Questions tagged [schrodinger-operators]

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When Schroedinger propagator commutes other operators?

Let $f\in \mathcal{S}(\mathbb R^d)$ (Schwartz Space). We know that $\widehat{\nabla f}(\xi)= 2 \pi i \xi \hat{f} (\xi). $ We define $$\widehat{|\nabla| f^{s}} (\xi) = (2 \pi |\xi|)^s \hat{f} (\xi), ...
XYZ's user avatar
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6 votes
2 answers
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Schrödinger eigenfunctions are bounded

Let $V:\mathbb{R}\rightarrow \mathbb{R}^{+ *}$ a real positive function such that $\displaystyle \lim_{ x \to \pm\infty} V(x)= +\infty $. Then the Schrödinger operator $H=-\frac{d^2}{dx^2}+V(x)$ has ...
M. Veruete's user avatar
2 votes
0 answers
207 views

Existence of solutions to time-dependent Schrödinger equations

I would like to know what is known about evolution equations of the form $$iy'(t)=H_0y(x,t)+u(t)V(x)y(x,t)$$ and $y(0)=y_0 \in D(H_0)$ where $V$ is not a bounded operator, but an unbounded one, $u \...
Landauer's user avatar
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3 votes
2 answers
393 views

Minimum eigenvalue of One-dimensional Schrodinger Operator

Consider the One dimensional Schrodinger Operator $$ -\frac{d^2}{dx^2} + V(x) $$ Where the Potential Function $V$ is of the form $V(x) = x^4 - a^2x^2$ , $a \in \mathbb{R} $. Now of course,the ...
Surajit's user avatar
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1 answer
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Spectral growth of One dimensional Schrodinger Operator

Conside the One dimentional Schrodinger Operator $$ -\frac{d^2}{dx^2} + ( V(x) + E ) $$ Where the Potential Function $V$ is of the form $V(x) = ax^2 + b^2x^4$ , $a,b \in \mathbb{R} $. What is known ...
Surajit's user avatar
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267 views

Invariance of sets under Schrödinger equations

We are considering the Schrödinger equation on $\mathbb{R}^d \times [0,T]$ $$i \partial_t \psi(x,t)=-\Delta \psi(x,t) + u(t)V(x) \psi(x,t), t>0$$ $$\psi(x,0):=\psi(x_0) \in L^2(\mathbb{R}^d)$$ ...
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117 views

Quantitative Approach to Existence of Minimal-Mass Blowup Solutions to NLS

Consider the mass-critical defocusing NLS in dimension $d\geq 1$: $$iu_{t}+\Delta u = |u|^{4/d}u, \quad (t,x) \in I\times\mathbb{R}^{2}$$ Define the mass $M(u)$ and scattering size $S(u)$ of the ...
Matt Rosenzweig's user avatar
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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
6 votes
1 answer
688 views

Resolvents of Schrodinger operators

In the free case one can compute the resolvents of the Laplacian $-\Delta$ in many cases explicitly, in the sense that they are given by an integral operator. Often, one uses the Hille-Yosida theorem ...
Kinzlin's user avatar
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Solutions to Schrödinger equation parameter dependence

This is somewhat unrelated to what I normally do in mathematics, which is why it may be obvious to some of you, but I was puzzled by this: If we look for classical solutions on $[0,1]$ to $$-y''(x) =...
Kinzlin's user avatar
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2 answers
182 views

Concerning the decay of the ground state of certain Schrodinger operators

Consider the Schrodinger operator in $n$ dimensions with a potential $V$, which grows rather quickly as $\mid x\mid$ tends to infinity, but with negative potential in a bounded region, for example, a ...
Wai's user avatar
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1 vote
1 answer
92 views

Dispersive estimates for one dimensional magnetic Schrodinger operators

I would like to know if there is any known result on dispersive estimates for Schrodinger operator with magnetic potential in one dimension. There is a lot of literature for three dimensional magnetic ...
Capublanca's user avatar
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1k views

