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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 ...
Eduardo Longa's user avatar
5 votes
0 answers
243 views

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 ...
yaoxiao's user avatar
  • 1,706
4 votes
0 answers
137 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 ...
Eduardo Longa's user avatar
4 votes
0 answers
127 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{...
Sanchit's user avatar
  • 81
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 (...
Tao's user avatar
  • 429
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 ...
Lwins's user avatar
  • 1,551
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 \\ \...
Du Xin's user avatar
  • 31
2 votes
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 ...
Joshua Isralowitz's user avatar
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}\...
energy's user avatar
  • 31
2 votes
0 answers
159 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 $...
Daniel's user avatar
  • 428
1 vote
0 answers
30 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 $N \geq 2$...
gpr1's user avatar
  • 144
1 vote
1 answer
254 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 ...
Ilya Kossovskiy's user avatar
1 vote
0 answers
126 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 ...
Leo Moos's user avatar
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1 vote
0 answers
64 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 + ...
Joshua Isralowitz's user avatar
1 vote
0 answers
72 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\...
Alan's user avatar
  • 111
1 vote
0 answers
89 views

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.
Fadil Kikawi's user avatar
0 votes
0 answers
78 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) = ...
Stack_Underflow's user avatar
0 votes
0 answers
67 views

Examples of symmetry-breaking solitons which retain a subgroup symmetry

There are many works on spontaneous symmetry breaking in the Nonlinear Schrödinger equation with asymmetric soliton solutions. However, all symmetry breaking soliton examples I have seen go from the ...
Leo Anibal's user avatar