Questions tagged [schrodinger-operators]
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165
questions
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When Schroedinger propagator commutes other operators?
Let $f\in \mathcal{S}(\mathbb R^d)$ (Schwartz Space).
We know that $\widehat{\nabla f}(\xi)= 2 \pi i \xi \hat{f} (\xi). $ We define $$\widehat{|\nabla| f^{s}} (\xi) = (2 \pi |\xi|)^s \hat{f} (\xi), ...
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2
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Schrödinger eigenfunctions are bounded
Let $V:\mathbb{R}\rightarrow \mathbb{R}^{+ *}$ a real positive function such that $\displaystyle \lim_{ x \to \pm\infty} V(x)= +\infty $.
Then the Schrödinger operator $H=-\frac{d^2}{dx^2}+V(x)$ has ...
2
votes
0
answers
207
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Existence of solutions to time-dependent Schrödinger equations
I would like to know what is known about evolution equations of the form
$$iy'(t)=H_0y(x,t)+u(t)V(x)y(x,t)$$
and $y(0)=y_0 \in D(H_0)$
where
$V$ is not a bounded operator, but an unbounded one, $u \...
3
votes
2
answers
393
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Minimum eigenvalue of One-dimensional Schrodinger Operator
Consider the One dimensional Schrodinger Operator
$$
-\frac{d^2}{dx^2} + V(x)
$$
Where the Potential Function $V$ is of the form $V(x) = x^4 - a^2x^2$ , $a \in \mathbb{R} $.
Now of course,the ...
4
votes
1
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268
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Spectral growth of One dimensional Schrodinger Operator
Conside the One dimentional Schrodinger Operator
$$
-\frac{d^2}{dx^2} + ( V(x) + E )
$$
Where the Potential Function $V$ is of the form $V(x) = ax^2 + b^2x^4$ , $a,b \in \mathbb{R} $.
What is known ...
0
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1
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267
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Invariance of sets under Schrödinger equations
We are considering the Schrödinger equation on $\mathbb{R}^d \times [0,T]$
$$i \partial_t \psi(x,t)=-\Delta \psi(x,t) + u(t)V(x) \psi(x,t), t>0$$
$$\psi(x,0):=\psi(x_0) \in L^2(\mathbb{R}^d)$$
...
2
votes
0
answers
117
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Quantitative Approach to Existence of Minimal-Mass Blowup Solutions to NLS
Consider the mass-critical defocusing NLS in dimension $d\geq 1$:
$$iu_{t}+\Delta u = |u|^{4/d}u, \quad (t,x) \in I\times\mathbb{R}^{2}$$
Define the mass $M(u)$ and scattering size $S(u)$ of the ...
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88
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Eignfunctions of an elliptic operator
I am looking for a reference which makes it possible to say that one can devellop $f$ in the form of the sentence underlined by the yellow.
Thank you in advance.
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688
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Resolvents of Schrodinger operators
In the free case one can compute the resolvents of the Laplacian $-\Delta$ in many cases explicitly, in the sense that they are given by an integral operator. Often, one uses the Hille-Yosida theorem ...
0
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141
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Solutions to Schrödinger equation parameter dependence
This is somewhat unrelated to what I normally do in mathematics, which is why it may be obvious to some of you, but I was puzzled by this:
If we look for classical solutions on $[0,1]$ to
$$-y''(x) =...
0
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2
answers
182
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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 ...
1
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1
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92
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Dispersive estimates for one dimensional magnetic Schrodinger operators
I would like to know if there is any known result on dispersive estimates for Schrodinger operator with magnetic potential in one dimension.
There is a lot of literature for three dimensional magnetic ...
4
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1
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1k
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Birman-Schwinger Principle
The Birman-Schwinger principle says that if $\Delta$ is the usual Laplacian on $\mathbb{R}^n$ and we consider the operator $H=-\Delta-V$ for a positive potential $V$, then, for any $\lambda>0$, the ...
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0
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Delta function propogation in the NLS regime
consider the $(1+1)D$ or $(2+1)D$ NLS:
$$ i\psi _t (t,{\bf x}) + \Delta \psi + |\psi|^2\psi = 0 \, ,$$
$$ \psi (t,|{\bf x}|\to \infty) =0 \, , \quad \psi(t=0,{\bf x}) = \psi_0 ({\bf x}) \, , $$
with $...
3
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1
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144
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Operator norm of almost mathieu operator
The almost Mathieu operator has become famous since it is the central object of the ten martini problem.
In this paper here a bound on the operator norm is given. Although the bound is of course ...
4
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1
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482
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Question about normalization factors in the direct integral of operators
So the original question I wanted to ask was this one:
I'm currently a bit puzzled about the normalization for the Gelfand transform $U$:
So if we have a periodic Schrödinger operator $H$, then we ...
