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3 votes
0 answers
102 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 ...
GSofer's user avatar
  • 251
2 votes
0 answers
145 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 ...
Frederik Ravn Klausen's user avatar
2 votes
0 answers
158 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 ...
asv's user avatar
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2 votes
0 answers
218 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 \...
Landauer's user avatar
  • 173
1 vote
0 answers
61 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\...
Matt Rosenzweig's user avatar
1 vote
0 answers
154 views

One-parameter group of unitary operators and Core

Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ ...
Alphabeta's user avatar
0 votes
0 answers
237 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 +\...
asv's user avatar
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