All Questions
26 questions
3
votes
0
answers
66
views
Effective action of unbounded operators on subspaces outside their domains of definition
Consider a densely defined, self-adjoint operator
$$
H: \mathcal{D} \rightarrow \mathscr{H}.
$$
Assume for simplicity that $H$ is nonnegative.
We want to effectively restrict this operator $H$ to a ...
- 133
1
vote
0
answers
127
views
Trace type convergence of the Laplacian on the box to the Laplacian on $\mathbb R^d$
Let $-\Delta \colon H^2(\mathbb R^d) \to \mathbb R^d$ be the (negative) Laplacian on the full space and $-\Delta_L$ the Laplacian acting on $L^2([-L,L]^d)$ with some boundary conditions making it self-...
- 61
0
votes
0
answers
66
views
Equality between operators, on dense subspace, from a quadratic form point of view
Let $L \ge 1$ and consider a finite box $\Omega = [0,L]^{d} \subset \mathbb{R}^{d}$. The set of functions:
$$\psi_{p}(x) = \frac{1}{L^{d/2}}e^{i\langle p,x\rangle} \quad p\in \frac{2\pi}{L}\mathbb{Z}^{...
- 1,305
0
votes
0
answers
138
views
Question about a step in the proof of the min-max principle
I honestly do not think this is a hard question, maybe it is even obvious but I tried MSE and had no success so far, so I am reproducing the question Question about the proof of the min-max principle ...
- 1,305
0
votes
0
answers
210
views
Reed-Simon Vol. IV: Question regarding convergence of eigenvalues
I am reading through Chapter XIII.16 of Reed and Simon's Methods of Modern Mathematical Physics IV: Analysis of Operators about Schrödinger operators with periodic potentials. Since the topic is kind ...
- 91
0
votes
1
answer
79
views
The operator of exponential derivative applied in quotients
I have an other question for a function different to the example given before in the link below:
Exponential derivative operator and continuous functions
We define for instance a function as:
$$H(y)=\...
- 159
0
votes
1
answer
258
views
Exponential derivative operator and continuous functions
I would like to know how to write down the following expression
$$f(y)=\frac{1}{y^{n} e^{\frac{d}{dy}}g(y)}$$
in the form of $e^{-\frac{d}{dy}}y^{-n}(\frac{1}{g(y)})$ where $n$ is an integer and $f,g: ...
- 159
2
votes
0
answers
45
views
Additivity of squared Schatten $p$-norm with respect to spatial partition
Consider a Hilbert-Schmidt operator $A$ on $L^2(\mathbb R^d)$ with integral kernel $A(x,y)$. Let $\Omega\subset \mathbb R^d$ and $1_{\Omega}(x)$ denote its characteristic function as well as the ...
- 91
5
votes
1
answer
552
views
Scattering theory for Coulomb potential
Both physical and mathematical theories of quantum scattering seem to be well developed in the case when the potential (or a more general perturbation of the Laplacian) decays fast enough at infinity ...
- 21.8k
3
votes
2
answers
271
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 ...
- 21.8k
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 +\...
- 21.8k
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 \...
- 173
7
votes
3
answers
1k
views
Simultaneous diagonalization of self-adjoint operators on Hilbert space
Suppose I'm given a finite set of possibly unbounded commuting self-adjoint operators $T_i : \mathfrak H \supset \mathscr D(T_i)\to \mathfrak H, i = 1 , \dots , N$ on a Hilbert space (in the sense of ...
- 203
0
votes
1
answer
267
views
Invariance of sets under Schrödinger equations
We are considering the Schrödinger equation on $\mathbb{R}^d \times [0,T]$
$$i \partial_t \psi(x,t)=-\Delta \psi(x,t) + u(t)V(x) \psi(x,t), t>0$$
$$\psi(x,0):=\psi(x_0) \in L^2(\mathbb{R}^d)$$
...
- 115
3
votes
0
answers
94
views
Multiplicativity of $\zeta$-function regularized determinant
Let $A$ be a selfadjoint elliptic differential operator on a compact manifold. In mathematical physics and differential topology one often defines its determinant using the $\zeta$-function ...
- 21.8k
9
votes
2
answers
848
views
$\zeta$-function regularized determinants
In (mathematical) physics in order to compute path integrals one often makes an infinite dimensional change of variables and uses infinite Jacobian as a purely formal expression. This step is done in ...
- 21.8k
5
votes
2
answers
2k
views
On the domains and extensions of unbounded operators
I am not an expert in functional analysis but I was studying some, motivated from some mathematical physics considerations. I am not quite sure whether this is research-level, but let me state some ...
- 7,746
11
votes
2
answers
714
views
A neat evaluation of an infinite matrix?
Let $M_n$ be an $n\times n$ matrix defined as
$$M_n
=\left[\frac{2i+1}{2(i+j+1)}\binom{i-1/2}i\binom{j-1/2}jx^{i+j+1}\right]_{i,j=0}^n.$$
With $I_n$ the identity matrix, consider $A_n:=I_n-M_n^2$. ...
- 43.2k
3
votes
1
answer
144
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 ...
- 31
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 $\mathbb{R}^n$ ...
- 11
1
vote
0
answers
576
views
Perturbation of Laplacian via Kato-Rellich theorem
Let's consider a potential $V(x)\in L^3(\mathbb{R}^3)$. I want to know if the following Hamiltonian
$$-\Delta+V(x)$$
is self-adjoint on $H^2(\mathbb{R}^3)$.
My idea is to use Kato-Rellich theorem; ...
- 11
2
votes
1
answer
136
views
Proper domain for operators
in this paper on arxiv in equation 27, two operators
$$A_m^* = (1-x^2)^{\frac{1}{2}} \frac{d}{dx} + \frac{mx}{\sqrt{1-x^2}}$$
and $$A_m = - \frac{d}{dx}(1-x^2)^{\frac{1}{2}} + \frac{mx}{\sqrt{1-x^2}...
1
vote
2
answers
352
views
Witten index non-trivial in the context of Quantum Mechanics?
Let $H$ be a self-adjoint Hamiltonian and $H$ admits a decomposition into closed operators $D,D^*$, such that we have $H = D^*D$.
I will now consider the one-dimensional case on a compact set:
So ...
6
votes
1
answer
353
views
Domains of raising and lowering operators in QM
Let $H : \operatorname{dom}(H) \subset L^2(\Omega) \rightarrow L^2(\Omega)$, where $dom(H) \subset H^2(\Omega)$, $\Omega \subset \mathbb{R}$ should be a bounded open interval(so 1-d setting(!)) and $H$...
user54300
4
votes
1
answer
275
views
Asymptotic behavior of Schrödinger operators
I am currently dealing with $1$ or at most $2$-dimensional Schrödinger operators on compact domains. A classical result of spectral theory is the Weyl approximations for this operator
$H = -\Delta +V$....
user54300
9
votes
2
answers
2k
views
Mathematical equivalent to ladder operators?
A powerful method in theoretical physics are ladder operators. They are used in QM to solve problems like the harmonic oscillator and the hydrogen atom. The idea is to solve with their help the ...
user37929