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Questions tagged [schrodinger-operators]

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1answer
345 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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1answer
105 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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0answers
123 views

Sudden appearance of an eigenvalue of a self-adjoint operator $H = H_0 + \lambda H_1$

In doing some numerical calculation in quantum mechanics, we found something surprising to us. Let the Hamiltonian be $$ H = H_0 + \lambda H_1 , $$ where both $H_0$ and $H_1$ are self-adjoint, and $...
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1answer
120 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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40 views

What are tools we need to prove scattering result for NLS when propagator is unitary?

Consider nonlinear Schrödinger equation (NLS) $$i\partial_t u + \Delta u =F(u) , u(x,0)=u_0$$ where $F$ is some nonlinearity. Assume that there exists Banach space $X$ so that $\|e^{it\Delta} f\|_{X}...
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0answers
68 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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70 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
137 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
214 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
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2answers
173 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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0answers
70 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
135 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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0answers
70 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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1answer
142 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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0answers
73 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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45 views

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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0answers
111 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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0answers
55 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
99 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
216 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
136 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
151 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
146 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
90 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 ...
0
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1answer
103 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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0answers
65 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 ...
2
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1answer
176 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{...
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0answers
58 views

When Schroedinger propagator commutes other operators?

Let $f\in \mathcal{S}(\mathbb R^d)$ (Schwartz Space). We know that $\widehat{\nabla f}(\xi)= 2 \pi i \xi \hat{f} (\xi). $ We define $$\widehat{|\nabla| f^{s}} (\xi) = (2 \pi |\xi|)^s \hat{f} (\xi), ...
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2answers
308 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 ...
2
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0answers
179 views

Existence of solutions to time-dependent Schrödinger equations

I would like to know what is known about evolution equations of the form $$iy'(t)=H_0y(x,t)+u(t)V(x)y(x,t)$$ and $y(0)=y_0 \in D(H_0)$ where $V$ is not a bounded operator, but an unbounded one, $u \...
3
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2answers
206 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 ...
4
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1answer
198 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 ...
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1answer
261 views

Invariance of sets under Schrödinger equations

We are considering the Schrödinger equation on $\mathbb{R}^d \times [0,T]$ $$i \partial_t \psi(x,t)=-\Delta \psi(x,t) + u(t)V(x) \psi(x,t), t>0$$ $$\psi(x,0):=\psi(x_0) \in L^2(\mathbb{R}^d)$$ ...
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0answers
94 views

Quantitative Approach to Existence of Minimal-Mass Blowup Solutions to NLS

Consider the mass-critical defocusing NLS in dimension $d\geq 1$: $$iu_{t}+\Delta u = |u|^{4/d}u, \quad (t,x) \in I\times\mathbb{R}^{2}$$ Define the mass $M(u)$ and scattering size $S(u)$ of the ...
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0answers
55 views

Eignfunctions of an elliptic operator

I am looking for a reference which makes it possible to say that one can devellop $f$ in the form of the sentence underlined by the yellow. Thank you in advance.
2
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1answer
158 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 ...
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1answer
113 views

Solutions to Schrödinger equation parameter dependence

This is somewhat unrelated to what I normally do in mathematics, which is why it may be obvious to some of you, but I was puzzled by this: If we look for classical solutions on $[0,1]$ to $$-y''(x) =...
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2answers
127 views

Concerning the decay of the ground state of certain Schrodinger operators

Consider the Schrodinger operator in $n$ dimensions with a potential $V$, which grows rather quickly as $\mid x\mid$ tends to infinity, but with negative potential in a bounded region, for example, a ...
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1answer
54 views

Dispersive estimates for one dimensional magnetic Schrodinger operators

I would like to know if there is any known result on dispersive estimates for Schrodinger operator with magnetic potential in one dimension. There is a lot of literature for three dimensional magnetic ...
3
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1answer
370 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 ...
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0answers
41 views

Delta function propogation in the NLS regime

consider the $(1+1)D$ or $(2+1)D$ NLS: $$ i\psi _t (t,{\bf x}) + \Delta \psi + |\psi|^2\psi = 0 \, ,$$ $$ \psi (t,|{\bf x}|\to \infty) =0 \, , \quad \psi(t=0,{\bf x}) = \psi_0 ({\bf x}) \, , $$ with $...
3
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1answer
129 views

Operator norm of almost mathieu operator

The almost Mathieu operator has become famous since it is the central object of the ten martini problem. In this paper here a bound on the operator norm is given. Although the bound is of course ...
4
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1answer
437 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 ...
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0answers
65 views

Spectrum analytic in quasimomenta?

This is rather a reference request than a research question. As far as I know the following statement is true: Let $ H(k)\psi = -\psi''+V \psi$ where $V \in L^2[0,1]$ and the domain of $H(k)$ is $$...
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50 views

Sub-matrices with a real spectrum

This question arises from the study of PT-symmetric quantum mechanics. Let $A$ be an $n\times n$ complex-valued matrix with a real spectrum. If $A$ is Hermitian, then any sub-matrix corresponding to ...
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2answers
191 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}...
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0answers
379 views

estimate of smallest eigenvalue of Schrodinger operator

I am looking for references on estimate of first nonzero Dirichlet eigenvalue for Schrodinger operator $-\Delta + V$, if sharp bounds exist, that would be better, here for simplicity, we can assume ...
3
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1answer
158 views

Bounded solutions for Schrödinger equation at the edge of the essential spectrum

Let $V:R^d\to R_+$ be with a compact support. The Schrödinger operator $H_a=-\Delta - a V$ acting in $L^2(R^d)$ has then (at most) finitely many negative eigenvalues. Denote the number of negative ...
10
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3answers
879 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 (...
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0answers
68 views

When does the ground state energy continuously depend on a parameter?

Given a family of Schrödinger operators $H_\gamma=-\Delta+V_\gamma$, under which condition is the map $\gamma\mapsto\inf\sigma(H_\gamma)$ continuous? This is surely the case for many textbook ...