Questions tagged [schrodinger-operators]
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84 questions with no upvoted or accepted answers
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Comparing spectra of Laplacian and Schrödinger operator
Let $M$ be a closed (compact without boundary) Riemannian manifold. Is there a body of results that compares the eigenvalues of the Laplace-Beltrami operator with that of Schrödinger operators $-\...
- 109
5
votes
0
answers
101
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 ...
- 2,095
5
votes
0
answers
144
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 ...
- 2,188
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119
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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
5
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0
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243
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introduction books for Dynamic systems of discrete Schrodinger operator for beginner
In this semester, I study in a class of dynamic system. recently the French professor turn to the dynamic system of discrete operator. I find it is difficult to find a book in English. (I have found ...
- 1,706
4
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0
answers
137
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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 ...
- 2,095
4
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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
4
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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
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0
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155
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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
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
4
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0
answers
433
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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
0
answers
1k
views
Resonance of Schrödinger operator
Consider the dispersive estimates for the Schrödinger flow
$$
e^{itH}P_{c},\quad H=-\Delta+V \quad \text{on}\quad \mathbb{R}^n,n\ge 1
$$
where $P_{c}$ is the projection onto the continuous spectrum ...
- 1,644
3
votes
0
answers
151
views
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 < \...
- 481
3
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0
answers
106
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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 $n\to\infty$ (...
- 891
3
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answers
102
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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
3
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0
answers
186
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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
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
3
votes
0
answers
119
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Quantitative Approach to Existence of Minimal-Mass Blowup Solutions to NLS
Consider the mass-critical defocusing NLS in dimension $d\geq 1$:
$$iu_{t}+\Delta u = |u|^{4/d}u, \quad (t,x) \in I\times\mathbb{R}^{2}$$
Define the mass $M(u)$ and scattering size $S(u)$ of the ...
- 873
3
votes
0
answers
80
views
When does the ground state energy continuously depend on a parameter?
Given a family of Schrödinger operators $H_\gamma=-\Delta+V_\gamma$, under which condition is the map $\gamma\mapsto\inf\sigma(H_\gamma)$ continuous?
This is surely the case for many textbook ...
- 428
2
votes
0
answers
175
views
Eigenvalues of Laplace operator and Schrödinger operator
When reading the paper Control for Schrödinger operators on tori by N.Burq and M.Zworski (Math. Res. Lett., 19(2):309–324, 2012), an inequality confused me:
Define the flat torus $\mathbb{T}^2=\mathbb{...
- 21
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
2
votes
0
answers
65
views
Generalized Fourier transforms associated to Schroedinger operators
Let $n\geq 1$. Let $q\in C^{\infty}_0(\mathbb R^n)$ be compactly supported and consider the operator $P= -\Delta+q(x)$ on $\mathbb R^n$. We will assume that $q$ is sufficiently small so that the ...
- 4,135
2
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0
answers
143
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Convergence of eigenfunctions
In their 1999 paper "Sturm-Liouville operators with singular potentials", Savchuk and Shkalikov prove the uniform resolvent convergence of an operator $L_\varepsilon \rightarrow L$ for $\...
2
votes
0
answers
192
views
A question about the regularity of the Schrödinger equation
While reading the article [1], I noticed I don't understand part of the proof of regularity. For the Schrödinger eigenvalue problem,
\begin{cases}
-\Delta u+Vu=\lambda u, &\text{in } \Omega \\
\...
- 31
2
votes
0
answers
67
views
Rotation number for multicomponent Schrödinger equation
Rotation number for Schrödinger equation of the form
\begin{equation}
-x''(t) +q(t) x(t) = E x(t)
\end{equation}
was defined in R. Johnson J. Moser "The rotation number for almost periodic ...
- 593
2
votes
0
answers
45
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 ...
- 91
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 ...
2
votes
0
answers
72
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
$$
\partial_t \psi(t,\vec{x}) = i\left(a\nabla^2 + \int_{\mathbb{R}^3} \frac{|\psi(t,\vec{y})|^2}{|...
- 121
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0
answers
93
views
How to use Fredholm alternative to check that there are only finite eigenvalues of $H$ on the imaginary axis?
On $\mathbb{R}^3$, we consider the operator
\begin{equation}
\mathcal{H}= \left( \begin{matrix}
-\Delta +1 -2 \phi^2 & -\phi^2 \\
\phi^2 & \Delta -1 +2 \phi^2
\end{matrix} \right) , D(...
- 429
2
votes
0
answers
145
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Are Weyl sequences polynomially bounded?
Look at the Hilbert space $l^2( \mathbb{Z}) $ and let $A$ be a translation invariant band operator. I.e. if $\{ e_n \}_{n \in \mathbb Z} $ is the standard basis for $l^2( \mathbb{Z}) $ then it holds ...
- 1,054
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0
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145
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When does a one-dimensional Schrödinger operator have a threshold resonance?
Consider the operator
$$ L = -\partial_x^2 + V(x),$$
for some bounded, decaying potential, i.e. $V(x)\to 0$ as $x\to \pm \infty$. I'm interested in the $L^2(\mathbb R)$ spectrum of $L$. We know that $...
