# Tagged Questions

**0**

votes

**0**answers

43 views

### Parabolic partial differential equation, initial conditions

Let $U\subset\mathbb{R}^n$ be open bounded, $T>0$.
Given the parabolic PDE $$\partial_tf+a\partial_xf+b\partial_{xx}f = g \qquad (1)$$ I'm interested in the initial and boundary conditions that ...

**2**

votes

**1**answer

91 views

### Practical way to check whether a distribution is conormal

Let $X$ be a $C^\infty$-manifold, $Y$ be its $C^\infty$-submanifold. HÃ¶rmander defines the set $I^m(X,Y)$ of conormal distributions as the set of all $u \in \mathscr D'(X)$ such that
$$
L_1 ...

**4**

votes

**1**answer

201 views

### Question about a (relatively simple looking) differential operator and its eigenvalues

A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form
$L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$
for ...

**2**

votes

**2**answers

174 views

### Schrodinger's equation via Spectral Theorem

How do you prove basic facts on the Schrodinger equation using the spectral theorem? More precisely, here is what I have in mind.
The version of the Spectral Theorem I am familiar with is the ...

**1**

vote

**1**answer

348 views

### Precise versions of “differential operators are unbounded but closed linear operators”

I am trying to understand to what extent the following result of Hille is an extension of the usual theorems on differentiation under the integral sign.
Theorem (Hille). Let $(\Omega,\Sigma,\mu)$ ...

**-1**

votes

**1**answer

126 views

### Stone Cech compactification for exponential map

Recently I met with a problem related to Stone-Cech Compactification theorem
in Furstenberg's famous paper "non-commuting product."
I try my best to understand Stone-Cech compactification theorem by ...

**2**

votes

**1**answer

253 views

### Mellin transform between heat kernel and zeta-function

For some notion of a "positive operator" $D$ of "Laplacian type" one seems to be able to define a notion of a zeta-function as $\xi(s,f,D) = Tr_{L^2}(f D^{-s})$ where $f \in L^2$ (the space of ...

**2**

votes

**1**answer

197 views

### Fourier transform and spectrum of PDOs in $L^p$

Let $K$ be a compact subset in $\mathbb{R}^n$ with $m(K)=0$, Suppose $supp\hat{u}\subset K$ for some $u\in L^p$,where $2\leq p\leq \frac{2n}{n-1}$,can we get $u\equiv 0$ ?
Motivation: If $K$ is a ...

**0**

votes

**1**answer

670 views

### Can we construct a Hilbert space where the operator following differencial operator is symmetric?

I'd like to know if one can define a pertinent Hilbert space where the operator
$$A_p v := -\frac{1}{2} v" + (vF + v\int_\mathbb{R} Sp + p\int_\mathbb{R} Sv )'$$ is symmetric. Here, $p$ satisfies ...

**2**

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**0**answers

477 views

### What essential property justifies the name “derivative”?

Most, if not all, of the notions of derivative that I have so far seen have the property that they are locally defined -- meaning that the derivative of a map-type object at a point depends on the map ...

**2**

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96 views

### Invariant linear manifolds for multiplication by the independent variable in L^2 (R)

In general I am trying to determine when the self-adjoint operator $M$ of multiplication by the independent variable in $L^2 (\mathbb{R})$ has a symmetric restriction to a dense linear manifold ...

**0**

votes

**2**answers

887 views

### Diagonalization of a matrix of differential operators

Dear community,
i have a question regarding differential operators acting on vector valued functions and how to "diagonalize" them.
To explain my question i will use an example:
Let $V^k$ be the ...

**3**

votes

**0**answers

355 views

### PDES - from Vector fields whose inner product with their vector Laplacian equals norm of the vector field

Let $g(x_{1},........,x_{n}) = \sum_{i=1}^{n}g_{i}(x_{1},\cdots,x_{n})e_{i}$ be a function in $\mathbb{C}^n$ ($e_{i}$ are the standard bases).
Let $\nabla^{2}$ be the vector Laplacian. Let ...

**7**

votes

**1**answer

834 views

### What is the “correct” generalization of operator norms for nonlinear operators?

I have been recently wondering what is a (or even the) "correct" generalization of the notion of an operator norm to nonlinear operators?
Please excuse the naivete of my question; if you think that ...

**3**

votes

**3**answers

820 views

### Can the adjoint of the exterior derivative in semi-Riemannian geometry be defined without the Hodge * operator?

The adjoint of the exterior derivarive is defined by
$\delta:=(-1)^k\ast^{-1}d\ast$,
but I need a way which avoids the Hodge $\ast$ operator.
Is there another definition?
For example, for ...

**4**

votes

**1**answer

711 views

### How “generalized eigenvalues” combine into producing the spectral measure?

Hi... I am wondering how 'eigenvalues' that don't lie in my Hilbert space combine into producing the spectral measure. I study probability and I am quite ignorant in the field of spectral analysis of ...

**3**

votes

**1**answer

346 views

### Are these operators defined on 2D surfaces self-adjoint?

My research group finds/proposes a fundamental operator in quantum mechanics, the Cartesian momentum as I called (I think for mathematician the ref. 2007 is sufficient). However, I do not know whether ...

**7**

votes

**2**answers

717 views

### What is the smallest $C^*$-algebra containing the “standard” pseudodifferential operators?

Is $\Psi^0(\mathbb{R})$ (pseudodifferential operators with symbols obeying
$
|\partial^\alpha_x \partial^\beta_\xi a(x,\xi)| \leq C_{\alpha,\beta} (1+|\xi|)^{-|\beta|}
$
) a $C^*$-algebra?
In other ...

**5**

votes

**1**answer

557 views

### Self-adjoint extension of locally defined differential operators

The following is well known. Given a symmetric differential operator, like $\partial_x^2$, defined on smooth functions of compact support on $\mathbb{R}$, $C_0^\infty(\mathbb{R})$, one can count the ...

**7**

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**0**answers

516 views

### How hard is it to make a differential operator Hermitian?

Let $M$ be a closed finite-dimensional smooth manifold (over $\mathbb R$). Let $C^\infty(M) = C^\infty(M,\mathbb C)$ be the algebra of smooth complex-valued functions on $M$, with the natural complex ...