The schrodinger-operators tag has no wiki summary.
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votes
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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:
...
6
votes
1answer
193 views
Domains of raising and lowering operators in QM
I should add that what is known in physics as SUSY QM, is often called the Crum-Darboux method in Mathematics, so don't be confused about this.
Let $H : \operatorname{dom}(H) \subset L^2(\Omega) ...
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vote
0answers
52 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 ...
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68 views
Dipole Transition Integrals - Acceleration Form, What's Wrong?
I should have posted this question in a physics forum, but I think by posting in MathOverflow I may get more responses.
The following question may sound stupid, since I'm sure I was wrong somewhere, ...
2
votes
2answers
279 views
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 ...
2
votes
1answer
88 views
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 ...
3
votes
1answer
153 views
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 ...
3
votes
2answers
167 views
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 ...
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0answers
42 views
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 ...
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0answers
60 views
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 ...
0
votes
1answer
63 views
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 ...
6
votes
1answer
228 views
Spectrum of this ODE
I noticed something interesting studying this Sturm-Liouville Problem:
$$ \frac{d}{dx}\left(\sqrt{(1-x^{2})}\frac{df}{dx} \right)+\frac{\left(n \alpha x+\alpha^2 x^{2} + ...
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0answers
73 views
Decay of Eigenfunctions for the 1D Discrete Random Schrodinger Operators
Consider the operator on $\ell^2(\mathbb{Z})$
$$
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
1answer
178 views
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 ...
4
votes
2answers
184 views
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] ...
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0answers
64 views
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 ...
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0answers
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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 ...
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99 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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votes
0answers
77 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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88 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 ...
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171 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
94 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
181 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 ...
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votes
2answers
244 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
262 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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vote
0answers
58 views
Scattering solutions for $L_2$ potentials
Consider the equation
$$
Lu = -\Delta u+v(x)u = Eu, \tag{1}
$$
where $x = (x_1,x_2) \in \mathbb R^2$, $v \in L_2(\mathbb R^2)$, $E>0$. Is it known that for almost any $E>0$ and for any fixed ...
-1
votes
3answers
394 views
problem related to Airy functions [closed]
I have solved the Schrödinger equation for a triangular well potential and the solution comes in terms of Airy functions...now i am facing the following problems:
What are the normalization ...
5
votes
2answers
299 views
Asymptotic Expansion of the Schrödinger kernel?
My stackexchange post [http://math.stackexchange.com/questions/275830/schrodinger-kernels-on-manifolds] was somewhat unsatisfactory (also because I may not have stated clear enough what my interest ...
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350 views
Resonance of Schrödinger operator
Consider the dispersive estimates for the Schrödinger flow
$$
e^{itH}P_{c},\quad H=-\Delta+V \quad \text{on}\quad \mathbb{R}^n,n\ge 1
$$
where $P_{c}$ is the projection onto the continuous spectrum ...
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0answers
65 views
Random Schrödinger operators with asymmetric Lifshitz tails?
For a quantum mechanical system with a periodic Hamiltonian (Schrödinger operator) $H$, let $N(E)$ be its integrated density of states, i.e. the fraction of eigenvalues in the spectrum $\sigma(H)$ ...
3
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2answers
355 views
First eigenvalue of Schrödinger operator is simple
I once read that the first eigenvalue of a Schrödinger operator always is simple, together with an easy proof of it. But I cannot remember where. Does anybody know a reference?
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votes
2answers
318 views
Semiclassical expansions of eigenvalues of Schrödinger operators
Considering Schrödinger operators
$$ 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 ...
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votes
2answers
247 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 ...
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votes
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 ...
