Search results for "Schrödinger's cat"
showing 10 items of 39 documents
The Dynamical Problem for a Non Self-adjoint Hamiltonian
2012
After a compact overview of the standard mathematical presentations of the formalism of quantum mechanics using the language of C*- algebras and/or the language of Hilbert spaces we turn attention to the possible use of the language of Krein spaces.I n the context of the so-called three-Hilbert-space scenario involving the so-called PT-symmetric or quasi- Hermitian quantum models a few recent results are reviewed from this point of view, with particular focus on the quantum dynamics in the Schrodinger and Heisenberg representations.
Multiple normalized solutions for a Sobolev critical Schrödinger-Poisson-Slater equation
2021
We look for solutions to the Schr\"{o}dinger-Poisson-Slater equation $$- \Delta u + \lambda u - \gamma (|x|^{-1} * |u|^2) u - a |u|^{p-2}u = 0 \quad \text{in} \quad \mathbb{R}^3, $$ which satisfy \begin{equation*} \int_{\mathbb{R}^3}|u|^2 \, dx = c \end{equation*} for some prescribed $c>0$. Here $ u \in H^1(\mathbb{R}^3)$, $\gamma \in \mathbb{R},$ $ a \in \mathbb{R}$ and $p \in (\frac{10}{3}, 6]$. When $\gamma >0$ and $a > 0$, both in the Sobolev subcritical case $p \in (\frac{10}{3}, 6)$ and in the Sobolev critical case $p=6$, we show that there exists a $c_1>0$ such that, for any $c \in (0,c_1)$, the equation admits two solutions $u_c^+$ and $u_c^-$ which can be characterized respectively…
Determining a Random Schrödinger Operator : Both Potential and Source are Random
2020
We study an inverse scattering problem associated with a Schr\"odinger system where both the potential and source terms are random and unknown. The well-posedness of the forward scattering problem is first established in a proper sense. We then derive two unique recovery results in determining the rough strengths of the random source and the random potential, by using the corresponding far-field data. The first recovery result shows that a single realization of the passive scattering measurements uniquely recovers the rough strength of the random source. The second one shows that, by a single realization of the backscattering data, the rough strength of the random potential can be recovered…
Resonances over a potential well in an island
2020
In this paper we study the distribution of scattering resonances for a multidimensional semi-classical Schr\"odinger operator, associated to a potential well in an island at energies close to the maximal one that limits the separation of the well and the surrounding sea.
Generation of Non-Classical States through QND-like Processes
2007
In the spirit of quantum nondemolition measurement we show that repeatedly measuring the atomic state of a trapped ion subjected to suitable vibronic couplings it is possible to extract interesting nonclassical states. The possibility of generating angular momentum Schrödinger cat is demonstrated.
Strong Instability of Ground States to a Fourth Order Schrödinger Equation
2019
Abstract In this note, we prove the instability by blow-up of the ground state solutions for a class of fourth order Schrödinger equations. This extends the first rigorous results on blowing-up solutions for the biharmonic nonlinear Schrödinger due to Boulenger and Lenzmann [8] and confirm numerical conjectures from [1–3, 11].
Resolvent estimates for the magnetic Schrödinger operator in dimensions $$\ge 2$$
2019
It is well known that the resolvent of the free Schr\"odinger operator on weighted $L^2$ spaces has norm decaying like $\lambda^{-\frac{1}{2}}$ at energy $\lambda$. There are several works proving analogous high-frequency estimates for magnetic Schr\"odinger operators, with large long or short range potentials, in dimensions $n \geq 3$. We prove that the same estimates remain valid in all dimensions $n \geq 2$.
PT-symmetry and Schrödinger operators. The double well case
2015
We study a class of $PT$-symmetric semiclassical Schrodinger operators, which are perturbations of a selfadjoint one. Here, we treat the case where the unperturbed operator has a double-well potential. In the simple well case, two of the authors have proved in [6] that, when the potential is analytic, the eigenvalues stay real for a perturbation of size $O(1)$. We show here, in the double-well case, that the eigenvalues stay real only for exponentially small perturbations, then bifurcate into the complex domain when the perturbation increases and we get precise asymptotic expansions. The proof uses complex WKB-analysis, leading to a fairly explicit quantization condition.
Preparation of macroscopically distinguishable superpositions of circular or linear oscillatory states of a bidimensionally trapped ion
2000
A simple scheme for the generation of two different classes of bidimensional vibrational Schrodinger cat-like states of an isotropically trapped ion is presented. We show that by appropriately adjusting an easily controllable parameter having a clear physical meaning, the states prepared by our procedure are quantum superpositions of either vibrational axial angular momentum eigenstates or Fock states along two orthogonal directions.
Free fields via canonical transformations of matter-coupled two-dimensional dilaton gravity models
1998
It is shown that the 1+1-dimensional matter-coupled Jackiw-Teitelboim model and the model with an exponential potential can be converted by means of appropriate canonical transformations into a bosonic string theory propagating on a flat target space with an indefinite signature. This makes it possible to consistently quantize these models in the functional Schroedinger representation thus generalizing recent results on CGHS theory.