Questions tagged [schrodinger-operators]
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127
questions
0
votes
0answers
34 views
References to the theory of scattering resonances in dimension 1
Does anyone know a good reference to study the theory of scattering resonances in dimension 1 ?
I would also like to add that I don't know anything about this theory, so I'm starting from zero. Thanks ...
2
votes
0answers
46 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(...
4
votes
1answer
101 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 ...
0
votes
1answer
70 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^{...
2
votes
0answers
128 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 ...
5
votes
0answers
102 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 ...
1
vote
0answers
41 views
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 ...
2
votes
1answer
95 views
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 ...
3
votes
0answers
81 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 ...
1
vote
0answers
43 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,...
4
votes
0answers
86 views
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 ...
2
votes
0answers
97 views
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 $...
1
vote
0answers
103 views
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....
3
votes
0answers
153 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 (...
2
votes
0answers
376 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 ...
0
votes
0answers
123 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 ...
2
votes
0answers
39 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
$$(\mbox{number of eigenvalue} \leq 0\mbox{ , counted with multiplicity, of }-\Delta+V\,)\...
2
votes
1answer
87 views
Resonances for Schrodinger operators with radial potentials
Let $V\in L^{\infty}(\mathbb{R}^3)$ be a radial, compactly supported potential, and consider the Schrodinger operator $H:=-\Delta + V$ on $L^2(\mathbb{R}^3)$. Let $\psi$ be a resonance for $H$, i.e. a ...
7
votes
1answer
334 views
Spectrum of “classical” operators
Lately, I've been reading a couple of papers from different one-dimensional PDE contexts on which operators like $\mathcal{L}:=-\partial_x^2+c_*+\Phi$ repeatedly appear. Usually, on these contexts $\...
2
votes
0answers
57 views
Proving that $(f,g)$ are Cauchy data for the Schrödinger equation iff $(f,g)$ satisfies an equation
I have to prove that if $f\in H^{1/2}(\partial\Omega)$ then $(f,g)$ are Cauchy
data for the Schrödinger equation if and only if
$$g=\gamma^{-1/2} \Lambda_{\gamma}(\gamma^{-1/2} f)+1/2 \gamma^{-1}\...
3
votes
0answers
66 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 $|...
4
votes
0answers
139 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\...
3
votes
0answers
63 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 ...
2
votes
0answers
83 views
Lippmann-Schwinger equation for the Coulomb potential
Let $H=H_0+V$ be a Hamiltonian on $\mathbb{R}^3$ where $H_0=-\frac{\Delta}{2m}$ is the free Hamiltonian and $V$ is a potential. Let us assume first that $V$ decays sufficiently fast at infinity and ...
1
vote
0answers
59 views
On a interpolation inequality for the Schrödinger unitary group (NLS)
I'm trying to understand scattering for the classical nonlinear Schrödinger equation and for that i'm studying a scattering criterion on Tao's paper. At Lema 3.1 he states that $$\left\|e^{it\Delta}f\...
1
vote
1answer
110 views
Anderson localization for Bernoulli potentials on half-line
Anderson localisation for (discrete) Schrödinger operators with Bernoulli potentials on $l^2(\mathbb{Z})$ was proven in
https://link.springer.com/article/10.1007/BF01210702
I am wondering if there ...
1
vote
0answers
103 views
Uniform boundedness for Strichartz constants in Tao's book
This is a question from Tao's book "Nonlinear Dispersive
Equations". We define the space $S^0(I\times R^d)$ as the closure of the Schwartz functions under the norm
$$\|u\|_{S^0}=\sup_{(q,r)\text{ ...
1
vote
0answers
54 views
The ill-posedness of $L^2$-super critical nonlinear Schrödinger equation
For nonlinear Schrodinger equation$$\begin{cases}iu_t+\Delta u+|u|^\alpha u=0\\u(0)=\phi\in H^1(\mathbb R^d)\end{cases}$$
where $\alpha>\frac 4d$.
In Christ, Colliander, Tao's paper Ill-posedness ...
1
vote
0answers
162 views
Schrodinger and Laplace operators with infinitely many common eigenvalues
Let $V>0$ be a non-constant polynomial and consider the one dimensional Schrodinger operator
$H=-\frac{d^2}{dx^2}+V$ on $[0 ,L]$ with Neumann boundary condition. Can $H$ and $T=-\frac{d^2}{dx^2}$ ...
1
vote
0answers
128 views
An optimization problem for one- dimensional Schrodinger operator
For a potential of the form $V(x)=ax^4+bx^2$, where $a,b>0$, let us consider the one dimensional Schrodinger operator $D=-\frac{d^2}{dx^2}+V$ with Dirichlet B.C on $[-L,L]$ and denote its first ...
2
votes
2answers
169 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 $...
3
votes
2answers
218 views
Explicit form of S-matrix on the line
Consider the Hamiltonian $H$ on functions on the line with
\begin{eqnarray}
H=H_0+V,\\
H_0=-\frac{1}{2m}\frac{d^2}{dx^2}
\end{eqnarray}
where $V$ is a potential vanishing outside of a bounded interval ...
0
votes
0answers
64 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}...