0
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0
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56
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Sub-matrices with a real spectrum
This question arises from the study of PT-symmetric quantum mechanics.
Let $A$ be an $n\times n$ complex-valued matrix with a real spectrum.
If $A$ is Hermitian, then any sub-matrix corresponding to ...
6
votes
2
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264
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Gap-opening perturbations of the periodic Schrödinger operator
I am trying to understand this short paper and I am getting stuck right at the end.
Let $V(x)$ be $C^\infty$ and 1-periodic (that is, $V(x)=V(x+1)$).
We are considering the operator
$$A=-\dfrac{d^2}...
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0
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433
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estimate of smallest eigenvalue of Schrodinger operator
I am looking for references on estimate of first nonzero Dirichlet eigenvalue for Schrodinger operator $-\Delta + V$, if sharp bounds exist, that would be better, here for simplicity, we can assume ...
3
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1
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199
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Bounded solutions for Schrödinger equation at the edge of the essential spectrum
Let $V:R^d\to R_+$ be with a compact support. The Schrödinger operator $H_a=-\Delta - a V$ acting in $L^2(R^d)$ has then (at most) finitely many negative eigenvalues. Denote the number of negative ...
12
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3
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Connection between solution for Schrödinger equation and solution for heat equation
It's known, that if you write imaginary unit into a heat equation you'll get time-dependent Schrödinger equation. Recently one guy discovered a connection between solutions for these two equations (...
3
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0
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79
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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 ...
1
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0
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191
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How to solve Schrödinger equations with potential $|x|^{2}$ [closed]
We consider the initial value problem (IVP):
$i \frac{\partial}{\partial } u(x,t) \pm (\Delta \pm |x|^2) u(x,t)=0$
and
$u(x,0)= u_0(x)$, where $x,t \in \mathbb R, \Delta$ is the Laplacian.
My ...
2
votes
1
answer
1k
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Solving a simple Schrödinger equation with Fast Fourier Transforms
While trying to solve a stochastic Gross-Pitaevskii equation I have found a problem that can be tracked down to something buggy occurring in the simplest Schrodinger equation possible:
$$\partial_t \...
6
votes
1
answer
351
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 ...
6
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1
answer
250
views
Is this function Schwartz?
I already asked this question here on MSE, didn't get an answer, and I'm still stuck with it.
Suppose I have a smooth function $\psi$ from $\mathbb{R}^n$ to $\mathbb{C}$, for which I know that
$$
\...
1
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0
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76
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method for global existance for the NLS
We consider the nonlinar Schr\"odinger equation(NLS):
$$i\frac{\partial u}{\partial t}+\Delta_{x}u +\lambda |u|^{2k}u =0, \ u(x, 0)= u_{0}(x);$$
where $\lambda \in \mathbb R, \ k \in \mathbb N,$ ...
2
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0
answers
219
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A construction of the fundamental solution for Schroedinger equations
Does someone know some book or lecture notes useful for the reading of the paper
"A construction of the fundamental solution for the Schrödinger equation", Fujiwara, Daisuke, J. Analyse Math. 35 (...
1
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0
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150
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One-parameter group of unitary operators and Core
Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ ...
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275
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Comparing spectra of Laplacian and Schrödinger operator
Let $M$ be a closed (compact without boundary) Riemannian manifold. Is there a body of results that compares the eigenvalues of the Laplace-Beltrami operator with that of Schrödinger operators $-\...
2
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0
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184
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The motivation of Weyl-Titchmarsh function
Given a second linear differential operator,
$(Hf)(x)=-\frac{d^2}{dx^2}(x)+V(x)f(x)$,
where $V$ is a bounded and
real valued function, $f$ lies in $L^2(\mathbb{R})$.
For an $z$ with $Im(z)\neq 0$,...
9
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1
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439
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Infinitesimal variation of spectrum of Schrödinger operator with changing domain
Suppose we have a Schrödinger operator
$$-\frac{d^2}{dx^2}+V(x)$$
defined on $[a,b]$ with Dirichlet boundary conditions. I am interested in whether there are any results for the variation of the ...
3
votes
1
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188
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Bounds for smallest eigenvalue of Schrödinger equation with a superpotential and periodic boundary conditions
A very specific question, but posted on the off-chance that someone may be able to help. If we have a Schrödinger equation with arbitrary "superpotential"
\begin{equation}
-\frac{d^2 \psi}{dx^2} ...
7
votes
2
answers
610
views
Decay of solutions to Schrodinger equation with local minimum in potential
Consider the one-dimensional Schrodinger operator on the real line $\mathbb{R}$ given by
$$ L = - \partial_x^2 + V $$
where $V$ is a potential with the following properties:
$V$ is non-negative, and ...
