3
votes
0answers
79 views
+50

Uniform continuity of the function $x(t)=e^{tA}x$

Let $A$ be a bounded operator on a Banach space $X$. Consider the exponential function $x(t)=e^{tA}x:=\sum_{n=0}^{+\infty}\dfrac{t^nA^n}{n!}x$, for all $t\in \mathbb{R}$, where $x\in X$. If the ...
1
vote
1answer
82 views

Mathieu differential equation

Given the operator $T (\psi)(x):= \psi''(x)-2q \cos(2x)\psi(x)$ with $T : D(T) \subset L^2[0,2\pi] $ I was wondering: What is the right domain $D(T)$ for this operator if we want to solve the ...
4
votes
0answers
97 views

Operators such that $\langle Ax,x \rangle=-\langle x,Ax \rangle$

Let $X$ be a Banach space. We consider the differential equation: $$x'(t)=Ax(t), \ \ \ t\in\mathbb{R}$$ where $A$ is a bounded operator on $X$. If $X$ is a Hilbert space, and $x(t)$ is a solution of ...
0
votes
1answer
45 views

Positive operators in Hilbert spaces

Let $H$ be a Hilbert space. I am just asking if there's some reference which studies operators $A$ with this property: $$\left\langle Ax,x\right\rangle \geq0,$$ for all $x\in H$. And $Ax=0$ whenever ...
1
vote
0answers
14 views

Is there any relation between positive definite operator and an operator that satisfies maximum principle?

Suppose $L$ is a self adjoint differential operator which satisfies maximum principle. Maximum principle: Assume that $u(x)$ satisfies $u(0)\geq 0$ and $u(1)\geq 0$. Now $L$ is said to satisfy ...
1
vote
0answers
25 views

Application of operator theory in ODE and PDE

I am looking for references of applications of operator theory (especially spectral theory) in ODE, PDE and possibly SDE. I have learnt operator theory in the general set up, but only know little ...
0
votes
0answers
25 views

Second Level Operators:

What would be an example of an Operator $$H$$ such that for any and all explicit functions U $$H[u] = I$$ where I is some other function However, for some other Operator W ex: [d/dx] ...
2
votes
1answer
49 views

Exponential of matrices and bounded operators

Let $A$ be a complex $n \times n$ matrix, such that the function $t\mapsto e^{tA}x$ is bounded on $\mathbb{R}$ and nonzero, for some vector $x\in \mathbb{C}$. How can we prove that $\inf_{t\in ...
0
votes
0answers
59 views

Derivation of Euler Lagrange Equation

I was reading on the derivation of the Euler Lagrange Equations (in the link: http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation focusing on: "Derivation of one-dimensional Euler–Lagrange ...
0
votes
1answer
19 views

The index of some perturbation about elliptic operator with Robin boundary condition

Let $I$ be an closed interval $[0, 1]$. $C^{2}(\bar{I})$ is the space of all $C^{2}$ functions on $(0, 1)$ with continuity at boundary and usual maximal norm. $C(\bar{I})$ is the space of all ...
0
votes
0answers
12 views

Fredholm operators and their applications

What are the applications (and possibly generalizations) of fredholm operators in partial differential equations?
0
votes
0answers
29 views

Eigenvalues and eigenvectors of a nonlinear operator

I have found a few nice answers to the question: "Why are eigenvalues and eigenvectors useful." I can imagine that knowledge of eigenvectors (-values) for a general nonlinear operator is worthless. ...
0
votes
0answers
18 views

Properties of solutions to an ODE

I have an ODE: $$ \frac{\mathrm{d}u}{\mathrm{d}t} + \mathcal{A}(t, u) = 0 $$ with final condition: $$ u(T)= \mathbf{1} $$ The function $u:\mathbb{R} \rightarrow \mathbb{R}^m$ is vectorial, and the ...
0
votes
1answer
28 views

