Questions tagged [schrodinger-operators]

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Understanding a Bessel function gluing argument of Simon

I would like to construct a real-valued function $f$ on $(0, \infty)$ with the following properties: $f(r)$ is $C^1$ on $(0,\infty)$ and $C^\infty$ on $(0,1) \cup (1, \infty)$, $-f'' + \tfrac{3}{4}r^...
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Question on a mixed-norm estimate

I am currently reading the paper Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on $\mathbb{R}$ by Colliander, Holmer, Visan, Zhang. In this article,...
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Mathematical study of dispersive PDEs [closed]

My understanding is that there have been a lot of activities in harmonic analysis and PDEs that use sophisticated tools to study dispersive PDEs like the Schrodinger's equation. E.g. the Strichartz ...
Simplyorange's user avatar
1 vote
1 answer
226 views

Physical relevancy of two curious PDE's

My research has brought me to the following linear parabolic second order PDE: $$ \frac{\partial^2}{\partial x^2}\Psi(t,x)=c(t,x)\frac{\partial}{\partial t}\Psi(t,x) $$ for $c(t,x)=-\frac{t}{x}$ and $...
53Demonslayer's user avatar
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Is there a space of smooth functions dense in the domain of Coulomb-like potentials in dimension two?

Let $V : \mathbb{R}^2 \to \mathbb{R}$ be compactly supported, bounded away from the origin, and obey $$ |V(x)| \lesssim r^{-\delta_0}, \qquad 0 < |x| \le 1, \qquad r : =|x|,$$ for some $0 < \...
JZS's user avatar
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Eigenvalues of Schrödinger operator with Robin condition on the boundary

Let $(M^2,g)$ be a compact Riemannian surface with boundary and let $L = \Delta_g + q$ be a Schrödinger operator, where $\Delta_g = -\operatorname{div} \nabla$ is the Laplacian with respect to the ...
Eduardo Longa's user avatar
4 votes
1 answer
141 views

Spectrum near zero of $-\partial^2_x + V : L^2(\mathbb{R}) \to L^2(\mathbb{R})$, where $V = O(|x|^{-2 - \delta})$

Let $H = -\partial^2_x + V(x) : L^2(\mathbb{R}) \to L^2(\mathbb{R})$ be a one dimensional Schrödinger operator, where the potential $V$ is real-valued, belongs to $L^\infty(\mathbb{R})$, and, as $|x| \...
JZS's user avatar
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Schrödinger equation with nonstandard boundary conditions

Consider the partial differential equation $$\psi_t(t,x)=i\kappa \psi_{xx}(t,x) ~\mbox{for}~ 0<(t,x)\in\mathbb{R}\times\mathbb{R}$$ with boundary conditions $$\psi(0,x)=0 ~\mbox{for}~ x>0,$$ $$\...
Arnold Neumaier's user avatar
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1 answer
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Is the extension (dual restriction) operator on any smooth hypersurface a solution to some PDE?

We know that the extension operator on paraboloids $\widehat{fd\sigma}(t,x)=\int_\mathbb{R}^nf(\xi)e^{i(t|\xi|^2+x\cdot\xi)}d\xi$ is a solution to the homogeneous Schrodinger equation with initial ...
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Counterexamples to weak dispersion for the Schrödinger group

Let $A$ be a selfadjoint operator on some Hilbert space $H$, let $U(t)=e^{itA}$ be the corresponding continuous group, and let $f\in H$ be orthogonal to all eigenvectors of $A$. Are there examples ...
Piero D'Ancona's user avatar
3 votes
2 answers
214 views

Change of variables for obtaining a unitary group

Consider the following NLS: $$i u_t + \Delta u- 2 \operatorname{Re} u = F(u),$$ where $F(u):=(u + \bar{u} + |u|^2)u.$ In Scattering for the Gross–Pitaevskii equation, the authors S. Gustafson, K. ...
Mr. Proof's user avatar
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Analyticity of solutions to Schrödinger's equation

Take Schrödinger's equation on $\mathbb{R}$, $i\partial_t\psi(x,t)=H\psi(x,t)$. Assume that $\psi(x,0)$ has compact support. Using known integral formulas for the propagators, it is fairly ...
J_P's user avatar
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regularity of ground state of Schrodinger operator

