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3 votes
2 answers
570 views

Inverted harmonic oscillator

I am looking for the spectrum of th inverted oscillator $H=-\frac{d^2}{dx^2}-x^2$. Thanks in advance.
Fadil Kikawi's user avatar
3 votes
1 answer
229 views

Reference request: Original source of Yosida approximation

Numerous papers/books(citation needed) refer to the operator $$A_\lambda := \lambda AR_\lambda (A) = \lambda^2 R_\lambda(A) - \lambda I$$ where $R_\lambda(A)=(I+\lambda A)^{-1}$ is the resolvent, as ...
user109413's user avatar
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}(...
B.Hueber's user avatar
  • 1,171
2 votes
2 answers
290 views

Domain of Schrödinger operators

Let $S$ be a Schrödinger operator on $\mathbb{R}$, $Su=-u''+Vu$ with $V\geq1$ continuous and going to $+\infty$ at infinity (you can think of it as $x^2+1$). I wondering which assumptions do I have to ...
BlueCharlie's user avatar
2 votes
1 answer
404 views

Feynman-Kac formula for lattice heat equation with non-diagonal potential

Suppose that $X$ is the continuous-time simple symmetric random walk on the lattice $\mathbb Z^d$ (i.e., a simple symmetric random walk with i.i.d. exponential jump times), and let $$u(t,x):=\mathbf E\...
user78370's user avatar
  • 891
2 votes
1 answer
143 views

Equivalent ways to study a semilinear parabolic equation as a perturbed abstract Cauchy problem

I am considering the following abstract Cauchy problem on Banach space $X$: \begin{cases} u'(t)=Au(t)+\big(f(t)+Bu(t)\big),&t\in[0,T],\\ u(0)=x_0, \end{cases} Suppose $A$ generates an analyitc ...
John's user avatar
  • 503
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 ...
boundary's user avatar
1 vote
1 answer
292 views

A property of one-parameter groups of operators

Let $X$ be a Banach space. We consider the evolution equation: $$x'(t)=Ax(t), \ \ \ \ \ \ \ t\in \mathbb{R},$$ where $A$ is a bounded operator. I know that if $X=\mathbb{R^n}$ and $A$ is a matrix, ...
user144542's user avatar
1 vote
0 answers
182 views

Solution of non-linear Fredholm (Hammerstein) equation with non-degenerate kernel and reciprocal non-linearity

I have asked this question at MSE but got no response. I have rephrased it so that anyone who knows operator theory and integral equations could help me out... I faced a problem in physics which is a ...
arsal's user avatar
  • 11
0 votes
0 answers
29 views

On constructing the canonical boundary operator for a given differential operator

Given an $n\times n$ matrix $$X=\begin{pmatrix} x_{11} & x_{12} & \cdots & x_{1n} \\ x_{21} & x_{22} & \cdots & x_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ x_{n1}...
Ryan Hendricks's user avatar