Questions tagged [schrodinger-operators]

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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 ...
David Feldman's user avatar
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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10 votes
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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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9 votes
1 answer
694 views

Counterexamples to weak dispersion for the Schrödinger group

Let $A$ be a selfadjoint operator on some Hilbert space $H$, let $U(t)=e^{itA}$ be the corresponding continuous group, and let $f\in H$ be orthogonal to all eigenvectors of $A$. Are there examples ...
Piero D'Ancona's user avatar
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
  • 1,038
7 votes
1 answer
365 views

Spectrum of "classical" operators

Lately, I've been reading a couple of papers from different one-dimensional PDE contexts on which operators like $\mathcal{L}:=-\partial_x^2+c_*+\Phi$ repeatedly appear. Usually, on these contexts $\...
Sharik's user avatar
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7 votes
2 answers
777 views

Asymptotic expansion of the Schrödinger kernel?

My stackexchange post was somewhat unsatisfactory (also because I may not have stated clear enough what my interest was). So here it goes! Let $M$ be a compact Riemannian manifold and $\Delta$ be the ...
Matthias Ludewig's user avatar
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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7 votes
1 answer
235 views

Lower estimate of the minimal eigenvalue of a Hamiltonian

Consider a linear operator $H\colon L^2(\mathbb{R}^3)\to L^2(\mathbb{R}^3)$ given by $$H(\psi)(x):=-\Delta\psi(x)+V(x)\cdot \psi(x),$$ where $V\colon \mathbb{R}^3\to \mathbb{R}$ is a continuous (or ...
asv's user avatar
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7 votes
1 answer
370 views

Monotonicity of Schrödinger Eigenvalues

Let us consider the Schrödinger operator $$ H_hf(x)=-\frac{d^2}{dx^2}f(x)+h(h\sin^2(x)-\cos(x))f(x) $$ on $L^2[-\pi,\pi]$ with Neumann boundary conditions $f^\prime(\pm\pi)=0$. Here, $h\geq 0$ is a ...
julian's user avatar
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2 answers
508 views

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
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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1 answer
176 views

Fractional derivative notation in wave turbulence

This is my first question in MathOverflow and I will do my best to format it correctly and make it clear. I am reading a paper on dispersive wave turbulence which introduces the following family of ...
Nick S's user avatar
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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
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
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
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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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
1 answer
263 views

Maximal operator estimates for the Schrödinger equation

Let $a>0$ and consider the operator $$Tf(t,x)= \int_{\mathbb{R}^{n}}e^{ i x\cdot \xi} e^{i t \lvert\xi\rvert^{a}} \widehat{f}(\xi) \, d\xi.$$ When $a=2$, the function $Tf$ solves the Cauchy problem ...
Medo'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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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
5 votes
1 answer
232 views

Anderson localization for fractional Laplacians

There is a vast literature on Anderson localization, namely, the study of decay of eigenfunctions of operators on $l^2(\mathbb{Z}^d)$ such as $$ -\Delta+\lambda V $$ where $\Delta$ is the discrete ...
Abdelmalek Abdesselam's user avatar
5 votes
1 answer
284 views

Recovering the nonlinear Schrödinger equation from its Lax pair

My question is less concerned with the physical aspects of the nonlinear Schrödinger equation and more with the mathematical mechanics of using a Lax pair. I am considering how to recover the ...
Talmsmen's user avatar
  • 577
5 votes
0 answers
94 views

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
128 views

Is there a discrete Schrödinger operator with empty spectrum?

