicative counterpart to the well-known Harish-Chandra Itzykson Zuber integral. This counterpart can also be expressed in terms the Heckman-Opdam hypergeometric…

We consider a magnetic Schr\"odinger operator $H^h$, depending on a semiclassical parameter $h>0$, on a compact Riemannian manifold. We assume that there is no electric field. We suppose that the minimal value $b_0$ of the intensity of the…

In this note we consider the self-adjoint Schr\"odinger operator $\mathsf{A}_\alpha$ in $L^2(\mathbb{R}^d)$, $d\geq 2$, with a $\delta$-potential supported on a Lipschitz hypersurface $\Sigma\subseteq\mathbb{R}^d$ of strength $\alpha\in…

In this paper we study on $L^2(\mathbb{R}^d)$ the quasi-periodic Schr\"odinger operator $H=-\Delta+ \lambda V(x),$ where $V$ is a real analytic quasi-periodic function and $\lambda>0$. We first show that $H$ has no eigenvalues in…

We consider a magnetic Schr\"odinger operator $H^h$, depending on the semiclassical parameter $h>0$, on a two-dimensional Riemannian manifold. We assume that there is no electric field. We suppose that the minimal value $b_0$ of the…

This paper proves an analogue of a result of Banuelos and Sa Barreto on the asymptotic expansion for the trace of Schrodinger operators on $\R^d$ when the Laplacian $\Delta$, which is the generator of the Brownian motion, is replaced by the…

We analyze the (discrete) spectrum of the semirelativistic ``spinless-Salpeter'' Hamiltonian H = \beta \sqrt{m^2 + p^2} + V(r), beta > 0, where V(r) represents an attractive, spherically symmetric potential in three dimensions. In order to…

We have obtained the metric operator $\Theta=\exp T$ for the non-Hermitian Hamiltonian model $H=\omega(a^{\dag}a+1/2)+\alpha(a^{2}-a^{\dag^{2}})$. We have also found the intertwining operator which connects the Hamiltonian to the adjoint of…

Consider the following Lane-Emden system with Dirichlet boundary conditions: \[ -\Delta U = |V|^{\beta-1}V,\ -\Delta V = |U|^{\alpha-1}U \text{ in }\Omega,\qquad U=V= 0 \text{ on }\partial \Omega, \] in a bounded domain $\Omega$, for…

In this paper we consider eigenvalues asymptotics of the energy operator in the one of the most interesting models of quantum physics, describing an interaction between two-level system and harmonic oscillator. The energy operator of this…

We consider a single band approximation to the random Schroedinger operator in an external magnetic field. The spectrum of such an operator has been characterized in the case where delta impurities are located on the sites of a lattice. In…

The Hamiltonian of a system of two quantum mechanical particles moving on the $d$-dimensional lattice $\Z^d$ and interacting via zero-range attractive pair potentials is considered. For the two-particle energy operator $H_{\mu}(K),$ $K\in…