12.4 Spectral analysis of Schrödinger operators

Schrödinger operators are key to quantum mechanics, describing energy and dynamics of quantum systems. They're defined as linear differential operators with specific properties, including self-adjointness and an unbounded domain.

The spectrum of Schrödinger operators reveals crucial information about quantum systems. It includes point, continuous, and residual spectra, each corresponding to different physical states. Understanding these spectra is vital for analyzing quantum behavior.

Schrödinger Operators and Their Spectrum

Definition of Schrödinger operators

• Linear differential operators used in quantum mechanics describe the energy and dynamics of a quantum system
• Defined as $H = -\frac{\hbar^2}{2m}\nabla^2 + V(x)$, where:
  • $\hbar$ reduced Planck's constant
  • $m$ mass of the particle
  • $\nabla^2$ Laplacian operator
  • $V(x)$ potential energy function
• Properties:
  • Self-adjoint $\langle Hf, g \rangle = \langle f, Hg \rangle$ for all $f, g$ in the domain of $H$
  • Unbounded domain of $H$ is a proper subset of the Hilbert space
  • Closed graph of $H$ is a closed subset of the product Hilbert space

Spectrum of Schrödinger operators

• Set of all $\lambda \in \mathbb{C}$ for which $H - \lambda I$ is not invertible
• Types of spectra:
  • Point spectrum (discrete spectrum) eigenvalues of $H$
  • Continuous spectrum $\lambda$ for which $H - \lambda I$ is not bounded below
  • Residual spectrum $\lambda$ for which $H - \lambda I$ is injective but not surjective
• Related to the physical properties of the quantum system
  • Eigenvalues correspond to bound states (hydrogen atom)
  • Continuous spectrum corresponds to scattering states (free particle)

Spectral theory for Schrödinger operators

• Spectral theorem self-adjoint operator has a unique spectral decomposition
  • $H = \int_{\sigma(H)} \lambda dE(\lambda)$, where $E(\lambda)$ is the spectral measure
• Resolvent operator-valued function $R(z) = (H - zI)^{-1}$ for $z \in \mathbb{C} \setminus \sigma(H)$
  • Analyzes the spectrum and provides information about the eigenfunctions
• Spectral measures projection-valued measures associated with self-adjoint operators
  • Construct the functional calculus and study the spectrum

Applications and Interpretation

Spectral methods in Schrödinger equation

• Time-independent Schrödinger equation $H\psi = E\psi$
  • $\psi$ wavefunction
  • $E$ energy eigenvalue
• Spectral methods expand the wavefunction in terms of the eigenfunctions of $H$
  • $\psi(x) = \sum_n c_n \phi_n(x)$
    • $\phi_n(x)$ eigenfunctions
    • $c_n$ expansion coefficients
• Eigenvalues and eigenfunctions determined by solving the eigenvalue problem
  • Variational methods (Rayleigh-Ritz method) approximate the eigenvalues and eigenfunctions

Physical interpretation of Schrödinger spectra

• Spectrum represents the possible energy values of the quantum system
  • Discrete eigenvalues correspond to bound states with well-defined energies (electron in an atom)
  • Continuous spectrum represents scattering states or unbound states (free electron)
• Eigenfunctions provide information about the spatial distribution of the particle
  • Probability of finding the particle at a given position is proportional to $|\psi(x)|^2$
  • Eigenfunctions with different eigenvalues are orthogonal, representing distinct quantum states
• Spectral decomposition allows for the calculation of expectation values and time evolution
  • Expectation values $\langle A \rangle = \langle \psi | A | \psi \rangle = \int_{\sigma(H)} a(\lambda) d\langle \psi | E(\lambda) | \psi \rangle$
    • $a(\lambda)$ observable associated with the operator $A$
  • Time evolution $\psi(x, t) = e^{-iHt/\hbar} \psi(x, 0)$
    • $\psi(x, 0)$ initial wavefunction