I have a probably naive question on the regularity of ground state of the Schrodinger operator: $\Delta u - Vu = Eu$, where $\lim_{|x|\rightarrow\infty}V(x) = +\infty$ and $V\in C^2$, and $E$ is the ...
PPDE's user avatar
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Uniform boundedness Schrödinger operator eigenfunctions with Dirichlet conditions

I would like to ask a question with possibly a reference. If we have a Schrödinger operator $-\Delta+V$ on an interval $[0,L]$ with $V$ continous and Dirichlet conditions, can we state that the ...
Paolo Bernuzzi's user avatar
3 votes
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376 views

Duality argument

$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$Throughout my studying for some papers, in particular, the proof of localized Strichartz estimates, I encountered a use of the ...
Mr. Proof's user avatar
2 votes
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36 views

Potential scattering for non-decaying potential

I am reading this note on potential scattering by Siu-Hung Tang and Maciej Zworski. Just wondering if there is a scattering theory for the Schrödinger with a non-decaying potential $- \Delta + V$ on ...
alby's user avatar
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Green's function for elliptic PDE with potential

$\newcommand{\div}{\operatorname{div}}$Suppose I have an elliptic operator $\mathcal{L} u = -\div (A \nabla u) $ on some open set $\Omega \subseteq \mathbb{R}^d$ where here $A$ is uniformly elliptic ...
Joshua Isralowitz's user avatar
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Time evolution of Wigner transform

I am studying the Hartree equation for N-particles for the first time and things are not clear to me. Given the density matrix $$\gamma_{N, t}^{(n)} (x_1,..,n_n; y_1,...,y_n) = \begin{cases} \int \...
Mr. Proof's user avatar
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1 answer
84 views

Integration of Wigner transform

I am a mathematician studying the dynamics of the $N$-Body density matrix $\rho_{N}(x;y)$ for $n$ particles, defined by $$\rho_{N, t}^{(n)} (x_1,..,n_n; y_1,...,y_n) = \begin{cases} \int \rho_{N,t}(...
Mr. Proof's user avatar
2 votes
0 answers
59 views

Maximal Lyapunov exponent of Schrödinger-Newton equation

I am trying to determine the sign of the maximal Lyapunov exponent of the Schrödinger-Newton equation $$ \partial_t \psi(t,\vec{x}) = i\left(a\nabla^2 + \int_{\mathbb{R}^3} \frac{|\psi(t,\vec{y})|^2}{|...
trillianhaze's user avatar
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89 views

When are nodal lines on a sphere geodesics?

Let $(S^2, g)$ be a Riemannian sphere and let $L := \Delta_{S^2} + q$ be a Schrödinger operator on $S^2$. Suppose that $L$ has index equal to one and that $u \in C^{\infty}(S^2)$ ($u \neq 0$) lies in ...
Eduardo Longa's user avatar
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110 views

Reference for global theory of Schrödinger operators

Question. What is a good reference to learn about the spectral properties of Schrödinger operators in $\mathbf{R}^n$? I am specifically interested in references that discuss examples where the ...
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Positive semidefinite fundamental solution to Schrodinger operator

Lets say $V : \mathbb{R}^n \rightarrow \mathbb{M}_d (\mathbb{R})$ is a $d \times d$ symmetric, positive semidefinite matrix function on $\mathbb{R}^n$ and consider the Schrodinger operator $- \Delta + ...
Joshua Isralowitz's user avatar
2 votes
1 answer
107 views

Intuition/references for understanding bound states/discrete spectrum relationship

I am trying to form intuition for the following `well-known' facts about spectrum of unbounded operators (Schrodinger/wave etc.) $L$ on $\mathbb{R}^n$. Let $\lambda\in\mathbb{R}$ satisfy $Lf=\lambda f$...
mathamphetamine's user avatar
3 votes
0 answers
90 views

Convergence of Schrödinger ground states in $L^p$ for $p\neq 2$

Suppose that $H=-\Delta+V$ is a Schrödinger operator with a unique ground state $\psi$. Suppose that $H_n=-\Delta+V_n$ is a sequence of operators such that $V_n\to V$ in some sense as $n\to\infty$ (...
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Even and odd solutions for the Schrödinger equation