2
votes
0answers
79 views
Spectrum of a Hamiltonian which is a perturbation of Laplacian
Let $\Delta =\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\frac{\partial^2}{\partial x_3^2}$ be the Laplacian on $\mathbb{R}^3$.
Consider a self adjoint operator $H$ on complex ...
2
votes
0answers
92 views
A generalization of scattering theory
In the quantum scattering theory one proves results of the following type.
Let $H_0$ be the Laplacian $\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\frac{\partial^2}{\partial ...
6
votes
1answer
305 views
Monotonicity of Schrödinger Eigenvalues
Let us consider the Schrödinger operator
$$
H_hf(x)=-\frac{d^2}{dx^2}f(x)+h(h\sin^2(x)-\cos(x))f(x)
$$
on $L^2[-\pi,\pi]$ with Neumann boundary conditions $f^\prime(\pm\pi)=0$. Here, $h\geq 0$ is a ...
3
votes
1answer
172 views
Non-isolated ground state of a Schrödinger operator
Question. Does there exist a dimension $d \in \mathbb{N}$ and a measurable function $V: \mathbb{R}^d \to [0,\infty)$ such that the smallest spectral value $\lambda$ of the Schrödinger operator $-\...
1
vote
1answer
402 views
A question of the Schrodinger Semigroup --By B. Simon
The question comes from the paper: B. Simon, Schrodinger Semigroups, Bull. A.M.S., (1982) Vol. 7 (3).
On the Theorem C.1.2(subsolution estimate) of the paper, it says that: If $Hu=0$, where $H=-\...
7
votes
1answer
219 views
Lower estimate of the minimal eigenvalue of a Hamiltonian
Consider a linear operator $H\colon L^2(\mathbb{R}^3)\to L^2(\mathbb{R}^3)$ given by
$$H(\psi)(x):=-\Delta\psi(x)+V(x)\cdot \psi(x),$$
where $V\colon \mathbb{R}^3\to \mathbb{R}$ is a continuous (or ...
5
votes
1answer
183 views
Anderson localization for fractional Laplacians
There is a vast literature on Anderson localization, namely, the study of decay of eigenfunctions of operators on $l^2(\mathbb{Z}^d)$ such as
$$
-\Delta+\lambda V
$$
where $\Delta$ is the discrete ...
1
vote
0answers
70 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 (...
4
votes
0answers
97 views
Anderson Localization and Homogenization theory
I originally asked this on Mathematics Stack Exchange but realized it might be better to ask it here.
The question is mostly related to homogenization theory in mathematical physics.
$\textbf{...
2
votes
2answers
195 views
Criteria for Schrödinger operator on real line to have simple spectrum
Consider a Schrödinger operator $H:=-\Delta+V$ on $\mathbb R$, where $V$ is such that $H$ has a purely discrete spectrum
$-\infty<\lambda_1\leq\lambda_2\leq\cdots$ converging to $+\infty$. Do there ...
0
votes
1answer
236 views
Elementary quantum scattering problem on the line.
Let us consider the quantum scattering problem on the line with the Hamiltonian
$$H=-\frac{d^2}{dx^2}+ V(x),$$
where $V(x)=1$ when $x\in (0,a)$, and $V(x)=0$ otherwise.
It is easy to see that $H$ ...
1
vote
2answers
191 views
Fourier transform of a generalized function on the plane
Is there an explicit formula for the Fourier transform of the generalized function of 2 variables
$$\frac{1}{x+y^2+i0}?$$
Remark. Equivalent question: consider the Schroedinger equation one the ...
4
votes
0answers
94 views
Difference Between Eigenvalues of Schrödinger Operator with Different Boundary Conditions
Consider a Schrödinger operator
$$H=-\Delta+V$$
on a nice bounded domain $\Omega\subset\mathbb R^d$ (say, a ball or a cube), and assume for simplicity that $V$ is smooth.
Let $\lambda_D,\lambda_N$ ...
0
votes
3answers
240 views
Single quantum particle entropy
Consider a wave function of a single particle in free space, whose evolution is described by the (non-dimensional) linear Schrodinger equation $$i\psi _t (t,\underline{x}) + \Delta \psi=V(\underline{x}...
0
votes
0answers
166 views
Spectrum of a Hamiltonian on the real line
Consider the following linear (Hamiltonian) operator on functions on the real line $\mathbb{R}$
$$H\psi(x)=-\frac{d^2}{dx^2}\psi(x)+V(x)\psi(x).$$
Assume that $V$ is a smooth function and $V(x)\to +\...
2
votes
1answer
207 views
Barry Simon's decay of eigenfunctions for pseudo differential operators
In his celebrated paper on Schrödinger semigroups, Barry Simon proves the following result.
Let $V_{-} \in K_\nu$, $V_{+} \in K_\nu^{\mathrm{loc}}$ and suppose that $Hu=Eu$ where $E$ is the ...
3
votes
1answer
158 views
Discrete spectrum of Schrodinger operator
Assume $\Omega$ is a non-compact region or manifold with dimension $\geq4$. Let $H=-\Delta+V$ be Schrodinger operator. Here $V$ is a (smooth)function.
I know that if $V\geq c>0$ or $V\to c>0$,...
