Questions tagged [schrodinger-operators]
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179 questions
0 votes
0 answers
163 views
How soon does an energy-critical NLS with a finite $L^1$-deficit blow up?
For spatial dimension $d \geq 3$ consider the energy-critical, focusing nonlinear Schrödinger equation $$i\partial_t u + \Delta u + c(t)|u|^{\frac{4}{d-2}}u = 0, \qquad (t,x) \in (0,\infty) \times \...
2 votes
0 answers
166 views
Spectrum of "tail" of harmonic oscillator
Define the "harmonic oscillator" $T = -\Delta + x^2$ on the domain $\mathcal D(T) = \{u \in H^2(\mathbb R) : -\Delta u + x^2 u \in L^2(\mathbb R)\}$. I am interested in analysing a "...
- 121
3 votes
0 answers
144 views
Dynamics of Coulomb potential
Consider the operator $$H=-\Delta + \frac{1}{\vert x \vert}.$$ It is known that this is a self-adjoint operator on $L^2(\mathbb R^3)$ with domain $H^2(\mathbb R^3),$ the second Sobolev space. Now ...
3 votes
0 answers
130 views
A version of unique continuation
There are many results of the type 'an eigenfunction of a Schrödinger operator can not be too small at infinity'. Smallness in this context may mean superexponential decay, or compact support, ...
- 9,142
7 votes
1 answer
288 views
Ground states of Schrödinger operators via the Fokker–Planck equation
I will introduce the problem set-up, but I assume most of this is likely well-known by the reader. Apologies if this is elementary, the main questions can be found at the bottom. The Fokker–Planck ...
- 171
2 votes
0 answers
79 views
Explicit rate of decay of the positive standing wave of the subcritical nonlinear Schrödinger equation
Consider the following semilinear problem: $$ \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} $$ where $N \geq 2$...
- 154
4 votes
1 answer
131 views
Mapping properties of the Schrödinger semigroup
The Schrödinger semigroup $e^{t(-\Delta +V(x))}$ for Kato class potentials is fairly well-understood. A classical reference is the AMS paper "Schrödinger Semigroups" by Barry Simon. I was ...
- 1,127
3 votes
1 answer
357 views
Eigenvalues of Laplace operator and Schrödinger operator
When reading the paper Control for Schrödinger operators on tori by N.Burq and M.Zworski (Math. Res. Lett., 19(2):309–324, 2012), an inequality confused me: Define the flat torus $\mathbb{T}^2=\mathbb{...
- 31
0 votes
0 answers
87 views
In what sense is a change of boundary conditions a finite rank perturbation?
Also asked on MSE (https://math.stackexchange.com/questions/4987654/in-what-sense-is-a-change-of-boundary-conditions-a-finite-rank-perturbation and https://math.stackexchange.com/questions/4875398/how-...
- 121
2 votes
0 answers
153 views
Semiclassical limit of spectral gap of Schrödinger operators with nonsmooth potential
Let $\Omega$ be a connected compact subset of $\mathbb{R}^d$. It is well known that for a smooth potential $V:\Omega \to \mathbb{R}$ that has a unique nondegenerate minimum $V(0) = 0$, the operator $H ...
- 1,551
1 vote
0 answers
68 views
Energy estimation of density operator to von Neumann equation
Consider the Schrödinger equation on $\mathbb R_+\times\mathbb R^n$ as follows: $$i\partial_t\varphi(t,x)=-\frac12\Delta_x\varphi(t,x),\quad \varphi(0,x)=\varphi_0(x).$$ Denote by $\varphi$ its ...
- 1,693
0 votes
0 answers
106 views
Nonlinear quadratic Schrödinger equation with variable coefficients
Consider the following quadratic Schrödinger equation in $Q_T = \Omega \times [0,T]$: $$\begin{cases} i \partial_t u + \kappa(x,t) \Delta u + \beta(x,t) u^2 = f(x,t)\\ u(x,0) = ...
- 111
2 votes
2 answers
327 views
Show that the kernel $|x -y|^{-1}$ on $\mathbb{R}^3 \times \mathbb{R}^3$ is Hilbert-Schmidt with respect to a weighted $L^2$ space
Let $\langle x \rangle := \left(1 + |x|^2\right)^{1/2}$, $x \in \mathbb{R}^3$. For $s > 1$, consider the weighted convolution operator \begin{equation*} T_s \varphi = \langle x \rangle^{-s} \int_{\...
- 491
2 votes
0 answers
97 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 ...
- 4,189
2 votes
1 answer
108 views
Eigenvectors of matrices and solutions of (finite dimensional) Schroedinger equation
I am trying to understand certain statement in physical literature (a reference is given below). My question is a finite dimensional version of what is really necessary. Let $A,B$ be Hermitian $n\...