We consider $2a$ - periodic smooth solutions for \begin{eqnarray*} -\Delta u+V(x)\,u=0\qquad\hbox{in}\:[-a,a] \end{eqnarray*} We assume that $V$ is smooth and even (i.e. $V(-x)=V(x)$). We also assume ...
guest61's user avatar
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6 votes
1 answer
162 views

Fractional derivative notation in wave turbulence

This is my first question in MathOverflow and I will do my best to format it correctly and make it clear. I am reading a paper on dispersive wave turbulence which introduces the following family of ...
Nick S's user avatar
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5 votes
1 answer
234 views

Recovering the nonlinear Schrödinger equation from its Lax pair

My question is less concerned with the physical aspects of the nonlinear Schrödinger equation and more with the mathematical mechanics of using a Lax pair. I am considering how to recover the ...
Talmsmen's user avatar
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0 answers
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How to use Fredholm alternative to check that there are only finite eigenvalues of $H$ on the imaginary axis?

On $\mathbb{R}^3$, we consider the operator \begin{equation} \mathcal{H}= \left( \begin{matrix} -\Delta +1 -2 \phi^2 & -\phi^2 \\ \phi^2 & \Delta -1 +2 \phi^2 \end{matrix} \right) , D(...
Tao's user avatar
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4 votes
1 answer
161 views

Nonlinear ODE to linear PDE?

I am interested in when and how one can trade a non-liner ODE for a linear PDE. To explain what this could look like here is a physics-inspired discussion. Consider a classical mechanical system with ...
Weather Report's user avatar
0 votes
1 answer
93 views

An inequality for uniqueness proof of NLS

Setting Although this detail is not relevant to my question, let me set the problem that my question arise. We are considering an initial value problem \begin{align*} \begin{cases} u\in L^\infty(I,H^{...
Lev Bahn's user avatar
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2 votes
0 answers
141 views

Are Weyl sequences polynomially bounded?

Look at the Hilbert space $l^2( \mathbb{Z}) $ and let $A$ be a translation invariant band operator. I.e. if $\{ e_n \}_{n \in \mathbb Z} $ is the standard basis for $l^2( \mathbb{Z}) $ then it holds ...
Frederik Ravn Klausen's user avatar
5 votes
0 answers
122 views

Is there a discrete Schrödinger operator with empty spectrum?

A relatively well-known example of (continuous) Schrödinger operator with empty spectrum is the complex Airy operator on the line, i.e., the operator acting on $L^{2}(\mathbb{R})$ given by the ...
Twi's user avatar
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1 vote
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Domain issues regarding the Duhamel formula for the linear Schrödinger equation

I have some questions in succession regarding the rigorous domain issues about a Duhamel expansion formula (stated near the end of my post) for the linear Schrödinger equation. Consider a linear ...
Cabbage's user avatar
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2 votes
1 answer
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An inequality of spacetime Banach space for non-linear Schrodinger equation

I am reading a proof for the existence of a solution to the Local Cauchy problem of the non-linear Schrodinger equation $$ i\partial_t u+\Delta u +\epsilon u |u|^{2} = 0 \\ u(x,0)=u_0(x) $$ The ...
Ma Joad's user avatar
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3 votes
0 answers
91 views

Determining what happens to the spectrum of Schrödinger operator as boundary condition changes

I recently came across a problem in research, and I'm asking about it here after trying in math stack exchange with no luck. Suppose I have a metric graph $G$ (or even a closed interval, to make ...
GSofer's user avatar
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1 vote
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55 views

Semigroup theory for non-symmetric Markov processes / complex-valued potentials

Let $X$ be a continuous-time Markov process on a countable state space $E$, and let $V:E\mapsto\mathbb C$ be some complex function. $X$ can be characterized by its transition rates $(\lambda_{xy})_{x,...
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4 votes
0 answers
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Looking for an electronic copy of Lebeau's paper

I would like to know if anyone has an electronic copy of the paper "Gilles Lebeau - Contrôle De L'Équation De Schrödinger"? This article appeared in Journal de Mathématiques Pures et ...
Math23's user avatar
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When does a one-dimensional Schrödinger operator have a threshold resonance?

