The schrodinger-operators tag has no wiki summary.
3
votes
1answer
136 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 ...
1
vote
2answers
161 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
125 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
47 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
332 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
votes
1answer
198 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 ...
3
votes
0answers
261 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
vote
0answers
55 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
259 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
votes
2answers
296 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
221 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
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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 ...