- 21
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0
answers
188
views
Schrodinger operator with matrix potential
This question is inspired by attempts to extend in some way Shen's 1999 "On fundamental solutions of generalized Schrödinger operators" to Schrödinger operators $- \Delta + V $ with some ...
2
votes
0
answers
44
views
Cwikel–Lieb–Rosenbljum inequality including zero resonances
The Cwikel–Lieb–Rosenbljum inequality asserts that, for any potential $V:\mathbb{R}^n\to\mathbb{R}$, we have
$$(\mbox{number of eigenvalue} \leq 0\mbox{ , counted with multiplicity, of }-\Delta+V\,)\...
- 943
2
votes
0
answers
62
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
$$g=\gamma^{-1/2} \Lambda_{\gamma}(\gamma^{-1/2} f)+1/2 \gamma^{-1}\...
- 31
2
votes
0
answers
158
views
Lippmann-Schwinger equation for the Coulomb potential
Let $H=H_0+V$ be a Hamiltonian on $\mathbb{R}^3$ where $H_0=-\frac{\Delta}{2m}$ is the free Hamiltonian and $V$ is a potential. Let us assume first that $V$ decays sufficiently fast at infinity and ...
- 21.8k
2
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answers
102
views
Spectrum of a Hamiltonian which is a perturbation of Laplacian
Let $\Delta =\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\frac{\partial^2}{\partial x_3^2}$ be the Laplacian on $\mathbb{R}^3$.
Consider a self adjoint operator $H$ on complex ...
- 21.8k
2
votes
0
answers
107
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A generalization of scattering theory
In the quantum scattering theory one proves results of the following type.
Let $H_0$ be the Laplacian $\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\frac{\partial^2}{\partial ...
- 21.8k
2
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0
answers
218
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Existence of solutions to time-dependent Schrödinger equations
I would like to know what is known about evolution equations of the form
$$iy'(t)=H_0y(x,t)+u(t)V(x)y(x,t)$$
and $y(0)=y_0 \in D(H_0)$
where
$V$ is not a bounded operator, but an unbounded one, $u \...
- 173
2
votes
0
answers
223
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A construction of the fundamental solution for Schroedinger equations
Does someone know some book or lecture notes useful for the reading of the paper
"A construction of the fundamental solution for the Schrödinger equation", Fujiwara, Daisuke, J. Analyse Math. 35 (...
- 21
2
votes
0
answers
190
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The motivation of Weyl-Titchmarsh function
Given a second linear differential operator,
$(Hf)(x)=-\frac{d^2}{dx^2}(x)+V(x)f(x)$,
where $V$ is a bounded and
real valued function, $f$ lies in $L^2(\mathbb{R})$.
For an $z$ with $Im(z)\neq 0$,...
- 1,706
2
votes
0
answers
159
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Regularity of Schrödinger Resolvent
The following problem keeps bothering me:
Let $H:=-\Delta+V$ be a Schrödinger-Operator in $\mathbb{R}^n$, where $V$ is a Kato-Potential of type $K_n$, which especially yields that $H$ is e.s.a. on $...
- 428
2
votes
0
answers
152
views
Finite Volume 1D Anderson Tight Binding Model
My question is about bounds on the number of eigenvalues in a microscopic interval for the random Schrodinger operator on $\mathbb{Z}_n$ for $n \in \mathbb{N}$. For my question, these are the ...
- 195
2
votes
0
answers
137
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eigenfunction of schrodinger operators
For a Schrodinger operator $H=\Delta+V$, with very nice potential, such as in Schwartz class, and if $0$ is an eigenvalue, furthermore, there exists a positive eigenfunction associated with 0, then my ...
- 879
1
vote
0
answers
28
views
Explicit rate of decay of the positive standing wave of the subcritical nonlinear Schrödinger equation
Consider the following semilinear problem:
$$
\begin{cases}
- \Delta u + u = u |u|^{p - 2}
&\text{in} ~ \mathbb{R}^N;
\\
u (x) \to 0 &\text{as} ~ |x| \to \infty,
\end{cases}
$$
where $N \geq 2$...
- 144
1
vote
1
answer
253
views
What can one say about the Dirichlet problem for Schrödinger equation with negative potential?
Consider the Schrödinger type equation in $\Bbb R^2$:
$$
\Delta f(x,y)+c(x,y)f(x,y)=0
$$
where $c(x,y)$ is a positive (!) function everywhere analytic on the plane, and $\Delta$ is the Laplace ...
1
vote
0
answers
88
views
Pointwise convergence of Schrodinger's equation with potential term
A famous problem of Carleson asks if $f\in H^s(\mathbb{R}^n)$, under what condition of $s$ do we have almost everywhere pointwise convergence of the solution to the Schrodinger's equation
$$iu_t-\...
- 265
1
vote
0
answers
43
views
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^...
- 481
1
vote
0
answers
47
views
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,...
- 840
1
vote
0
answers
150
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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 ...
- 439