Schrödinger operator

The Schrödinger operator is an operator from quantum mechanics . It gives a simplified description of a non-relativistic movement of a quantum mechanical particle in an external potential.

The negative eigenvalues ​​of the Schrödinger operator correspond to the so-called bound states , such as the energies of the electrons that are bound to an atomic nucleus.

The spectral theory of the Schrödinger operator has been intensively developed since 1950 due to its mathematical abundance and its physical importance.

Definition and introduction

The Schrödinger operator for a quantum system is the linear , partial differential operator

${\displaystyle H=-{\frac {\hbar ^{2}}{2m}}\Delta +V}$

in the space of square-integrable functions . The constant is the reduced mass of the system and is the reduced Planck quantum of action . The real-valued function is often called the potential, the Laplace operator is called the operator of kinetic energy. This family of linear operators describes different quantum systems for different potentials . ${\displaystyle L^{2}(\mathbb {R} ^{n})}$ ${\displaystyle m}$ ${\displaystyle \hbar }$ ${\displaystyle V}$ ${\displaystyle V}$

Elements of the Hilbert space , which are also called wave functions, represent different states of the system. The time evolution of a wave function for a quantum system with the Schrödinger operator is described by the Schrödinger equation ${\displaystyle L^{2}(\mathbb {R} ^{n})}$ ${\displaystyle H}$

${\displaystyle i\hslash {\frac {\partial }{\partial t}}\psi _{t}=H\psi _{t}}$.

For every reasonable initial value of the system the solution of the Schrödinger equation has the form ${\displaystyle \psi _{0}}$

${\displaystyle \psi _{t}=U_{H}(t)\psi _{0}}$,

where the mapping is the expansion operator for the Schrödinger equation. ${\displaystyle U_{H}(t)\colon \psi _{0}\to \psi _{t}}$

A requirement from quantum mechanics is that

${\displaystyle \parallel \psi _{0}\parallel =\parallel U_{H}(t)\psi _{0}\parallel =\parallel \psi _{t}\parallel \qquad \qquad (1)}$

applies. Another requirement for the uniqueness of solutions to the Schrödinger equation is that for all ${\displaystyle s,t\in \mathbb {R} }$

${\displaystyle U_{H}(s)U_{H}(t)=U_{H}(s+t)\qquad \qquad \quad \;(2)}$

applies.

example

As potential we consider the Coulomb potential : ${\displaystyle V}$

${\displaystyle V\colon \mathbb {R} ^{3}\to \mathbb {R} ,x\mapsto {\frac {-Z}{|x|}},Z>0}$

where the constant stands for the atomic number . ${\displaystyle Z}$

With this potential, hydrogen-like atoms or ions can be modeled. B. a single electron is bound to an atomic nucleus.

The Schrödinger operator thus has the shape ${\displaystyle H}$

${\displaystyle H=-{\frac {\hbar ^{2}}{2m}}\Delta -{\frac {Z}{|x|}}}$

properties

This section summarizes some of the results of the Schrödinger operator. Important aspects of the Schrödinger operator are self-adjointness , the negative, the discrete and the essential spectrum .

Substantial self adjointness

The self-adjointness of the Schrödinger operator is a necessary and sufficient condition for the existence and uniqueness of solutions to the Cauchy problem of the Schrödinger equation, which also satisfy requirements (1) and (2). The question of whether the Schrödinger operator is self-adjoint to a given potential V is not easy to answer.

• If and are semi-constrained down to (that is, there is one with for all ), it is essentially self-adjoint up . ${\displaystyle V\in C^{\infty }(\mathbb {R} ^{n})}$ ${\displaystyle H}$ ${\displaystyle C_{0}^{\infty }(\mathbb {R} ^{n})}$ ${\displaystyle \alpha \in \mathbb {R} }$ ${\displaystyle \langle u,Hu\rangle \geq \alpha \langle u,u\rangle }$ ${\displaystyle u\in C_{0}^{\infty }(\mathbb {R} ^{n})}$ ${\displaystyle H}$ ${\displaystyle C_{0}^{\infty }(\mathbb {R} ^{n})}$