Conditions for an Operator to Map Onto

Let operator $A[f(x)]=g(x)f(x)$ such that $A:C[a,b] \rightarrow C[a,b]$. I'm trying to think of the necessary and sufficient conditions needed on $g(x)$ such that the map is onto. Obviously it needs ...
0
votes
0answers
84 views

adjoint of a differential equation

Suppose we have the following differential equation $$ \psi''(y)- k^2 \psi(y) - \frac{U''(y)\phi(y)}{V(y)-c}=0$$ where $\psi $ and $\phi$ are complex valued functions, say on the interval $[0,1]$. ...
1
vote
0answers
54 views

Smoothing operator on manifolds

I am reading John Roe "Elliptic operators, topology and asymptotic methods". On page 79 there is the definition 5.20 of smoothing operators. The definition is the following: "A bounded operator on ...
3
votes
1answer
75 views

Spectrum of the Hill Operator $L(y)= -y''+ v(x) y $

Consider the eigenvalue equation for the Hill operator $$L(y)= -y''+ v(x) y = \lambda y, \quad x\in \mathbb{R},$$ where $v(x)$ is any potential and $\lambda$ is the spectral parameter. If $v(x) ...
2
votes
1answer
145 views

Compact resolvent

Given that the operator $$ Hf(x) = -xf''(x) + (x - 1)f'(x) $$ on the Hilbert space $L^2([0,\infty),e^{-x}dx)$ possesses, for each $n \in \mathbb{N}$, an eigenvalue $\lambda_n = n$ with eigenvector ...
6
votes
1answer
226 views

Kernel of adjoint operator

This problem is puzzling me, even though it should be really simple. Let $L=-\partial_x^2 + \frac 1 2 x^{-2}$ be an operator defined on $D(L)=C^\infty_c(0,+\infty)\subset L^2(0,+\infty)$. Its adjoint ...
1
vote
1answer
109 views

Self-adjointness of $D=\frac{d^2}{dx^2}-1$ with boundary conditions $u'(0) = 0 = u'(a)$ on $[0,a]$.

Im trying to show that $$D=\frac{d^2}{dx^2}-1$$ is self adjoint on $[0,a]$ subject to $u'(0)=u'(a)=0$. I think I need to use integration by parts but I'm not sure how to do that.
0
votes
1answer
185 views

Differential operators: elliptic vs strongly elliptic

This morning a collegue of mine came to me with the following question: does there exist any elliptic operator of order $2m$ with real (variable) coefficients that is not strongly elliptic? After ...
4
votes
0answers
353 views

Confused by a proof in Rudin *Functional Analysis*

I am reading Rudin's Functional Analysis and got quite confused by his proof of Thm 8.5, that is, the existence of fundamental solutions for differential operator $P(D)$, where $P$ is a polynomial. ...
1
vote
0answers
83 views

positively homogeneous asymptotic expansion associated to the symbol of a pseudodifferential operator

I am currently reading about pseudodifferential operators and their symbols, and I came across the notion of classical pseudodifferential operators. For these it is possible to find an asymptotic ...
3
votes
2answers
260 views

Asymptotic Expansion for heat operator $e^{-t\triangle}$

I'm afraid the question below might turn out to be very stupid - I just don't know how to make sense of two asymptotic expansions, given the heat operator $e^{-t\triangle}$ with $\triangle$ a ...
14
votes
3answers
4k views

Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
1
vote
2answers
74 views

The inequality with a differential operator

Let $A f = a f'''$ for $f\in C_0^3(\mathbb R)$ where $a$ - some constant. Is it possible to find $a$ such that $$ \|\lambda f-A f\|\geq \|\lambda f\| $$ for all $f\in C_0^3(\mathbb R)$ ...
0
votes
1answer
199 views

When is this Self-adjoint?

When is the following operator self-adjoint?(Is there a difference between self-adjoint and Hermitian?) $O:= \sum_{n=0}^4 f_n(x){d^n\over dx^n}$ subjected to boundary conditions ...
5
votes
3answers
587 views

Square root of differential operator

If $D_x$ is the differential operator. eg. $D_x x^3=3 x^2$. How can I find out what the operator $Q_x=(1+(k D_x)^2)^{(-1/2)}$ does to a (differentiable) function $f(x)$? ($k$ is a real number) For ...