All Questions
35 questions with no upvoted or accepted answers
4
votes
0
answers
169
views
Inverse Laplacian and convolution in Albeverio's “Solvable Models in quantum mechanics”
I asked this question on math.stackexchange.com two weeks ago but got no answers so far and I got no clues from literature, so maybe someone here knows a reference. I hope it is ok to ask this ...
- 141
3
votes
0
answers
108
views
A question on essentially self-adjoint differential operators of the type $\Delta=D^{\ast}D$
Let $(M,g)$ be a smooth (connected, complete, oriented) Riemannian manifold and let $D:C^{\infty}(M)\to C^{\infty}(M)$ be a linear partial differential operator, which I view as an operator in $L^{2}(...
- 1,171
3
votes
0
answers
322
views
Heat equation damps backward heat equation?
In a previous question on mathoverflow, I was wondering about the following:
Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions. ...
- 536
2
votes
0
answers
94
views
Nemytskij operator for Lebesgue variable UNBOUNDED exponent spaces
Let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function (i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is measurable for all $t\in\mathbb{R}$), where $\Omega\...
- 1,759
2
votes
0
answers
143
views
How to prove $\operatorname{Id}-K$ is a proper map when $K$ is a $C^1$ compact operator?
Assume $X$ is a Banach space, $\Omega \subseteq X$ is an open set, $K\in {C}^{1}( \overline{\Omega}, X)$ is a nonlinear compact map, Is $\operatorname{Id}-K$ a proper map? I think maybe it has ...
- 21
2
votes
0
answers
180
views
Approximating $L^p$ functions by eigenfunctions of Laplacian
I'm reading a paper https://www.sciencedirect.com/science/article/pii/S0022039608004932.
In this paper, the authors assume that $\mathcal{O}$ is a bounded domain of $\mathbb{R}^N$ with $C^m$ boundary ...
- 121
2
votes
0
answers
49
views
Determining a space of differentiability
I have a questions and maybe you are able to assist with this? Let us consider the space $X:=\mathrm{L}^2[0,\pi]$. On $X$ we consider the family of operators $(P(t,s))_{t\geq s}$ defined by
$$
P(t,s)f:...
2
votes
0
answers
654
views
Convergence of operator in norm resolvent sense and their eigenvectors
Let $\{T_n\}_{n=1}^\infty$ and $T$ be (unbounded) self-adjoint operators and $T_n\to T$ in norm resolvent sense, that is, for some $z\in \mathbb{C} \setminus \mathbb{R}$, $\|(zI- T_n)^{-1}- (zI- T)^{-...
- 21
2
votes
0
answers
101
views
Strongly continuous semigroups on weighted $\ell^1$ space
Let $x=(x_i)$ be a sequence in $\ell^1$ such that all $x_i>0.$
Let $T(t):\ell^1 \rightarrow \ell^1$ be a strongly continuous semigroup of, i.e. $t \mapsto T(t)y$ is continuous for every $y \in \ell^...
- 536
2
votes
0
answers
159
views
Principal symbol of a non-local operator and Atiyah–Singer index formula
I am trying to understand the Atiyah–Singer index formula for pseudo-differential operators. As far as I understood, the Fredholm index of the operator $A$ on a manifold can be computed just from the ...
- 41
2
votes
0
answers
190
views
Absence of fixed points
Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function
$$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \frac{xy}{(x^2+y^2+1)} \ dx$$
where $x_0$ is an arbitrary but fixed ...
2
votes
0
answers
149
views
Projection semigroup of an isolated eigenvalue
I'm currently working with a paper and I don't get something there. Let $A$ be a closed operator on a Banach space $X$ and $\lambda \in \sigma(A)$ an isolated eigenvalue, i.e. there is a $r > 0$ ...
- 381
2
votes
0
answers
78
views
Generalization of supersymmetry to dimension 3
in two dimensions there is a simple trick to study the spectrum of operators of the form
$$\textbf{A}:=\left( \begin{matrix}0 && A^* \\ A && 0 \end{matrix}\right)$$
The trick is to ...
- 167
2
votes
0
answers
218
views
Existence of solutions to time-dependent Schrödinger equations
I would like to know what is known about evolution equations of the form
$$iy'(t)=H_0y(x,t)+u(t)V(x)y(x,t)$$
and $y(0)=y_0 \in D(H_0)$
where
$V$ is not a bounded operator, but an unbounded one, $u \...
- 173
2
votes
0
answers
77
views
When do finite dimensional approximations approximate the spectral absicssa of a linear operator?
I apologize if the following is trivial for experts in the field. If so, please feel free to refer me instead to any proper references.
I would like to compute the spectrum of a known non-normal, ...
- 121
2
votes
0
answers
116
views
The composition of a dissipative operator and a positive operator is dissipative?
Consider the following bilinear system on a open and bounded domain $\Omega$
\begin{equation}
\left\{\begin{array}{r c l}
\displaystyle\frac{dy(t)}{dt} &=& Ay(t)+u(t)By(t)\\
y(0) &...
- 23
2
votes
0
answers
86
views
I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection
I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...
user74319
2
votes
0
answers
144
views
Is logarithmic convexity of the heat kernel with complex time a general fact?
Suppose $H$ is a non-negative self-adjoint operator acting on $L^2(\mathbb{R}^n)$, and generates an analytic semigroup with kernel satisfying a Gaussian upper bound, i.e., if we denote $K(t,x,y)$ the ...
