Mage's Rhizome hamburgminicourse2017

hamburgminicourse2017:classes_of_static_potentials

Anonymous (anonymous@undisclosed.example.com) — 2017-04-04T09:27:52+00:00

Classes of static potentials " Give a mathematician a situation which is the least bit ill-defined -- he will first of all make it well defined. Perhaps appropriately, but perhaps also inappropriately. The hydrogen atom illustrates this process nicely. The physicist asks: $i\frac{d}{dt}\psi=(-\Delta+V)\psi$$V$$V=0$$D(H)=H^2(\mathbb{R}^n)$$D(H)=H^2(\Omega)\cap H^1_0(\Omega)$$\Delta$$A,B$$B$$A$$D(A)\subseteq D(B)$$a,b>0$$\varphi \in D(A)$$\|B\varphi\| \leq a\|A\varphi\| + b\|\varphi\|$$a$$A$$B$$…

hamburgminicourse2017:regularity_of_schroedinger_solutions_and_functional_differentiability

Anonymous (anonymous@undisclosed.example.com) — 2017-04-02T22:59:27+00:00

Regularity of Schrödinger solutions and functional differentiability In the proof for Schrödinger solutions with time-dependent potentials it was already noted, that $H^2$ regularity is guaranteed, i.e. initial states in $D(H)=H^2$ stay in this set during evolution. A natural question is now for which potential classes $H^{2m}$$H(t)^{-1}H(t)$$H(t)^{-m}H(t)^m$$H(t)^{m}H(s)^{-m} = K_m(t,s) + \mathrm{id}$$\mathrm{Lip}([0,T],K^m)$\[ -\nabla\cdot (n \nabla v) = q - \partial_t^2 n. \]$q$$\psi(t)$$H^4…

hamburgminicourse2017:start

Anonymous (anonymous@undisclosed.example.com) — 2017-04-02T20:32:04+00:00

Hamburg mini course on Schrödinger dynamics Notes for a mini course held in Hamburg 3-4 Apr 2017 on some mathematical details of the Schrödinger equation. Contents: * Why Hilbert space? * Why self-adjoint operators? * The spectrum of operators * The abstract Cauchy problem in Banach space * Stone's theorem * Classes of static potentials * Time-dependent potentials * Regularity of Schrödinger solutions and functional differentiability Author: Markus Penz (2017) m.penz@inter.at…

hamburgminicourse2017:stone_s_theorem

Anonymous (anonymous@undisclosed.example.com) — 2017-04-02T15:26:10+00:00

Stone's theorem This is a generation theorem in the setting of Hilbert spaces that links self-adjoint operators to unitary semigroups. It simply states that the Cauchy problem \[ i\frac{d}{dt}\psi(t) = H\psi(t) \] with self-adjoint but possibly unbounded Hamiltonian $H$ and initial state $\psi(0)=\psi_0\in \mathcal{H}$$t\in\mathbb{R}$$U(t)$$\exp(-iHt):=U(t)$$H_\lambda = -i\lambda - \lambda^2 (i \lambda - H)^{-1}$$\lambda>0$$t>0$$\|(i\lambda - H)^{-1}\| \leq |\lambda|^{-1}$$H_\lambda \rightarrow…

hamburgminicourse2017:the_abstract_cauchy_problem_in_banach_space

Anonymous (anonymous@undisclosed.example.com) — 2017-04-02T15:57:39+00:00

The abstract Cauchy problem in Banach space The setting consists of a Banach space $X$, a possibly unbounded, densely defined operator $A:X \rightarrow X$ with domain $D(A)$ and an evolution equation \[ \frac{d}{dt}u(t) = Au(t) \] for all $t>0$ and initial condition $u(0)=x \in X$ (not only $D(A)$). A solution to this problem is denoted $u(t)=T(t)x$$A$$T(t)$$t \geq 0$$\mathcal{B}(X,X)$$\mathcal{C}^0$$T(0)=\mathrm{id}$$T(s+t)=T(t)T(s)$$t,s \geq 0$$\lim_{t \rightarrow 0} \|T(t)x-x\| = 0$$x\in X$$…

hamburgminicourse2017:the_spectrum_of_operators

Anonymous (anonymous@undisclosed.example.com) — 2017-04-04T17:38:35+00:00

The spectrum of operators About the terminology “spectrum” in mathematics and physics Dieudonné writes: " It finally dawned upon them [the physicists in the 1920s] that their “observables” had properties which made them look like hermitian operators in Hilbert space, and that, by an extraordinary coincidence, the $T = -\Delta$$\mathcal{C}^\infty_0(\Omega)$$H^2(\Omega)\cap H^1_0(\Omega)$$H^2_p(\Omega)$$H^2(\mathbb{R}^n)$$H^2(\mathbb{R}^n)$$[0,\infty)$$L^2(\mathbb{R}^n)$$\langle \varphi,-\Delta…

hamburgminicourse2017:time-dependent_potentials

Anonymous (anonymous@undisclosed.example.com) — 2017-04-02T22:06:27+00:00

Time-dependent potentials If the Hamiltonian $H(t)=-\Delta+V(t)$ is time-dependent, the approach involving evolution semigroups has to be modified and below we give a proof for the existence of solutions to the Schrödinger equation with potential $V(t)$. An obvious restriction is $V(t) \in K$$\{H(t)\}_t$$D(H)=D(-\Delta)$$U(t,s)$$U(t,s), 0\leq s,t \leq T$$U(t,t) = \mathrm{id}$$U(t,r)U(r,s) = U(t,s)$$U(t,s)^{-1} = U(t,s)^* = U(s,t)$$\partial_t U(t,s) = -i H(t) U(t,s)$$\partial_s U(t,s) = i U(t,s)…

hamburgminicourse2017:why_hilbert_space

Anonymous (anonymous@undisclosed.example.com) — 2017-04-03T16:11:09+00:00

Why Hilbert space? Pragmatic answer. A Hilbert space is like an euclidean space with possibly infinite dimensions, i.e. it has a lot of structure, like that of a vector space, a scalar product, and completeness! Sometimes a Banach space (with just a norm and completeness) is enough.\[ \partial_x^n \quad\longleftrightarrow\quad \delta^{(n)}(x-x') \]\[ \int |\psi|^2 dx < \infty \quad\longleftrightarrow\quad \sum |x_n|^2 < \infty. \]$x_n$$H$$\mathbb{N} \rightarrow \mathbb{C}$\[ L^2(\Omega) \simeq …

hamburgminicourse2017:why_self-adjoint_operators

Anonymous (anonymous@undisclosed.example.com) — 2017-04-03T16:12:34+00:00

Why self-adjoint operators? What is self-adjoint anyway? The adjoint $A^*$ of an operator $A$ (always densely defined on a Hilbert space $\mathcal{H}$) is given by all pairs $(y,z)$ (the graph of $A^*$) that obey \[ \langle y,Ax \rangle = \langle z,x \rangle \quad\forall x \in D(A) \] and thus it holds $z = A^*y$ and $y \in D(A^*)$. If $A=A^*$ on $D(A)$ the operator is called symmetric and one easily gets $D(A) \subseteq D(A^*)$$D(A) = D(A^*)$$D(A)=\mathcal{H}$$i\partial_x$$H^1([0,1]) = \{ f \i…