Consider the operator $$ L = -\partial_x^2 + V(x),$$ for some bounded, decaying potential, i.e. $V(x)\to 0$ as $x\to \pm \infty$. I'm interested in the $L^2(\mathbb R)$ spectrum of $L$. We know that $...
user's user avatar
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0 answers
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Angular excitations and Schrodinger operators with radial potential in N-dimensions

Can someone please explain the following in mathematical language? "First of all, angular excitations only push the energy up, never down, so it is enough to analyze spherically symmetric s-waves....
user2002's user avatar
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3 votes
0 answers
174 views

How to prove the following linearized operator is positive?

In $L^2(\mathbb{R}^d)$, let $Q$ be the solution to \begin{equation} -\Delta Q+\alpha^2 Q = |Q|^{2\sigma} Q, \end{equation} and $Q$ satisfies that it is positive, radial, and exponentially decaying (...
Tao's user avatar
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3 votes
0 answers
1k views

The Node Theorem - an argument from physics

The theorem on the number of zeros of a solution to a Sturm-Liouville equation is a well-know result in quantum mechanics. It doesn't seem to have a special name in the mathematics literature, but it ...
Benjamin T's user avatar
2 votes
0 answers
161 views

Schrodinger operator with matrix potential

This question is inspired by attempts to extend in some way Shen's 1999 "On fundamental solutions of generalized Schrödinger operators" to Schrödinger operators $- \Delta + V $ with some ...
Joshua Isralowitz's user avatar
2 votes
0 answers
42 views

Cwikel–Lieb–Rosenbljum inequality including zero resonances

The Cwikel–Lieb–Rosenbljum inequality asserts that, for any potential $V:\mathbb{R}^n\to\mathbb{R}$, we have $$(\mbox{number of eigenvalue} \leq 0\mbox{ , counted with multiplicity, of }-\Delta+V\,)\...
Capublanca's user avatar
2 votes
1 answer
95 views

Resonances for Schrodinger operators with radial potentials

Let $V\in L^{\infty}(\mathbb{R}^3)$ be a radial, compactly supported potential, and consider the Schrodinger operator $H:=-\Delta + V$ on $L^2(\mathbb{R}^3)$. Let $\psi$ be a resonance for $H$, i.e. a ...
Capublanca's user avatar
7 votes
1 answer
355 views

Spectrum of "classical" operators

Lately, I've been reading a couple of papers from different one-dimensional PDE contexts on which operators like $\mathcal{L}:=-\partial_x^2+c_*+\Phi$ repeatedly appear. Usually, on these contexts $\...
Sharik's user avatar
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2 votes
0 answers
61 views

Proving that $(f,g)$ are Cauchy data for the Schrödinger equation iff $(f,g)$ satisfies an equation

I have to prove that if $f\in H^{1/2}(\partial\Omega)$ then $(f,g)$ are Cauchy data for the Schrödinger equation if and only if $$g=\gamma^{-1/2} \Lambda_{\gamma}(\gamma^{-1/2} f)+1/2 \gamma^{-1}\...
energy's user avatar
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4 votes
0 answers
125 views

What's the essential definition of resonance of Schrodinger operator?

Rencently, I am reading some articles about time decay estimates or Strichartz estimates for Schrodinger equations with potential. When considering Strichartz estimates for potential $V$ with decay $|...
Tao's user avatar
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4 votes
0 answers
148 views

Schrodinger operator with magnetic field: eigenvalues

Consider the self-adjoint operator on $L^{2}(\mathbb{R}^{N})$, $$H=-\frac{1}{2}(\nabla-iA)^{2}+V,$$ where $A\in C^{\infty}(\mathbb{R}^{N}, \mathbb{R}^{N} )$, $V\in C^{\infty}(\mathbb{R}^{N})$, $V\...
user152385's user avatar
3 votes
0 answers
74 views

Logarithmic Sobolev growth of time-space-periodic Schrödinger solutions

Consider the following Schrödinger equation $$i\partial_t \psi (t,x) + \Delta \psi - V(t,x) \psi = 0 \, ,$$ where $x\in \mathbb{T}^d$ and $V(t,\cdot)$ is real, smooth, and periodic (with a diophantine ...
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