4
votes
2
answers
558
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Examples of potentials for which Schrödinger equation lacks discrete points in continuous spectrum
In Landau, Lifshitz, "Quantum Mechanics, non-relativistic theory" in $\S18$ "The fundamental properties of Schrödinger's equation" the following is said about potential $U(x,y,z)$ in a footnote:
it ...
6
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1
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346
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Domains of raising and lowering operators in QM
Let $H : \operatorname{dom}(H) \subset L^2(\Omega) \rightarrow L^2(\Omega)$, where $dom(H) \subset H^2(\Omega)$, $\Omega \subset \mathbb{R}$ should be a bounded open interval(so 1-d setting(!)) and $H$...
2
votes
0
answers
151
views
Regularity of Schrödinger Resolvent
The following problem keeps bothering me:
Let $H:=-\Delta+V$ be a Schrödinger-Operator in $\mathbb{R}^n$, where $V$ is a Kato-Potential of type $K_n$, which especially yields that $H$ is e.s.a. on $...
2
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2
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403
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Geometrical interpretation of a Schrödinger operator
Consider a $2 \times 2$ Hermitian (or symmetric) matrix-valued function
$$g(x) = \{ g_{jk}(x)\}_{j,k=1,2}, \quad x \in \mathbb{R}^{2},$$
such that $0 < m_{-}I \leq g(x) \leq m_{+}I$, for some $m_{-}...
1
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1
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391
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Limit-circle and limit-point at endpoints
I was wondering if the following holds:
If you have an ODE $$-y''(x) + q(x) y(x) = \lambda y(x)$$ on a finite interval $(a,b)$ and you know that this equation is limit-circle or limit-point at the ...
5
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1
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495
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The complex heat kernel on a Riemann manifold
There is a vast literature available for the heat kernel. Nevertheless, I haven't been able to find almost anything useful about the kernel of the equation $\frac{1}{\mathbb{i}} \frac{\partial u}{\...
3
votes
2
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402
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Schrödinger operators on a sphere
if you have a Schrödinger operator on a sphere ( $\mathbb{S}^2$) $-\Delta_{\theta,\phi} \psi(\theta,\phi) + V(\theta) \psi(\theta,\phi) = E\psi(\theta,\phi),$ where the potential does not depend on ...
3
votes
1
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224
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Self-adjointness of the components of the magnetic derivative
On $L^{2}(\mathbb{R}^{n})$ define the operator $\Pi_{j} u := (-i\partial/\partial x_{j} - A_{j})u$, where $A_{j} \in L^{2}_{loc}(\mathbb{R}^{n})$ represents the $j$-th component of the magnetic ...
1
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0
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157
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Asymptotic decay for the inhomogeneous Schrödinger equation
Let $H$ be an Schrödinger operator $(-\Delta+V(x))$ with a radially symmetric and smooth potential and $H$ is e.s.a. on $C^\infty_0(\mathbb{R}^n)$, furthermore let $\lambda$ be an element of the ...
1
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1
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359
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Zeroes of Sturm-Liouville solutions as a function of the (complex) eigenvalue
Given the Sturm-Liouville type (time independent Schroedinger) equation
\begin{equation}
\frac{d^2 y}{d x^2} - \left(\mu + V(x)\right) y = \lambda \, y,\quad x \in \mathbb{R}
\end{equation}
where $V(...
5
votes
1
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477
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Spectrum of this ODE
I noticed something interesting studying this Sturm-Liouville Problem:
$$ \frac{d}{dx}\left(\sqrt{(1-x^{2})}\frac{df}{dx} \right)+\frac{\left(n \alpha x+\alpha^2 x^{2} + \lambda\right)f}{\sqrt{(1-x^{...
0
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0
answers
198
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Decay of Eigenfunctions for the 1D Discrete Random Schrodinger Operators
Consider the operator on $\ell^2(\mathbb{Z})$
$$
H = \Delta + v.
$$ Here $\Delta$ is the nearest neighbour Laplacian on $\mathbb{Z}$, $\Delta_{k, \ell} =1 $ if $|k - \ell| =1 $ and zero otherwise, ...
4
votes
1
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268
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Asymptotic behavior of Schrödinger operators
I am currently dealing with $1$ or at most $2$-dimensional Schrödinger operators on compact domains. A classical result of spectral theory is the Weyl approximations for this operator
$H = -\Delta +V$....
4
votes
2
answers
563
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Solution to Schrödinger equation
I asked this question already on stackexchange, but I did not get any resonance at all, so maybe anybody here can give me a few hints about my problem.
My goal is to solve this PDE for $f:[-1,1] \...
6
votes
1
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238
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Time decay for Hartree equation with Coulomb potential
Are there any time-decay results for the solution of the Hartree equation
\begin{equation}\frac{1}{i}\partial_t\phi-\Delta\phi=-(|x|^{-1}\ast|\phi|^2)\phi,\quad x\in\mathbb{R}^3\end{equation} in $L^p$...
5
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0
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241
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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 ...