Birman-Schwinger Principle

The Birman-Schwinger principle says that if $\Delta$ is the usual Laplacian on $\mathbb{R}^n$ and we consider the operator $H=-\Delta-V$ for a positive potential $V$, then, for any $\lambda>0$, the ...
cav's user avatar
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Delta function propogation in the NLS regime

consider the $(1+1)D$ or $(2+1)D$ NLS: $$ i\psi _t (t,{\bf x}) + \Delta \psi + |\psi|^2\psi = 0 \, ,$$ $$ \psi (t,|{\bf x}|\to \infty) =0 \, , \quad \psi(t=0,{\bf x}) = \psi_0 ({\bf x}) \, , $$ with $...
Amir Sagiv's user avatar
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3 votes
1 answer
144 views

Operator norm of almost mathieu operator

The almost Mathieu operator has become famous since it is the central object of the ten martini problem. In this paper here a bound on the operator norm is given. Although the bound is of course ...
Yurisov's user avatar
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4 votes
1 answer
482 views

Question about normalization factors in the direct integral of operators

So the original question I wanted to ask was this one: I'm currently a bit puzzled about the normalization for the Gelfand transform $U$: So if we have a periodic Schrödinger operator $H$, then we ...
plain's user avatar
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0 answers
56 views

Sub-matrices with a real spectrum

This question arises from the study of PT-symmetric quantum mechanics. Let $A$ be an $n\times n$ complex-valued matrix with a real spectrum. If $A$ is Hermitian, then any sub-matrix corresponding to ...
Lior Eldar's user avatar
6 votes
2 answers
264 views

Gap-opening perturbations of the periodic Schrödinger operator

I am trying to understand this short paper and I am getting stuck right at the end. Let $V(x)$ be $C^\infty$ and 1-periodic (that is, $V(x)=V(x+1)$). We are considering the operator $$A=-\dfrac{d^2}...
Darren Ong's user avatar
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0 answers
433 views

estimate of smallest eigenvalue of Schrodinger operator

I am looking for references on estimate of first nonzero Dirichlet eigenvalue for Schrodinger operator $-\Delta + V$, if sharp bounds exist, that would be better, here for simplicity, we can assume ...
Yimin's user avatar
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3 votes
1 answer
199 views

Bounded solutions for Schrödinger equation at the edge of the essential spectrum

Let $V:R^d\to R_+$ be with a compact support. The Schrödinger operator $H_a=-\Delta - a V$ acting in $L^2(R^d)$ has then (at most) finitely many negative eigenvalues. Denote the number of negative ...
poupy's user avatar
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12 votes
3 answers
2k views

Connection between solution for Schrödinger equation and solution for heat equation

It's known, that if you write imaginary unit into a heat equation you'll get time-dependent Schrödinger equation. Recently one guy discovered a connection between solutions for these two equations (...
Glinka's user avatar
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When does the ground state energy continuously depend on a parameter?

Given a family of Schrödinger operators $H_\gamma=-\Delta+V_\gamma$, under which condition is the map $\gamma\mapsto\inf\sigma(H_\gamma)$ continuous? This is surely the case for many textbook ...
Daniel's user avatar
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0 answers
191 views

How to solve Schrödinger equations with potential $|x|^{2}$ [closed]

We consider the initial value problem (IVP): $i \frac{\partial}{\partial } u(x,t) \pm (\Delta \pm |x|^2) u(x,t)=0$ and $u(x,0)= u_0(x)$, where $x,t \in \mathbb R, \Delta$ is the Laplacian. My ...
Inquisitive's user avatar
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2 votes
1 answer
1k views

Solving a simple Schrödinger equation with Fast Fourier Transforms

While trying to solve a stochastic Gross-Pitaevskii equation I have found a problem that can be tracked down to something buggy occurring in the simplest Schrodinger equation possible: $$\partial_t \...
Carlos_San's user avatar
6 votes
1 answer
351 views

Travelling waves for nonlinear Schrödinger equation

Consider the following nonlinear Schrödinger equation: $$ -\Delta \Phi - i\frac{\partial \Phi}{\partial t} = f(|\Phi|^2)\Phi, $$ where $\Delta$ is the Laplacian on $\mathbb{R}^n$, $f$ gives the ...
user84867's user avatar
6 votes
1 answer
250 views

Is this function Schwartz?

