Related papers: On eigenstates for some $sl_2$ related Hamiltonian

This paper is devoted to the construction of what we will call {\em exactly solvable models}, i.e. of quantum mechanical systems described by an Hamiltonian $H$ whose eigenvalues and eigenvectors can be explicitly constructed out of some…

We consider a Hamiltonian $\hat H$ with a (partially) continuous spectrum and examine the zero-overlap condition which involves the projection onto exact continuum eigenstates of a set of pseudostates obtained from the diagonalization of…

We show that if we start from a symmetric lower semi-bounded Schr\"odinger operator $\mathcal{H}$ on finitely supported functions on a discrete weighted graph (satisfying certain conditions), apply the Friedrichs construction to get a…

In this article a study was made of the conditions under which a Hamiltonian which is an element of the complex $ \left\{ h (1) \oplus h(1) \right\} \uplus u(2) $ Lie algebra admits ladder operators which are also elements of this algebra.…

Consider a semiclassical Hamiltonian \begin{equation*} H_{V, h} := h^{2} \Delta + V - E \end{equation*} where $h > 0$ is a semiclassical parameter, $\Delta$ is the positive Laplacian on $\mathbb{R}^{d}$, $V$ is a smooth, compactly supported…

We build the coherent states for a family of solvable singular Schr\"odinger Hamiltonians obtained through supersymmetric quantum mechanics from the truncated oscillator. The main feature of such systems is the fact that their…

We consider the 1D Schr\"odinger operator $Hy=-y''+(p+q)y$ with a periodic potential $p$ plus compactly supported potential $q$ on the real line. The spectrum of $H$ consists of an absolutely continuous part plus a finite number of simple…

The spectral problem $(A + V(z))\psi=z\psi$ is considered where the main Hamiltonian $A$ is a self-adjoint operator of sufficiently arbitrary nature. The perturbation $V(z)=-B(A'-z)^{-1}B^{*}$ depends on the energy $z$ as resolvent of…

We study the existence of solutions in Hilbert space $H$ of the semilinear equation \[ L u+N(u)=h, \] where $L$ is linear self-adjoint, $N$ is a nonlinear operator and $h\in H$. We concentrate on the case when $0$ is a right boundary point…

In this survey article we consider the operator pair $(H,H_0)$ where $H_0$ is the shifted 3D Schr\"odinger operator with constant magnetic field, $H : = H_0 + V$, and $V$ is a short-range electric potential of a fixed sign. We describe the…

We compute the deficiency spaces of operators of the form $H_A{\hat{\otimes}} I + I{\hat{\otimes}} H_B$, for symmetric $H_A$ and self-adjoint $H_B$. This enables us to construct self-adjoint extensions (if they exist) by means of von…

We consider the Schrodinger operator with a constant magnetic field in the exterior of a compact domain on the plane. The spectrum of this operator consists of clusters of eigenvalues around the Landau levels. We discuss the rate of…

We show that the algebra of commuting Hamiltonians of the homogeneous XXX Heisenberg model has simple spectrum on the subspace of singular vectors of the tensor product of two-dimensional $gl_2$-modules. As a byproduct we show that there…

For nonrelativistic Hamiltonians which are shape invariant, analytic expressions for the eigenvalues and eigenvectors can be derived using the well known method of supersymmetric quantum mechanics. Most of these Hamiltonians also possess…

Let $H:D(H)\subseteq{\mathscr F}\to{\mathscr F}$ be self-adjoint and let $A:D(H)\to{\mathscr F}$ (playing the role of the annihilator operator) be $H$-bounded. Assuming some additional hypotheses on $A$ (so that the creation operator…

Using an isomorphism between Hilbert spaces $L^2$ and $\ell^{2}$ we consider Hamiltonians which have tridiagonal matrix representations (Jacobi matrices) in a discrete basis and an eigenvalue problem is reduced to solving a three term…

We consider the one-dimensional discrete Schr\"odinger operator $$ \bigl[H(x,\omega)\varphi\bigr](n)\equiv -\varphi(n-1)-\varphi(n+1) + V(x + n\omega)\varphi(n)\ , $$ $n \in \mathbb{Z}$, $x,\omega \in [0, 1]$ with real-analytic potential…

We analyze the spectrum of the generalized Schrodinger operator in $L^2(R^\nu) \nu \geq 2$, with a general local, rotationally invariant singular interaction supported by an infinite family of concentric, equidistantly spaced spheres. It is…

In this paper, we consider Schr\"odinger operators on $L^2(0,\infty)$ given by \begin{align} Hu=(H_0+V)u=-u^{\prime\prime}+V_0u+Vu,\nonumber \end{align} where $V_0$ is real, $1$-periodic and $V$ is the perturbation. It is well known that…

We introduce a self-adjoint time operator $T_w = i\hbar\bigl(\partial_E + \tfrac12\,\partial_E\ln w(E)\bigr)$ on the weighted energy space $L^2(\mathbb R,\,w(E)\,dE)$. Under mild conditions on the weight $w$ (positivity, local absolute…