A relatively well-known example of (continuous) Schrödinger operator with empty spectrum is the complex Airy operator on the line, i.e., the operator acting on $L^{2}(\mathbb{R})$ given by the ...
Twi's user avatar
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5 votes
0 answers
119 views

Difference Between Eigenvalues of Schrödinger Operator with Different Boundary Conditions

Consider a Schrödinger operator $$H=-\Delta+V$$ on a nice bounded domain $\Omega\subset\mathbb R^d$ (say, a ball or a cube), and assume for simplicity that $V$ is smooth. Let $\lambda_D,\lambda_N$ ...
user78370's user avatar
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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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4 votes
2 answers
967 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?
Matthias Ludewig's user avatar
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
  • 285
4 votes
1 answer
207 views

Non-isolated ground state of a Schrödinger operator

Question. Does there exist a dimension $d \in \mathbb{N}$ and a measurable function $V: \mathbb{R}^d \to [0,\infty)$ such that the smallest spectral value $\lambda$ of the Schrödinger operator $-\...
Jochen Glueck's user avatar
4 votes
1 answer
154 views

Spectrum near zero of $-\partial^2_x + V : L^2(\mathbb{R}) \to L^2(\mathbb{R})$, where $V = O(|x|^{-2 - \delta})$

Let $H = -\partial^2_x + V(x) : L^2(\mathbb{R}) \to L^2(\mathbb{R})$ be a one dimensional Schrödinger operator, where the potential $V$ is real-valued, belongs to $L^\infty(\mathbb{R})$, and, as $|x| \...
JZS's user avatar
  • 469
4 votes
1 answer
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
  • 63
4 votes
1 answer
268 views

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
  • 73
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
4 votes
1 answer
297 views

Nonlinear ODE to linear PDE?

I am interested in when and how one can trade a non-liner ODE for a linear PDE. To explain what this could look like here is a physics-inspired discussion. Consider a classical mechanical system with ...
Weather Report's user avatar
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
1 answer
349 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 ...
Michael Tinker's user avatar
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
  • 95
4 votes
0 answers
123 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
111 views

Looking for an electronic copy of Lebeau's paper

I would like to know if anyone has an electronic copy of the paper "Gilles Lebeau - Contrôle De L'Équation De Schrödinger"? This article appeared in Journal de Mathématiques Pures et ...
Math's user avatar
  • 481
4 votes
0 answers
154 views

What's the essential definition of resonance of Schrodinger operator?

Rencently, I am reading some articles about time decay estimates or Strichartz estimates for Schrodinger equations with potential. When considering Strichartz estimates for potential $V$ with decay $|...
Tao's user avatar
  • 419
4 votes
0 answers
154 views

Schrodinger operator with magnetic field: eigenvalues

Consider the self-adjoint operator on $L^{2}(\mathbb{R}^{N})$, $$H=-\frac{1}{2}(\nabla-iA)^{2}+V,$$ where $A\in C^{\infty}(\mathbb{R}^{N}, \mathbb{R}^{N} )$, $V\in C^{\infty}(\mathbb{R}^{N})$, $V\...
user152385's user avatar
4 votes
0 answers
119 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
4 votes
0 answers
429 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} ...
Felice Iandoli's user avatar
4 votes
0 answers
1k 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 ...
user23078's user avatar
  • 1,614
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
  • 73
3 votes
1 answer
429 views

Ground state for a double well potential (Schrödinger)

Consider $V(x)$ a one dimensional polynomial, confining, symmetric double well potential i.e. $V(x)=V(-x)$ for all $x\in\mathbb{R}$ $\displaystyle \lim_{x\to\pm\infty} V(x)=+\infty$ $V(x)\in \mathbb{...
M. Veruete's user avatar
3 votes
2 answers
221 views

Change of variables for obtaining a unitary group

Consider the following NLS: $$i u_t + \Delta u- 2 \operatorname{Re} u = F(u),$$ where $F(u):=(u + \bar{u} + |u|^2)u.$ In Scattering for the Gross–Pitaevskii equation, the authors S. Gustafson, K. ...
Mr. Proof's user avatar
  • 159
3 votes
1 answer
403 views

Duality argument

$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$Throughout my studying for some papers, in particular, the proof of localized Strichartz estimates, I encountered a use of the ...
Mr. Proof's user avatar
  • 159
3 votes
2 answers
261 views

Explicit form of S-matrix on the line

Consider the Hamiltonian $H$ on functions on the line with \begin{eqnarray} H=H_0+V,\\ H_0=-\frac{1}{2m}\frac{d^2}{dx^2} \end{eqnarray} where $V$ is a potential vanishing outside of a bounded interval ...
asv's user avatar
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