All Questions
23 questions
2
votes
0
answers
102
views
Semiclassical limit of spectral gap of Schrödinger operators with nonsmooth potential
Let $\Omega$ be a connected compact subset of $\mathbb{R}^d$. It is well known that for a smooth potential $V:\Omega \to \mathbb{R}$ that has a unique nondegenerate minimum $V(0) = 0$, the operator $H ...
- 1,551
5
votes
1
answer
311
views
Maximal operator estimates for the Schrödinger equation
Let $a>0$ and consider the operator
$$Tf(t,x)= \int_{\mathbb{R}^{n}}e^{ i x\cdot \xi} e^{i t \lvert\xi\rvert^{a}} \widehat{f}(\xi) \, d\xi.$$
When $a=2$, the function $Tf$ solves the Cauchy problem ...
- 852
1
vote
0
answers
150
views
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 ...
- 439
3
votes
1
answer
453
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 ...
- 159
2
votes
0
answers
166
views
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 ...
1
vote
1
answer
93
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
$$\rho_{N, t}^{(n)} (x_1,..,n_n; y_1,...,y_n) =
\begin{cases}
\int \rho_{N,t}(...
- 159
2
votes
0
answers
93
views
How to use Fredholm alternative to check that there are only finite eigenvalues of $H$ on the imaginary axis?
On $\mathbb{R}^3$, we consider the operator
\begin{equation}
\mathcal{H}= \left( \begin{matrix}
-\Delta +1 -2 \phi^2 & -\phi^2 \\
\phi^2 & \Delta -1 +2 \phi^2
\end{matrix} \right) , D(...
- 429
3
votes
0
answers
102
views
Determining what happens to the spectrum of Schrödinger operator as boundary condition changes
I recently came across a problem in research, and I'm asking about it here after trying in math stack exchange with no luck.
Suppose I have a metric graph $G$ (or even a closed interval, to make ...
- 251
4
votes
0
answers
164
views
What's the essential definition of resonance of Schrodinger operator?
Rencently, I am reading some articles about time decay estimates or Strichartz estimates for Schrodinger equations with potential. When considering Strichartz estimates for potential $V$ with decay $|...
- 429
2
votes
2
answers
360
views
Estimate of a solution of Schroedinger equation for a free particle
Let $\psi(x,t)$ be a solution of the Schroedinger on the line
$$i\frac{\partial \psi}{\partial t}=-\frac{1}{2m}\frac{\partial^2 \psi}{\partial x^2}.$$
One assumes that $\psi(x,0)$ "behaves well" as $...
- 21.8k
0
votes
0
answers
75
views
Partial well-posedness results on Schrödinger operators?
Set $ A_i:= -\Delta + V_i :H^2(\mathbb{R}^3) \subseteq L^2(\mathbb{R}^3) \to L^2(\mathbb{R}^3), \ i =1,2 $, where
\begin{equation*}
V_1 = 0, \ \ (\textrm{No interaction}) \\
V_2 = - \frac{\gamma}...
- 269
1
vote
0
answers
73
views
relative compact on nonlinear term
On the paper: Decay of Solutions to Nonlinear Schrodinger
Equations.
Let $u$ be a solution of the equation
$$Hu+|u|^2u=0,$$
where $H$ is a Schrodinger operator, i.e. $-\Delta+V$ and $V$ is a (...
- 1,915
1
vote
0
answers
77
views
Zero energy resonances for scaling critical Schrodinger operators
Given a real valued potential $V\in L^1(\mathbb{R}^3)$, we say that the Schrodinger operator $-\Delta + V$ has a zero-energy resonance if there exists $\psi\in L^2_{loc}(\mathbb{R}^3)\setminus L^2(\...
- 943
1
vote
2
answers
219
views
Ground state has always constant sign?
Question: What hypothesis on the potential $V$ are required such that the ground state $\phi_0$ has constant sign?
Consider the Schrödinger operator in 1 dimension with potential $V$:
$$\mathcal{H}=-...
- 129
1
vote
0
answers
79
views
Time dependent Hamiltonians
I'm studying time dependent perturbation theory on Reed-Simon book "Method of modern mathematical physics, II". If one considers an Hamiltonian of the form
$$H(t)=H_0+V(t)$$
the corresponding formal ...
- 11
6
votes
2
answers
529
views
Schrödinger eigenfunctions are bounded
Let $V:\mathbb{R}\rightarrow \mathbb{R}^{+ *}$ a real positive function such that $\displaystyle \lim_{ x \to \pm\infty} V(x)= +\infty $.
Then the Schrödinger operator $H=-\frac{d^2}{dx^2}+V(x)$ has ...
- 129
2
votes
0
answers
218
views
Existence of solutions to time-dependent Schrödinger equations
I would like to know what is known about evolution equations of the form
$$iy'(t)=H_0y(x,t)+u(t)V(x)y(x,t)$$
and $y(0)=y_0 \in D(H_0)$
where
$V$ is not a bounded operator, but an unbounded one, $u \...
- 173
0
votes
2
answers
197
views
Concerning the decay of the ground state of certain Schrodinger operators
Consider the Schrodinger operator in $n$ dimensions with a potential $V$, which grows rather quickly as $\mid x\mid$ tends to infinity, but with negative potential in a bounded region, for example, a ...
- 219
6
votes
1
answer
357
views
Travelling waves for nonlinear Schrödinger equation
Consider the following nonlinear Schrödinger equation:
$$
-\Delta \Phi - i\frac{\partial \Phi}{\partial t} = f(|\Phi|^2)\Phi,
$$
where $\Delta$ is the Laplacian on $\mathbb{R}^n$, $f$ gives the ...
- 61
1
vote
0
answers
154
views
One-parameter group of unitary operators and Core
Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ ...
- 11
7
votes
2
answers
641
views
Decay of solutions to Schrodinger equation with local minimum in potential
Consider the one-dimensional Schrodinger operator on the real line $\mathbb{R}$ given by
$$ L = - \partial_x^2 + V $$
where $V$ is a potential with the following properties:
$V$ is non-negative, and ...
- 39k
4
votes
0
answers
433
views
Scattering for rapidly decaying solutions of NLS
Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions
of the Nonlinear Schrödinger Equation" the following property.
Given the problem
\begin{equation}
\left\{
\begin{array}{rl}
...
- 403
4
votes
1
answer
355
views
(Ref req) Schrödinger heat kernel is a weak solution of parabolic Schrödinger equation
If we have nonnegative $V \in L^1_{\textrm{loc}}(\mathbb{R}^{n})$, then the operator $H = -\Delta + V$ can be defined on $L^{2}(\mathbb{R}^{n})$ via quadratic form methods. This is done by, for ...
- 363