# Questions tagged [schrodinger-operators]

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### 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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### 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 \mathcal{H}= \left( \begin{matrix} -\Delta +1 -2 \phi^2 & -\phi^2 \\ \phi^2 & \Delta -1 +2 \phi^2 \end{matrix} \right) , D(...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### How to prove the following linearized operator is positive?

In $L^2(\mathbb{R}^d)$, let $Q$ be the solution to $$-\Delta Q+\alpha^2 Q = |Q|^{2\sigma} Q,$$ and $Q$ satisfies that it is positive, radial, and exponentially decaying (...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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. $\textbf{... 2answers 195 views ### Criteria for Schrödinger operator on real line to have simple spectrum Consider a Schrödinger operator$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 ... 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$H$... 2answers 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 ... 0answers 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$\lambda_D,\lambda_N$... 3answers 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 $$i\psi _t (t,\underline{x}) + \Delta \psi=V(\underline{x}... 0answers 166 views ### Spectrum of a Hamiltonian on the real line Consider the following linear (Hamiltonian) operator on functions on the real line \mathbb{R}$$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 +\...
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 ...
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 $V\to c>0$,...