- 23k
0 votes
1 answer
146 views
Spatially localised solution to the Schrödinger equation with potential is a combination of eigenfunctions
In this Terry Tao's blog post, he claims that if one has a solution to the Schrödinger equation $$i\,\partial_t u +\Delta u=Vu $$ with a "reasonably smooth and localised $V$", $u$ has ...
- 166
2 votes
1 answer
243 views
Difference in essential spectrum between Schrodinger operators
I am considering two Schrodinger operators on $\mathbb{Z}^2$ and compare their essential spectrum. The operators are both of the form $H=A+V$ where $A$ is the adjacency operator on the $\mathbb{Z}^2$-...
- 703
2 votes
0 answers
174 views
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 $\...
1 vote
1 answer
425 views
Eigenvalues of a Schrödinger operator
I'm interested in the existence of eigenfunctions and finding eigenvalues of the following operator $$L(\varphi) = \varphi_{rr} - \frac{1}{r} \varphi_r - [V + \frac{m}{r^2}] \varphi$$ $$\varphi(0) = \...
- 453
2 votes
0 answers
268 views
A question about the regularity of the Schrödinger equation
While reading the article [1], I noticed I don't understand part of the proof of regularity. For the Schrödinger eigenvalue problem, \begin{cases} -\Delta u+Vu=\lambda u, &\text{in } \Omega \\ \...
- 31
0 votes
0 answers
54 views
Eigenvalues of minors to Schrodinger matrices
Suppose that we have a graph $G$, define the hamiltonian $H$ on it as $$Hu(x) = \sum_{y\sim x}u(y).$$ Consider the operator $H+V$ where $V$ multiplies the value $u(x)$ in any vertex by the potential ...
1 vote
1 answer
113 views
Spectrum below zero for $-\beta(x) \partial^2_x : L^2(\mathbb{R}) \to L^2(\mathbb{R})$
Let $\beta \in L^\infty(\mathbb{R} ; (0, \infty))$ be bounded from above and below by positive constants. Consider the self-adjoint operator $ -\beta^{-1} \partial^2_x : L^2(\mathbb{R}; \beta dx) \to ...
- 491
1 vote
1 answer
393 views
What can one say about the Dirichlet problem for Schrödinger equation with negative potential?
Consider the Schrödinger type equation in $\Bbb R^2$: $$ \Delta f(x,y)+c(x,y)f(x,y)=0 $$ where $c(x,y)$ is a positive (!) function everywhere analytic on the plane, and $\Delta$ is the Laplace ...
2 votes
0 answers
98 views
Rotation number for multicomponent Schrödinger equation
Rotation number for Schrödinger equation of the form \begin{equation} -x''(t) +q(t) x(t) = E x(t) \end{equation} was defined in R. Johnson J. Moser "The rotation number for almost periodic ...
- 613
6 votes
1 answer
484 views
Maximal operator estimates for the Schrödinger equation
Let $a>0$ and consider the operator $$Tf(t,x)= \int_{\mathbb{R}^{n}}e^{ i x\cdot \xi} e^{i t \lvert\xi\rvert^{a}} \widehat{f}(\xi) \, d\xi.$$ When $a=2$, the function $Tf$ solves the Cauchy problem ...
- 790
1 vote
0 answers
124 views
Pointwise convergence of Schrodinger's equation with potential term
A famous problem of Carleson asks if $f\in H^s(\mathbb{R}^n)$, under what condition of $s$ do we have almost everywhere pointwise convergence of the solution to the Schrodinger's equation $$iu_t-\...
- 265
1 vote
0 answers
63 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^...
- 491
1 vote
0 answers
63 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,...
- 936
2 votes
0 answers
175 views
Mathematical study of dispersive PDEs [closed]
My understanding is that there have been a lot of activities in harmonic analysis and PDEs that use sophisticated tools to study dispersive PDEs like the Schrodinger's equation. E.g. the Strichartz ...
- 265
1 vote
1 answer
302 views
Physical relevancy of two curious PDE's
My research has brought me to the following linear parabolic second order PDE: $$ \frac{\partial^2}{\partial x^2}\Psi(t,x)=c(t,x)\frac{\partial}{\partial t}\Psi(t,x) $$ for $c(t,x)=-\frac{t}{x}$ and $...
- 621
3 votes
0 answers
175 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 < \...
- 491
4 votes
0 answers
164 views
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 ...
- 2,127
4 votes
1 answer
219 views
Spectrum near zero of $-\partial^2_x + V : L^2(\mathbb{R}) \to L^2(\mathbb{R})$, where $V = O(|x|^{-2 - \delta})$
Let $H = -\partial^2_x + V(x) : L^2(\mathbb{R}) \to L^2(\mathbb{R})$ be a one dimensional Schrödinger operator, where the potential $V$ is real-valued, belongs to $L^\infty(\mathbb{R})$, and, as $|x| \...