I already asked this question here on MSE, didn't get an answer, and I'm still stuck with it. Suppose I have a smooth function $\psi$ from $\mathbb{R}^n$ to $\mathbb{C}$, for which I know that $$ \...
Daniel's user avatar
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method for global existance for the NLS

We consider the nonlinar Schr\"odinger equation(NLS): $$i\frac{\partial u}{\partial t}+\Delta_{x}u +\lambda |u|^{2k}u =0, \ u(x, 0)= u_{0}(x);$$ where $\lambda \in \mathbb R, \ k \in \mathbb N,$ ...
Inquisitive's user avatar
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2 votes
0 answers
219 views

A construction of the fundamental solution for Schroedinger equations

Does someone know some book or lecture notes useful for the reading of the paper "A construction of the fundamental solution for the Schrödinger equation", Fujiwara, Daisuke, J. Analyse Math. 35 (...
Stan's user avatar
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1 vote
0 answers
150 views

One-parameter group of unitary operators and Core

Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ ...
Alphabeta's user avatar
10 votes
0 answers
275 views

Comparing spectra of Laplacian and Schrödinger operator

Let $M$ be a closed (compact without boundary) Riemannian manifold. Is there a body of results that compares the eigenvalues of the Laplace-Beltrami operator with that of Schrödinger operators $-\...
noname's user avatar
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2 votes
0 answers
184 views

The motivation of Weyl-Titchmarsh function

Given a second linear differential operator, $(Hf)(x)=-\frac{d^2}{dx^2}(x)+V(x)f(x)$, where $V$ is a bounded and real valued function, $f$ lies in $L^2(\mathbb{R})$. For an $z$ with $Im(z)\neq 0$,...
yaoxiao's user avatar
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9 votes
1 answer
439 views

Infinitesimal variation of spectrum of Schrödinger operator with changing domain

Suppose we have a Schrödinger operator $$-\frac{d^2}{dx^2}+V(x)$$ defined on $[a,b]$ with Dirichlet boundary conditions. I am interested in whether there are any results for the variation of the ...
Austen's user avatar
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3 votes
1 answer
188 views

Bounds for smallest eigenvalue of Schrödinger equation with a superpotential and periodic boundary conditions

A very specific question, but posted on the off-chance that someone may be able to help. If we have a Schrödinger equation with arbitrary "superpotential" \begin{equation} -\frac{d^2 \psi}{dx^2} ...
Gerry's user avatar
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7 votes
2 answers
610 views

Decay of solutions to Schrodinger equation with local minimum in potential

Consider the one-dimensional Schrodinger operator on the real line $\mathbb{R}$ given by $$ L = - \partial_x^2 + V $$ where $V$ is a potential with the following properties: $V$ is non-negative, and ...
Willie Wong's user avatar
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4 votes
2 answers
558 views

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: it ...
Ruslan's user avatar
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6 votes
1 answer
346 views

Domains of raising and lowering operators in QM

Let $H : \operatorname{dom}(H) \subset L^2(\Omega) \rightarrow L^2(\Omega)$, where $dom(H) \subset H^2(\Omega)$, $\Omega \subset \mathbb{R}$ should be a bounded open interval(so 1-d setting(!)) and $H$...
user avatar
2 votes
0 answers
151 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
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2 votes
2 answers
403 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 $m_{-}...
Geno Whirl's user avatar
1 vote
1 answer
391 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 ...
Fabiano's user avatar
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5 votes
1 answer
495 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 u}{\...
Alex M.'s user avatar
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3 votes
2 answers
402 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 ...
user avatar
3 votes
1 answer
224 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 ...
Geno Whirl's user avatar
1 vote
0 answers
157 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 ...
Daniel's user avatar
  • 428
1 vote
1 answer
359 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 $V(...
Frits Veerman's user avatar
5 votes
1 answer
477 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} + \lambda\right)f}{\sqrt{(1-x^{...
user avatar
0 votes
0 answers
198 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, ...
Ben's user avatar
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4 votes
1 answer
268 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 $H = -\Delta +V$....
user avatar
4 votes
2 answers
563 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] \...
user avatar
6 votes
1 answer
238 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 $L^p$...
user49794's user avatar
5 votes
0 answers
241 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
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