Questions tagged [schrodinger-operators]

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98 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 ...
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22 views

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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32 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\,)\...
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63 views

How to prove $U(t,s)$ is unitary if potential $V(t,x)$ is smooth in both variables?

In $\mathbb{R}^3$, let $V(t,x)$ be a time-dependent potential such that $\lVert V \rVert_{L_t^\infty L_x^{3/2}}$ sufficiently small, and $V(t,x)$ be smooth in both invariables. Consider equation \...
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1answer
78 views

Resonances for Schrodinger operators with radial potentials

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 ...
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1answer
316 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 $\...
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56 views

Proving that $(f,g)$ are Cauchy data for the Schrödinger equation iff $(f,g)$ satisfies an equation

I have to prove that if $f\in H^{1/2}(\partial\Omega)$ then $(f,g)$ are Cauchy data for the Schrödinger equation if and only if $$g=\gamma^{-1/2} \Lambda_{\gamma}(\gamma^{-1/2} f)+1/2 \gamma^{-1}\...
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52 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 $|...
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138 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\...
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58 views

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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77 views

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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43 views

On a interpolation inequality for the Schrödinger unitary group (NLS)

I'm trying to understand scattering for the classical nonlinear Schrödinger equation and for that i'm studying a scattering criterion on Tao's paper. At Lema 3.1 he states that $$\left\|e^{it\Delta}f\...
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1answer
65 views

Anderson localization for Bernoulli potentials on half-line

Anderson localisation for (discrete) Schrödinger operators with Bernoulli potentials on $l^2(\mathbb{Z})$ was proven in https://link.springer.com/article/10.1007/BF01210702 I am wondering if there ...
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95 views

Uniform boundedness for Strichartz constants in Tao's book

This is a question from Tao's book "Nonlinear Dispersive Equations". We define the space $S^0(I\times R^d)$ as the closure of the Schwartz functions under the norm $$\|u\|_{S^0}=\sup_{(q,r)\text{ ...
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47 views

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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158 views

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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128 views

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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167 views

Estimate of a solution of Schroedinger equation for a free particle

Let $\psi(x,t)$ be a solution of the Schroedinger on the line $$i\frac{\partial \psi}{\partial t}=-\frac{1}{2m}\frac{\partial^2 \psi}{\partial x^2}.$$ One assumes that $\psi(x,0)$ "behaves well" as $...
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213 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 ...
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60 views

Partial well-posedness results on Schrödinger operators?

Set $ A_i:= -\Delta + V_i :H^2(\mathbb{R}^3) \subseteq L^2(\mathbb{R}^3) \to L^2(\mathbb{R}^3), \ i =1,2 $, where \begin{equation*} V_1 = 0, \ \ (\textrm{No interaction}) \\ V_2 = - \frac{\gamma}...
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71 views

Schrödinger operators : current status between ballistic motion and absolutely continuous spectrum?

Regarding Schrödinger operators $H= \Delta + V$ on $\mathbb{R}^d$ or $\mathbb{Z}^d$, it is said that $H$ manifests ballistic motion if the square root of the second moment of the position operator ...
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74 views

Spectrum of a Hamiltonian which is a perturbation of Laplacian

Let $\Delta =\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\frac{\partial^2}{\partial x_3^2}$ be the Laplacian on $\mathbb{R}^3$. Consider a self adjoint operator $H$ on complex ...
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86 views

A generalization of scattering theory

In the quantum scattering theory one proves results of the following type. Let $H_0$ be the Laplacian $\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\frac{\partial^2}{\partial ...
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52 views

Construction of a special sequence [duplicate]

I m looking for a sequence $(f_j)\in C^\infty(\Bbb{R})$ such that $$ \int^\infty_0\Big|4r\partial^2_r f_j(r)+4\partial_r f_j(r)+rf_j(r)\Big|^2dr\to 0, $$ and $$\int_{\Bbb{R^+}}|f_j(r)|^2 dr=1\quad\...
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1answer
279 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 ...
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1answer
167 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 $-\...
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1answer
396 views

A question of the Schrodinger Semigroup --By B. Simon

The question comes from the paper: B. Simon, Schrodinger Semigroups, Bull. A.M.S., (1982) Vol. 7 (3). On the Theorem C.1.2(subsolution estimate) of the paper, it says that: If $Hu=0$, where $H=-\...
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217 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 ...
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1answer
164 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 ...
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70 views

relative compact on nonlinear term

On the paper: Decay of Solutions to Nonlinear Schrodinger Equations. Let $u$ be a solution of the equation $$Hu+|u|^2u=0,$$ where $H$ is a Schrodinger operator, i.e. $-\Delta+V$ and $V$ is a (...
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89 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{...
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2answers
177 views

Criteria for Schrödinger operator on real line to have simple spectrum

Consider a Schrödinger operator $H:=-\Delta+V$ on $\mathbb R$, where $V$ is such that $H$ has a purely discrete spectrum $-\infty<\lambda_1\leq\lambda_2\leq\cdots$ converging to $+\infty$. Do there ...
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1answer
236 views

Elementary quantum scattering problem on the line.