- 879
1
vote
0
answers
153
views
Maximal domain of an unbounded linear operator in a weighted Hilbert-space
Let's consider the following (unbounded) linear operator. (So called transport operator in some context.)
$$ \mathrm{T}: \mathcal{H} \supset \mathcal{D}(\mathrm{T}) \to \mathcal{H} , f \mapsto \mathrm{...
- 185
1
vote
0
answers
33
views
Limiting absorption principle for higher powers of resolvents
Let $H$, $A$ be self-adjoint operators on a Hilbert space. Moreover, let $I$ be a bounded open interval contained in the spectrum of $H$. Assume that $H$,$A$ satisfy the following positive commutator ...
- 141
1
vote
0
answers
122
views
Series and solution of $-\Delta u + \lambda u = f(x)$
Consider a bounded smooth set $\Omega \subset \mathbb R^n$ (for example, we can take a ball). Can we write down the solution of
\begin{align*}
-\Delta u(x) + \lambda u(x) &= f(x), & x \in \...
- 131
1
vote
0
answers
64
views
Bound on number of linearly independent eigenvectors of adjoint of composition operator
Fix $N>1$. Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ be such that the composition operator via
$$
\begin{aligned}
C_f:C^{\infty}(\mathbb{R},\mathbb{R}) &\rightarrow C^{\infty}(\mathbb{R},\...
- 5,405
1
vote
0
answers
78
views
Inequality for fractional power norm (sectorial operators)
How could we prove following inequality:
$\int\limits_{0}^{l} u^{3}(x) dx \leq \sqrt{l} \cdot|| u||_{\frac{1}{2}}^{3}$
where
$ || u ||_{\frac{1}{2}} = ||A^{\frac{1}{2}}(u)||_{L^{2}} + || u ||_{L^{...
- 31
1
vote
0
answers
196
views
Compact embedding result
Let $\tau$ and $\ell$ be positive numbers. We know that the space $H^2(0,\ell)\cap H^1_0(0,\ell)$ is compactly embedded into $L^6(0,\ell)$. Now, is the space $L^2(0,\tau;H^2(0,\ell)\cap H^1_0(0,\ell))$...
- 395
1
vote
0
answers
93
views
Relative boundedness of the adjoint
Let $X$ be a separable Banach space and $T_1:D(T_1) \subset X \rightarrow X$ and $T_2:D(T_2) \subset X \rightarrow X$ two closed operators with $D(T_2)\subset D(T_1)$ and $D(T_2^*) \subset D(T_1^*).$
...
- 33
1
vote
0
answers
210
views
Propagation of singularities and the Schrodinger equation
I always thought that the propagation of singularities theorem by Hörmander says (on $\mathbb R^n$ for a classical symbol $p(x,\xi)=\xi^2+V(x)$) that for a Schrödinger equation
$$(i \partial_t-p(x,D))...
- 11
1
vote
0
answers
76
views
Which sets support which spectra?
I know (and this is of course rather elementary) that an isolated point in the spectrum of a self-adjoint operator $T$ always belongs to the point-spectrum.
I would like to ask: Are there similar ...
- 173
1
vote
0
answers
124
views
Singular value decomposition in two spaces (reference in Russian paper?)
Let $H$ be a Hilbert space and $X$ be a Banach space such that $H \cap X$ is dense in both.
Now, let $T$ be an operator such that $T: H \rightarrow H$ and $T:X \rightarrow X$ exists in the sense that ...
1
vote
0
answers
351
views
Heat semigroup ultracontractive?
Let $g(x,t)= \frac{1}{(4 \pi t)^{\frac{n}{2}}}e^{\frac{-|x|^2}{4t}}$ be the heat kernel on $\mathbb{R}^{n}.$ Is the standard definition now to say that this heat-semigroup $T(t)(f):=g *f(.,t)$ is ...
- 11
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
0
votes
0
answers
111
views
Characterization of the adjoint of a $C_0$-Semigoup infinitesimal generator
I am looking for characterizations of the adjoint operator of the infinitesimal generator of a $C_0$-semigoup in Hilbert spaces. All I could find in the literature is that if $(A, D(A))$ is the ...
- 101
0
votes
0
answers
326
views
Measurability of the heat semigroup in $L^\infty$
Let $S(t)$ be the $C_0$-semigroup generated by the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega)$, where $\Omega$ is a bounded open subset of $R^n$.
It is known that $S(t)$ ...
- 1
0
votes
0
answers
81
views
Differential operator and equivalence
Here is the problem:
I have a certain PDE and there is the nonlinear terme $h$, I have as data:
$f \in H_0^2(0,L)$,,,$g \in {H^1}(0,L)$ with ${g_x}(0) = {g_x}(L) = 0$
Now on consider the fnction $$h(...
- 617
0
votes
0
answers
137
views
Heat asymptotics
Consider a compact manifold $M$ with smooth boundary, with either the Dirichlet or the Neumann boundary conditions. Consider a (time-dependent) open ball $B_t \subset M$. Given a fixed $u \in L^1(M)$, ...
- 11
0
votes
0
answers
358
views
Boundedness of heat semigroup on $L^1(\Omega)$
On a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary, consider the Laplacian $-\Delta$ with either the Dirichlet or Neumann boundary conditions. More generally, one can also consider ...
- 1