# Questions tagged [schrodinger-operators]

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### Uniform boundedness Schrödinger operator eigenfunctions with Dirichlet conditions

I would like to ask a question with possibly a reference. If we have a Schrödinger operator $-\Delta+V$ on an interval $[0,L]$ with $V$ continous and Dirichlet conditions, can we state that the ...
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### Duality argument

$\newcommand\norm{\lVert#1\rVert}\newcommand\abs{\lvert#1\rvert}$Throughout my studying for some papers, in particular, the proof of localized Strichartz estimates, I encountered a use of the ...
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### Potential scattering for non-decaying potential

I am reading this note on potential scattering by Siu-Hung Tang and Maciej Zworski. Just wondering if there is a scattering theory for the Schrödinger with a non-decaying potential $- \Delta + V$ on ...
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### Green's function for elliptic PDE with potential

$\newcommand{\div}{\operatorname{div}}$Suppose I have an elliptic operator $\mathcal{L} u = -\div (A \nabla u)$ on some open set $\Omega \subseteq \mathbb{R}^d$ where here $A$ is uniformly elliptic ...
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I am trying to determine the sign of the maximal Lyapunov exponent of the Schrödinger-Newton equation \partial_t \psi(t,\vec{x}) = i\left(a\nabla^2 + \int_{\mathbb{R}^3} \frac{|\psi(t,\vec{y})|^2}{|... 5 votes 0 answers 81 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 ... 1 vote 0 answers 100 views ### Reference for global theory of Schrödinger operators Question. What is a good reference to learn about the spectral properties of Schrödinger operators in \mathbf{R}^n? I am specifically interested in references that discuss examples where the ... 1 vote 0 answers 49 views ### Positive semidefinite fundamental solution to Schrodinger operator Lets say V : \mathbb{R}^n \rightarrow \mathbb{M}_d (\mathbb{R}) is a d \times d symmetric, positive semidefinite matrix function on \mathbb{R}^n and consider the Schrodinger operator - \Delta + ... 2 votes 1 answer 98 views ### Intuition/references for understanding bound states/discrete spectrum relationship I am trying to form intuition for the following `well-known' facts about spectrum of unbounded operators (Schrodinger/wave etc.) L on \mathbb{R}^n. Let \lambda\in\mathbb{R} satisfy Lf=\lambda f... 3 votes 0 answers 88 views ### Convergence of Schrödinger ground states in L^p for p\neq 2 Suppose that H=-\Delta+V is a Schrödinger operator with a unique ground state \psi. Suppose that H_n=-\Delta+V_n is a sequence of operators such that V_n\to V in some sense as n\to\infty (... 2 votes 1 answer 198 views ### Even and odd solutions for the Schrödinger equation We consider 2a - periodic smooth solutions for \begin{eqnarray*} -\Delta u+V(x)\,u=0\qquad\hbox{in}\:[-a,a] \end{eqnarray*} We assume that V is smooth and even (i.e. V(-x)=V(x)). We also assume ... 6 votes 1 answer 142 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 ... 5 votes 1 answer 195 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 ... 2 votes 0 answers 58 views ### How to use Fredholm alternative to check that there are only finite eigenvalues of H on the imaginary axis? On \mathbb{R}^3, we consider the operator \begin{equation} \mathcal{H}= \left( \begin{matrix} -\Delta +1 -2 \phi^2 & -\phi^2 \\ \phi^2 & \Delta -1 +2 \phi^2 \end{matrix} \right) , D(... 4 votes 1 answer 120 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 ... 0 votes 1 answer 81 views ### An inequality for uniqueness proof of NLS Setting Although this detail is not relevant to my question, let me set the problem that my question arise. We are considering an initial value problem \begin{align*} \begin{cases} u\in L^\infty(I,H^{... 2 votes 0 answers 134 views ### Are Weyl sequences polynomially bounded? Look at the Hilbert space l^2( \mathbb{Z})  and let A be a translation invariant band operator. I.e. if \{ e_n \}_{n \in \mathbb Z}  is the standard basis for l^2( \mathbb{Z})  then it holds ... 5 votes 0 answers 112 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 ... 1 vote 0 answers 64 views ### Domain issues regarding the Duhamel formula for the linear Schrödinger equation I have some questions in succession regarding the rigorous domain issues about a Duhamel expansion formula (stated near the end of my post) for the linear Schrödinger equation. Consider a linear ... 2 votes 1 answer 104 views ### An inequality of spacetime Banach space for non-linear Schrodinger equation I am reading a proof for the existence of a solution to the Local Cauchy problem of the non-linear Schrodinger equation i\partial_t u+\Delta u +\epsilon u |u|^{2} = 0 \\ u(x,0)=u_0(x) $$The ... 3 votes 0 answers 84 views ### Determining what happens to the spectrum of Schrödinger operator as boundary condition changes I recently came across a problem in research, and I'm asking about it here after trying in math stack exchange with no luck. Suppose I have a metric graph G (or even a closed interval, to make ... 1 vote 0 answers 47 views ### Semigroup theory for non-symmetric Markov processes / complex-valued potentials Let X be a continuous-time Markov process on a countable state space E, and let V:E\mapsto\mathbb C be some complex function. X can be characterized by its transition rates (\lambda_{xy})_{x,... 4 votes 0 answers 98 views ### Looking for an electronic copy of Lebeau's paper I would like to know if anyone has an electronic copy of the paper "Gilles Lebeau - Contrôle De L'Équation De Schrödinger"? This article appeared in Journal de Mathématiques Pures et ... 2 votes 0 answers 119 views ### When does a one-dimensional Schrödinger operator have a threshold resonance? Consider the operator$$ L = -\partial_x^2 + V(x),$$for some bounded, decaying potential, i.e. V(x)\to 0 as x\to \pm \infty. I'm interested in the L^2(\mathbb R) spectrum of L. We know that ... 1 vote 0 answers 113 views ### Angular excitations and Schrodinger operators with radial potential in N-dimensions Can someone please explain the following in mathematical language? "First of all, angular excitations only push the energy up, never down, so it is enough to analyze spherically symmetric s-waves.... 3 votes 0 answers 162 views ### How to prove the following linearized operator is positive? In L^2(\mathbb{R}^d), let Q be the solution to \begin{equation} -\Delta Q+\alpha^2 Q = |Q|^{2\sigma} Q, \end{equation} and Q satisfies that it is positive, radial, and exponentially decaying (... 3 votes 0 answers 677 views ### The Node Theorem - an argument from physics The theorem on the number of zeros of a solution to a Sturm-Liouville equation is a well-know result in quantum mechanics. It doesn't seem to have a special name in the mathematics literature, but it ... 2 votes 0 answers 151 views ### Schrodinger operator with matrix potential This question is inspired by attempts to extend in some way Shen's 1999 "On fundamental solutions of generalized Schrödinger operators" to Schrödinger operators - \Delta + V  with some ... 2 votes 0 answers 40 views ### Cwikel–Lieb–Rosenbljum inequality including zero resonances The Cwikel–Lieb–Rosenbljum inequality asserts that, for any potential V:\mathbb{R}^n\to\mathbb{R}, we have$$(\mbox{number of eigenvalue} \leq 0\mbox{ , counted with multiplicity, of }-\Delta+V\,)\...
Let $V\in L^{\infty}(\mathbb{R}^3)$ be a radial, compactly supported potential, and consider the Schrodinger operator $H:=-\Delta + V$ on $L^2(\mathbb{R}^3)$. Let $\psi$ be a resonance for $H$, i.e. a ...