# Questions tagged [schrodinger-operators]

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### 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 ...
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### stability of NLS solutions with $N$ nodes

I have a question regarding the stability of standing wave solutions to nonlinear Schrodinger equations (NLS) with $N$ nodes on the real line. In case $N=1$, the stability result is known due result ...
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### Logarithmic Sobolev growth of time-space-periodic Schrödinger solutions

Consider the following Schrödinger equation $$i\partial_t \psi (t,x) + \Delta \psi - V(t,x) \psi = 0 \, ,$$ where $x\in \mathbb{T}^d$ and $V(t,\cdot)$ is real, smooth, and periodic (with a diophantine ...
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### Lippmann-Schwinger equation for the Coulomb potential

Let $H=H_0+V$ be a Hamiltonian on $\mathbb{R}^3$ where $H_0=-\frac{\Delta}{2m}$ is the free Hamiltonian and $V$ is a potential. Let us assume first that $V$ decays sufficiently fast at infinity and ...
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### The ill-posedness of $L^2$-super critical nonlinear Schrödinger equation

For nonlinear Schrodinger equation$$\begin{cases}iu_t+\Delta u+|u|^\alpha u=0\\u(0)=\phi\in H^1(\mathbb R^d)\end{cases}$$ where $\alpha>\frac 4d$. In Christ, Colliander, Tao's paper Ill-posedness ...
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### Schrodinger and Laplace operators with infinitely many common eigenvalues

Let $V>0$ be a non-constant polynomial and consider the one dimensional Schrodinger operator $H=-\frac{d^2}{dx^2}+V$ on $[0 ,L]$ with Neumann boundary condition. Can $H$ and $T=-\frac{d^2}{dx^2}$ ...
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### An optimization problem for one- dimensional Schrodinger operator

For a potential of the form $V(x)=ax^4+bx^2$, where $a,b>0$, let us consider the one dimensional Schrodinger operator $D=-\frac{d^2}{dx^2}+V$ with Dirichlet B.C on $[-L,L]$ and denote its first ...
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### Lower bound of the spectrum of a Schrodinger operator on a bounded domain

I am trying to look for references on estimate of the lower bound of the spectrum of a Schrodinger operator $-\Delta + V$ on a bounded domain in three-dimensional space. For simplicity, we can take ...
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### Pseudo-polynomial potentials for Schrödinger operators

Consider the one dimensional Schrödinger hamiltonian $\mathcal{H}=-\frac{\hbar^2}{2} \frac{d^2}{dx^2} + V(x)$. Suppose that $V:\mathbb{R} \rightarrow \mathbb{R}^+$ is a continuous and confining ...
I'm studying time dependent perturbation theory on Reed-Simon book "Method of modern mathematical physics, II". If one considers an Hamiltonian of the form $$H(t)=H_0+V(t)$$ the corresponding formal ...