# Questions tagged [schrodinger-operators]

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127
questions

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34 views

### References to the theory of scattering resonances in dimension 1

Does anyone know a good reference to study the theory of scattering resonances in dimension 1 ?
I would also like to add that I don't know anything about this theory, so I'm starting from zero. Thanks ...

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46 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

101 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 ...

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**1**answer

70 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^{...

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128 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 ...

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102 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 ...

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41 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 ...

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**1**answer

95 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 ...

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81 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 ...

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43 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,...

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86 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 ...

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97 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 $...

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103 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....

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153 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 (...

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376 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 ...

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123 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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39 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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**1**answer

87 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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334 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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57 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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66 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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139 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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63 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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83 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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59 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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**1**answer

110 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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103 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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54 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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162 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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169 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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218 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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64 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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79 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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92 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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**1**answer

305 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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**1**answer

172 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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**1**answer

402 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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votes

**1**answer

219 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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**1**answer

183 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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**0**answers

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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97 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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195 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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**1**answer

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 $H$ ...

**1**

vote

**2**answers

191 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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votes

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94 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 $\lambda_D,\lambda_N$ ...

**0**

votes

**3**answers

240 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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166 views

### 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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votes

**1**answer

207 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 ...

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**1**answer

158 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 $V\to c>0$,...