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3
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1answer
43 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 ...
2
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0answers
70 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 ...
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0answers
60 views

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 ...
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0answers
58 views

Weighted energy estimate for the heat equation of higher order

The question is originally related to Hardy's uncertainty principle, convexity and Schrodinger evolutions. In this work the authors deduce a convex property of Schrodinger equation by doing it first ...
2
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0answers
129 views

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} ...
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0answers
72 views

Wave operator for focusing NLS

Consider the NLS equation \begin{equation} \left\{ \begin{array}{rl} iu_t + \Delta u+u|u|^{\alpha}=0\\ u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\ \end{array}\right. \end{equation} where ...
3
votes
1answer
143 views

(Ref req) Schrödinger heat kernel is a weak solution of parabolic Schrödinger equation

If we have nonnegative $V \in L^1_{\textrm{loc}}(\mathbb{R}^{n})$, then the operator $H = -\Delta + V$ can be defined on $L^{2}(\mathbb{R}^{n})$ via quadratic form methods. This is done by, for ...
2
votes
2answers
174 views

Schrodinger's equation via Spectral Theorem

How do you prove basic facts on the Schrodinger equation using the spectral theorem? More precisely, here is what I have in mind. The version of the Spectral Theorem I am familiar with is the ...
0
votes
1answer
157 views

direct proof that schrodinger's equation kernel corresponds to delta-function initial value [closed]

I want to show directly, that the kernel for the n-dimensional free linear schrodinger equation, if taken to time t=0, is dirac's $\delta $ function. I can show that the integral is constant, but it ...
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0answers
53 views

Scattering solutions for $L_2$ potentials

Consider the equation $$ Lu = -\Delta u+v(x)u = Eu, \tag{1} $$ where $x = (x_1,x_2) \in \mathbb R^2$, $v \in L_2(\mathbb R^2)$, $E>0$. Is it known that for almost any $E>0$ and for any fixed ...
0
votes
3answers
340 views

problem related to Airy functions [closed]

I have solved the Schrödinger equation for a triangular well potential and the solution comes in terms of Airy functions...now i am facing the following problems: What are the normalization ...
5
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1answer
203 views

Asymptotic Expansion of the Schrödinger kernel?

My stackexchange post [http://math.stackexchange.com/questions/275830/schrodinger-kernels-on-manifolds] was somewhat unsatisfactory (also because I may not have stated clear enough what my interest ...
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274 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 ...
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vote
0answers
56 views

Random Schrödinger operators with asymmetric Lifshitz tails?

For a quantum mechanical system with a periodic Hamiltonian (Schrödinger operator) $H$, let $N(E)$ be its integrated density of states, i.e. the fraction of eigenvalues in the spectrum $\sigma(H)$ ...
3
votes
2answers
269 views

First eigenvalue of Schrödinger operator is simple

I once read that the first eigenvalue of a Schrödinger operator always is simple, together with an easy proof of it. But I cannot remember where. Does anybody know a reference?
2
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2answers
300 views

Semiclassical expansions of eigenvalues of Schrödinger operators

Considering Schrödinger operators $$ H(\hbar) = \hbar \Delta + V $$ where $V$ is some potential, perturbation theory tells that the eigenvalues of $H(\hbar)$ are holomorphic on some region containing ...
2
votes
2answers
225 views

Eigenvalues in the semiclassical limit

Consider the Schrödinger operator $H_\hbar = -\hbar^2\Delta + V$ on $M=\mathbb{R}^n$, where $V$ is a potential that behaves well in a certain sense ($C^\infty$, bounded from below, going to infinity ...
25
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1answer
2k views

A puzzling remark of Manin (ICM 1978)

Manin ends his 1978 ICM talk with this remark: I would also like to mention I. M. Gel'fand's suggestion that the $\zeta$-functions of certain special differential operators should have an arithmetic ...