• If is, where the space is locally integrable functions , then it is essentially self-adjoint to . ${\displaystyle 0\leq V\in L_{\operatorname {loc} }^{2}(\mathbb {R} ^{n})}$ ${\displaystyle L_{\operatorname {loc} }^{2}(\mathbb {R} ^{n})}$ ${\displaystyle H}$ ${\displaystyle C_{0}^{\infty }(\mathbb {R} ^{n})}$

• If and is real, then is self adjoint with . ${\displaystyle V\in L^{2}(\mathbb {R} ^{3})+L^{\infty }(\mathbb {R} ^{3})}$ ${\displaystyle H}$ ${\displaystyle D(H)=D(\Delta )=H^{2}(\mathbb {R} ^{3})}$

• If is measurable with and with for , for , then it is self adjoint on . ${\displaystyle V\colon \mathbb {R} ^{n}\to \mathbb {R} }$ ${\displaystyle V_{+}:=\max\{V,0\}\in L_{\operatorname {loc} }^{2}(\mathbb {R} ^{n})}$ ${\displaystyle V_{-}:=\max\{-V,0\}\in L^{p}(\mathbb {R} ^{n})+L^{\infty }(\mathbb {R} ^{n})}$ ${\displaystyle p=2}$ ${\displaystyle n\leq 3}$ ${\displaystyle p>{\frac {n}{2}}}$ ${\displaystyle n\geq 4}$ ${\displaystyle H}$ ${\displaystyle C_{0}^{\infty }(\mathbb {R} ^{n})}$

Discrete spectrum

• If and , the spectrum of in is discrete for each . ${\displaystyle V\in L_{\operatorname {loc} }^{\infty }(\mathbb {R} ^{n})}$ ${\displaystyle a:=\lim _{R\to \infty }\inf _{|x|\geq R}V(x)>-\infty }$ ${\displaystyle a^{'}<a}$ ${\displaystyle H}$ ${\displaystyle (-\infty ,a^{'})}$

Negative spectrum

From the above result we know that the negative spectrum is discrete: nevertheless, the question arises whether there are any negative eigenvalues ​​at all.

• For with , and the Schrödinger operator has at least one negative eigenvalue. ${\displaystyle H=-{\frac {\hbar ^{2}}{2m}}\Delta +V}$ ${\displaystyle V\in C_{0}^{\infty }(\mathbb {R} ^{n})}$ ${\displaystyle \int Vdx=:V_{0}<0}$ ${\displaystyle n\leq 2}$

• Be . Then there is a constant so that the estimate holds for all ${\displaystyle n\geq 3}$ ${\displaystyle L_{n}}$ ${\displaystyle V\in C^{\infty }(\mathbb {R} ^{n})\cap L^{\frac {n}{2}}(\mathbb {R} ^{n}),\lim _{x\to \infty }V(x)=0}$

${\displaystyle N(H)\leq L_{n}\int \limits _{V(x)\leq 0}(-V)^{\frac {n}{2}}dx}$,

where is the number of negative eigenvalues ​​of . ${\displaystyle N(H)}$ ${\displaystyle H}$

Essential spectrum

• Be the essential spectrum of . If is self-adjoint then: ${\displaystyle \sigma _{ess}}$ ${\displaystyle H}$ ${\displaystyle H}$

${\displaystyle \lambda \in \sigma _{ess}}$is equivalent to a Weyl sequence to and to . ${\displaystyle H}$ ${\displaystyle \lambda }$

• If and , then is . ${\displaystyle V\in L_{\operatorname {loc} }^{\infty }(\mathbb {R} ^{n},\mathbb {C} )}$ ${\displaystyle \lim _{|x|\to \infty }V(x)=0}$ ${\displaystyle \sigma _{ess}=[0,\infty )}$

literature

• Andrey Tyukin: The eigenvalue asymptotics for Schrödinger operators , written elaboration on the 13th lecture of the advanced seminar, University of Mainz 2009
• A. Pankov: Introduction to Spectral Theory of Schrödinger Operators , Vorlesungsnotizen, Vinnitsa State Pedagogical University 1999/2000.
• Konstantin Pankrashkin: Schrödinger operators , lecture notes, HU Berlin WS 2005/2006.
• Rupert L. Frank: Hardy-Lieb-Thirring inequalities for eigenvalues ​​of Schrödinger operators , Doctoral thesis, Stockholm 2007.
• PD Hislop: Introduction to Spectral theory With Applications to Schrödinger Operators , Springer Verlag, New York 1996.