- 491
1 vote
1 answer
136 views
Schrödinger equation with nonstandard boundary conditions
Consider the partial differential equation $$\psi_t(t,x)=i\kappa \psi_{xx}(t,x) ~\mbox{for}~ 0<(t,x)\in\mathbb{R}\times\mathbb{R}$$ with boundary conditions $$\psi(0,x)=0 ~\mbox{for}~ x>0,$$ $$\...
- 4,122
1 vote
1 answer
199 views
Is the extension (dual restriction) operator on any smooth hypersurface a solution to some PDE?
We know that the extension operator on paraboloids $\widehat{fd\sigma}(t,x)=\int_\mathbb{R}^nf(\xi)e^{i(t|\xi|^2+x\cdot\xi)}d\xi$ is a solution to the homogeneous Schrodinger equation with initial ...
- 275
9 votes
1 answer
769 views
Counterexamples to weak dispersion for the Schrödinger group
Let $A$ be a selfadjoint operator on some Hilbert space $H$, let $U(t)=e^{itA}$ be the corresponding continuous group, and let $f\in H$ be orthogonal to all eigenvectors of $A$. Are there examples ...
- 9,142
3 votes
2 answers
262 views
Change of variables for obtaining a unitary group
Consider the following NLS: $$i u_t + \Delta u- 2 \operatorname{Re} u = F(u),$$ where $F(u):=(u + \bar{u} + |u|^2)u.$ In Scattering for the Gross–Pitaevskii equation, the authors S. Gustafson, K. ...
- 159
1 vote
0 answers
189 views
Analyticity of solutions to Schrödinger's equation
Take Schrödinger's equation on $\mathbb{R}$, $i\partial_t\psi(x,t)=H\psi(x,t)$. Assume that $\psi(x,0)$ has compact support. Using known integral formulas for the propagators, it is fairly ...
- 439
2 votes
1 answer
132 views
Uniform boundedness Schrödinger operator eigenfunctions with Dirichlet conditions
I would like to ask a question with possibly a reference. If we have a Schrödinger operator $-\Delta+V$ on an interval $[0,L]$ with $V$ continous and Dirichlet conditions, can we state that the ...
3 votes
1 answer
563 views
Duality argument
$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$Throughout my studying for some papers, in particular, the proof of localized Strichartz estimates, I encountered a use of the ...
- 159
2 votes
0 answers
68 views
Potential scattering for non-decaying potential
I am reading this note on potential scattering by Siu-Hung Tang and Maciej Zworski. Just wondering if there is a scattering theory for the Schrödinger with a non-decaying potential $- \Delta + V$ on ...
- 91
2 votes
0 answers
232 views
Green's function for elliptic PDE with potential
$\newcommand{\div}{\operatorname{div}}$Suppose I have an elliptic operator $\mathcal{L} u = -\div (A \nabla u) $ on some open set $\Omega \subseteq \mathbb{R}^d$ where here $A$ is uniformly elliptic ...
1 vote
0 answers
71 views
Time evolution of Wigner transform
I am studying the Hartree equation for N-particles for the first time and things are not clear to me. Given the density matrix $$\gamma_{N, t}^{(n)} (x_1,..,n_n; y_1,...,y_n) = \begin{cases} \int \...
- 159
1 vote
1 answer
124 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 $$\rho_{N, t}^{(n)} (x_1,..,n_n; y_1,...,y_n) = \begin{cases} \int \rho_{N,t}(...
- 159
2 votes
0 answers
92 views
Maximal Lyapunov exponent of Schrödinger-Newton equation
I am trying to determine the sign of the maximal Lyapunov exponent of the Schrödinger-Newton equation $$ \partial_t \psi(t,\vec{x}) = i\left(a\nabla^2 + \int_{\mathbb{R}^3} \frac{|\psi(t,\vec{y})|^2}{|...
- 121
5 votes
0 answers
122 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 ...
- 2,127
1 vote
0 answers
171 views
Reference for global theory of Schrödinger operators
Question. What is a good reference to learn about the spectral properties of Schrödinger operators in $\mathbf{R}^n$? I am specifically interested in references that discuss examples where the ...
- 5,163
1 vote
0 answers
87 views
Positive semidefinite fundamental solution to Schrodinger operator
Lets say $V : \mathbb{R}^n \rightarrow \mathbb{M}_d (\mathbb{R})$ is a $d \times d$ symmetric, positive semidefinite matrix function on $\mathbb{R}^n$ and consider the Schrodinger operator $- \Delta + ...
2 votes
1 answer
195 views
Intuition/references for understanding bound states/discrete spectrum relationship
I am trying to form intuition for the following `well-known' facts about spectrum of unbounded operators (Schrodinger/wave etc.) $L$ on $\mathbb{R}^n$. Let $\lambda\in\mathbb{R}$ satisfy $Lf=\lambda f$...
- 116
3 votes
0 answers
135 views
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