Related papers: Sturm-Schroedinger equations: formula for metric

For a given subcritical discrete Schr\"odinger operator $H$ on a weighted infinite graph $X$, we construct a Hardy-weight $w$ which is optimal in the following sense. The operator $H - \lambda w$ is subcritical in $X$ for all $\lambda < 1$,…

In quantum mechanics (formulated, say, in Schr\"{o}dinger picture) only the knowledge of a complete set of observables $\Lambda_j$ enables us to declare the related physical inner product (i.e., the Hilbert-space metric $\Theta$ such that…

Operating just once the naive Foldy-Wouthuysen-Tani transformation on the Schr\"odinger equation for $Q\bar q$ bound states described by a hamiltonian, we systematically develop a perturbation theory in $1/m_Q$ which enables one to solve…

We study the scattering problem for the nonlinear Schr\"odinger equation $i\partial_t u + \Delta u = |u|^p u$ on $\mathbb{R}^d$, $d\geq 1$, with a mass-subcritical nonlinearity above the Strauss exponent. For this equation, it is known that…

We consider the Schr\"odinger equation with singular position dependent effective mass and prove that it is very weakly well posed. A uniqueness result is proved in an appropriate sense, moreover, we prove the consistency with the classical…

We provide necessary and sufficient geometric conditions for the exact observability of the Schr\"odinger equation with inverse-square potentials on the half-line. These conditions are derived from a Logvinenko-Sereda type theorem for…

This paper is dedicated to investigating the $L^p$-bounds of wave operators $W_\pm(H,\Delta^2)$ associated with fourth-order Schr\"odinger operators $H=\Delta^2+V$ on $\mathbb{R}^3$. We consider that real potentials satisfy $|V(x)|\lesssim…

We define angle $\Theta_{X,Y}$ between non-zero Hilbert--Schmidt operators $X$ and $Y$ by $\cos\Theta_{_{X,Y}} = \frac{{\rm Re}{\rm Tr}(Y^*X)}{{\|X\|}_{_2}{\|Y\|}_{_2}}$, and give some of its essentially properties. It is shown, among other…

We propose a technical reformulation of the measurement problem of quantum mechanics, which is based on the postulate that the final state of a measurement is classical. Unlike the usual formulation (in which the post-measurement state is a…

In conventional Schr\"{o}dinger representation the unitarity of the evolution of bound states is guaranteed by the Hermiticity of the Hamiltonian. A non-unitary isospectral simplification of the Hamiltonian, $\mathfrak{h} \to…

Let $M=(0,\infty)_r\times Y$ be a $d$-dimensional ($d\ge 3$) metric cone with metric<br/>$g=dr^2+r^2h$, where $(Y,h)$ is a closed Riemannian manifold. Let<br/>$H=\Delta+V_0/r^2$ be the associated Schrodinger operator, with<br/>$V_0\in…

Let $A$ be a positive bounded operator on a Hilbert space $\big(\mathcal{H}, \langle \cdot, \cdot\rangle \big)$. The semi-inner product ${\langle x, y\rangle}_A := \langle Ax, y\rangle$, $x, y\in\mathcal{H}$ induces a semi-norm…

The determination of the spectrum of a Schr\"odinger operator is a fundamental problem in mathematical quantum mechanics. We discuss a series of results showing that Schr\"odinger operators can exhibit spectra that are remarkably thin in…

Recently, Steinbach et al. introduced a novel operator $\mathcal{H}_T: L^2(0,T) \to L^2(0,T)$, known as the modified Hilbert transform. This operator has shown its significance in space-time formulations related to the heat and wave…

We solve the stationary Schr\"odinger equation for a particle confined to a 3D spherical wedge -- the region $\{(r,\theta,\phi): 0 \leq r \leq R,\, 0 \leq \theta \leq \pi,\, 0 \leq \phi \leq \Phi\}$ with Dirichlet BCs on all surfaces. This…

We undertake a comprehensive study of the nonlinear Schr\"odinger equation $$ i u_t +\Delta u = \lambda_1|u|^{p_1} u+ \lambda_2 |u|^{p_2} u, $$ where $u(t,x)$ is a complex-valued function in spacetime $\R_t\times\R^n_x$, $\lambda_1$ and…

We establish necessary and sufficient conditions for the boundedness of the relativistic Schr\"odinger operator $\mathcal{H} = \sqrt{-\Delta} + Q$ from the Sobolev space $W^{1/2}_2 (\R^n)$ to its dual $W^{-1/2}_2 (\R^n)$, for an arbitrary…

Let $A$ be a positive bounded linear operator acting on a complex Hilbert space $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle \big)$. Let $\omega_A(T)$ and ${\|T\|}_A$ denote the $A$-numerical radius and the $A$-operator seminorm of an…

Let $H(\hbar)=-\hbar^2d^2/dx^2+V(x)$ be a Schr\"odinger operator on the real line, $W(x)$ be a bounded observable depending only on the coordinate and $k$ be a fixed integer. Suppose that an energy level $E$ intersects the potential $V(x)$…

Stark-Heegner points are conjectural substitutes for Heegner points when the imaginary quadratic field of the theory of complex multiplication is replaced by a real quadratic field $K$. They are constructed analytically as local points on…