Let us consider the quantum scattering problem on the line with the Hamiltonian $$H=-\frac{d^2}{dx^2}+ V(x),$$ where $V(x)=1$ when $x\in (0,a)$, and $V(x)=0$ otherwise. It is easy to see that $H$ ...
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2answers
184 views

Fourier transform of a generalized function on the plane

Is there an explicit formula for the Fourier transform of the generalized function of 2 variables $$\frac{1}{x+y^2+i0}?$$ Remark. Equivalent question: consider the Schroedinger equation one the ...
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80 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 $\lambda_D,\lambda_N$ ...
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3answers
191 views

Single quantum particle entropy

Consider a wave function of a single particle in free space, whose evolution is described by the (non-dimensional) linear Schrodinger equation $$i\psi _t (t,\underline{x}) + \Delta \psi=V(\underline{x}...
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Spectrum of a Hamiltonian on the real line

Consider the following linear (Hamiltonian) operator on functions on the real line $\mathbb{R}$ $$H\psi(x)=-\frac{d^2}{dx^2}\psi(x)+V(x)\psi(x).$$ Assume that $V$ is a smooth function and $V(x)\to +\...
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1answer
182 views

Barry Simon's decay of eigenfunctions for pseudo differential operators

In his celebrated paper on Schrödinger semigroups, Barry Simon proves the following result. Let $V_{-} \in K_\nu$, $V_{+} \in K_\nu^{\mathrm{loc}}$ and suppose that $Hu=Eu$ where $E$ is the ...
3
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1answer
144 views

Discrete spectrum of Schrodinger operator

Assume $\Omega$ is a non-compact region or manifold with dimension $\geq4$. Let $H=-\Delta+V$ be Schrodinger operator. Here $V$ is a (smooth)function. I know that if $V\geq c>0$ or $V\to c>0$,...
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Realizing $N$-body Hamiltonian operator from $2$-body operator

Let $N\in\mathbb{N}$, and consider the formal $N$-body Schrodinger operator $$\sum_{j=1}^{N}-\partial_{x_{j}}^{2}+2c\sum_{1\leq j_{1}<j_{2}\leq N}\delta(X_{j_{1}}-X_{j_{2}}), \tag{1}$$ where $c\in\...
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142 views

Perturbation of Elliptic operator

Let $\Omega$ be an open region or a non-compact complete manifold, $L$ be an elliptic operator with possibly non vanishing zero-order term, e.g. $-\Delta+q$. Suppose $W$ is an operator such that $W(...
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62 views

Zero energy resonances for scaling critical Schrodinger operators

Given a real valued potential $V\in L^1(\mathbb{R}^3)$, we say that the Schrodinger operator $-\Delta + V$ has a zero-energy resonance if there exists $\psi\in L^2_{loc}(\mathbb{R}^3)\setminus L^2(\...
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1answer
115 views

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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1answer
305 views

Spectrum of magnetic Laplacian

Consider the discrete magnetic Laplacian on $\mathbb Z^2.$ $$(\Delta_{\alpha,\lambda}\psi)(n_1,n_2) = e^{-i \pi \alpha n_2} \psi(n_1+1,n_2) + e^{i\pi \alpha n_2} \psi(n_1-1,n_2) + \lambda \left(e^{i ...
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2answers
172 views

Limit (at infinity) for the lowest eigenvalue of a perturbed harmonic oscillator

Let $\epsilon \in [0, \infty[$. Consider the following operator on $L^2(\mathbb{R})$: \begin{equation} H(\epsilon) = -\frac{d^2}{dx^2} + x^2 + \epsilon |x|. \end{equation} How does one show that the ...
2
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1answer
159 views

the asymptotic behaviour of function as $\lambda \to -\infty$

Let's consider the following differential equation on $\mathbb{R}$: $$-u''(x)+u(x)-V(x)u(x)=\lambda u(x),$$ where $\lambda<1$ and $V$ is a bounded. We consider only that solution $u(x) \in C^1$ ...
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2answers
154 views

Ground state has always constant sign?

Question: What hypothesis on the potential $V$ are required such that the ground state $\phi_0$ has constant sign? Consider the Schrödinger operator in 1 dimension with potential $V$: $$\mathcal{H}=-...
2
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1answer
97 views

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 ...
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1answer
109 views

Why does it hold: the “observation” from Error Bounds for Exponential Operator Splittings by Jahnke & Lubich

In the paper, Tobias Jahnke and Christian Lubich (2000), "Error bounds for exponential operator splittings." BIT Numerical Mathematics, (Here is the link for the paper: http://www.math.kit.edu/ianm3/~...
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66 views

Time dependent Hamiltonians

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 ...