Questions tagged [schrodinger-operators]
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175 questions
4
votes
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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 ...
- 319
0
votes
1
answer
99
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^{...
- 239
2
votes
0
answers
145
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 ...
- 1,054
5
votes
0
answers
144
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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 ...
- 2,188
1
vote
0
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140
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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 ...
- 183
2
votes
1
answer
130
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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 ...
- 1,755
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
1
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0
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92
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
117
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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 ...
- 509
2
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0
answers
145
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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 $...
- 21
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0
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128
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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....
- 141
3
votes
0
answers
186
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 (...
- 429
3
votes
0
answers
2k
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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 ...
- 131
2
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0
answers
188
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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
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0
answers
44
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
$$(\mbox{number of eigenvalue} \leq 0\mbox{ , counted with multiplicity, of }-\Delta+V\,)\...
- 943
2
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1
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101
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Resonances for Schrodinger operators with radial potentials
Let $V\in L^{\infty}(\mathbb{R}^3)$ be a radial, compactly supported potential, and consider the Schrodinger operator $H:=-\Delta + V$ on $L^2(\mathbb{R}^3)$. Let $\psi$ be a resonance for $H$, i.e. a ...
- 943
7
votes
1
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Spectrum of "classical" operators
Lately, I've been reading a couple of papers from different one-dimensional PDE contexts on which operators like $\mathcal{L}:=-\partial_x^2+c_*+\Phi$ repeatedly appear. Usually, on these contexts $\...
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votes
0
answers
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Proving that $(f,g)$ are Cauchy data for the Schrödinger equation iff $(f,g)$ satisfies an equation
I have to prove that if $f\in H^{1/2}(\partial\Omega)$ then $(f,g)$ are Cauchy
data for the Schrödinger equation if and only if
$$g=\gamma^{-1/2} \Lambda_{\gamma}(\gamma^{-1/2} f)+1/2 \gamma^{-1}\...
- 31
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0
answers
164
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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
4
votes
0
answers
155
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\...
- 41
3
votes
0
answers
78
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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 ...
- 3,574
2
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0
answers
158
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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 ...
- 21.8k
1
vote
0
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72
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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 $$\left\|e^{it\Delta}f\...
- 111
2
votes
1
answer
165
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Anderson localization for Bernoulli potentials on half-line
Anderson localisation for (discrete) Schrödinger operators with Bernoulli potentials on $l^2(\mathbb{Z})$ was proven in
https://link.springer.com/article/10.1007/BF01210702
I am wondering if there ...
- 223
1
vote
0
answers
151
views
Uniform boundedness for Strichartz constants in Tao's book
This is a question from Tao's book "Nonlinear Dispersive
Equations". We define the space $S^0(I\times R^d)$ as the closure of the Schwartz functions under the norm
$$\|u\|_{S^0}=\sup_{(q,r)\text{ ...
- 333
1
vote
0
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76
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 ...
- 938
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0
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170
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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 $T=-\frac{d^2}{dx^2}$ ...
- 1,583
1
vote
0
answers
134
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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 ...
- 1,583
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2
answers
360
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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
3
votes
2
answers
271
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 ...
- 21.8k
0
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0
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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
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votes
0
answers
102
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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 ...
- 21.8k
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answers
107
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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 ...
- 21.8k
7
votes
1
answer
381
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 ...
- 93
4
votes
1
answer
220
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 $-\...
- 12.5k
1
vote
1
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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=-\...
- 1,915
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1
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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 ...
- 21.8k
5
votes
1
answer
241
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
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
4
votes
0
answers
127
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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{...
- 81
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votes
2
answers
380
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 ...
- 891
0
votes
1
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250
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 $H$ ...
- 21.8k
1
vote
3
answers
307
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 ...
- 21.8k
5
votes
0
answers
119
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 $\lambda_D,\lambda_N$ ...
- 891
0
votes
3
answers
389
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 $$i\psi _t (t,\underline{x}) + \Delta \psi=V(\underline{x}...
- 3,574
0
votes
0
answers
237
views
Spectrum of a Hamiltonian on the real line
Consider the following linear (Hamiltonian) operator on functions on the real line $\mathbb{R}$
$$H\psi(x)=-\frac{d^2}{dx^2}\psi(x)+V(x)\psi(x).$$
Assume that $V$ is a smooth function and $V(x)\to +\...
- 21.8k
3
votes
1
answer
469
views
Barry Simon's decay of eigenfunctions for pseudo differential operators
In his celebrated paper on Schrödinger semigroups, Barry Simon proves the following result.
Let $V_{-} \in K_\nu$, $V_{+} \in K_\nu^{\mathrm{loc}}$ and suppose that $Hu=Eu$ where $E$ is the ...
- 459
3
votes
1
answer
215
views
Discrete spectrum of Schrodinger operator
Assume $\Omega$ is a non-compact region or manifold with dimension $\geq4$. Let $H=-\Delta+V$ be Schrodinger operator. Here $V$ is a (smooth)function.
I know that if $V\geq c>0$ or $V\to c>0$,...
- 1,915
1
vote
0
answers
61
views
Realizing $N$-body Hamiltonian operator from $2$-body operator
Let $N\in\mathbb{N}$, and consider the formal $N$-body Schrodinger operator
$$\sum_{j=1}^{N}-\partial_{x_{j}}^{2}+2c\sum_{1\leq j_{1}<j_{2}\leq N}\delta(X_{j_{1}}-X_{j_{2}}), \tag{1}$$
where $c\in\...
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vote
0
answers
201
views
Perturbation of Elliptic operator
Let $\Omega$ be an open region or a non-compact complete manifold, $L$ be an elliptic operator with possibly non vanishing zero-order term, e.g. $-\Delta+q$. Suppose $W$ is an operator such that $W(...
- 1,915