Glossary

12.4 Spectral analysis of Schrödinger operators

3 min read•july 22, 2024

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 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

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  • Linear differential operators used in quantum mechanics describe the energy and dynamics of a quantum system
  • Defined as H=−ℏ22m∇2+V(x)H = -\frac{\hbar^2}{2m}\nabla^2 + V(x), where:
    • ℏ\hbar reduced Planck's constant
    • mm mass of the particle
    • ∇2\nabla^2 Laplacian operator
    • V(x)V(x) potential energy function
  • Properties:
    • Self-adjoint ⟨Hf,g⟩=⟨f,Hg⟩\langle Hf, g \rangle = \langle f, Hg \rangle for all f,gf, g in the domain of HH
    • Unbounded domain of HH is a proper subset of the
    • Closed graph of HH is a closed subset of the product Hilbert space

Spectrum of Schrödinger operators

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

Spectral theory for Schrödinger operators

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

Applications and Interpretation

Spectral methods in Schrödinger equation

  • Time-independent Schrödinger equation Hψ=EψH\psi = E\psi
    • ψ\psi wavefunction
    • EE energy eigenvalue
  • Spectral methods expand the wavefunction in terms of the eigenfunctions of HH
    • ψ(x)=∑ncnϕn(x)\psi(x) = \sum_n c_n \phi_n(x)
      • ϕn(x)\phi_n(x) eigenfunctions
      • cnc_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 ∣ψ(x)∣2|\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 ⟨A⟩=⟨ψ∣A∣ψ⟩=∫σ(H)a(λ)d⟨ψ∣E(λ)∣ψ⟩\langle A \rangle = \langle \psi | A | \psi \rangle = \int_{\sigma(H)} a(\lambda) d\langle \psi | E(\lambda) | \psi \rangle
      • a(λ)a(\lambda) observable associated with the operator AA
    • Time evolution ψ(x,t)=e−iHt/ℏψ(x,0)\psi(x, t) = e^{-iHt/\hbar} \psi(x, 0)
      • ψ(x,0)\psi(x, 0) initial wavefunction

Key Terms to Review (26)

Banach Space: A Banach space is a complete normed linear space where every Cauchy sequence converges within the space. This completeness property is vital in functional analysis as it ensures that limits of sequences remain within the space, allowing for robust analysis of functional properties and the behavior of operators.
Banach space: A Banach space is a complete normed vector space, meaning it is a vector space equipped with a norm such that every Cauchy sequence converges to a limit within the space. This property of completeness is crucial for ensuring the convergence of sequences, which allows for more robust analysis and applications in functional analysis.
Bertschinger's Theorem: Bertschinger's Theorem is a result in the spectral analysis of Schrödinger operators that provides insights into the essential spectrum of certain classes of operators. It connects the properties of these operators with the nature of their potential, particularly when dealing with potentials that exhibit specific growth conditions. This theorem plays a critical role in understanding how perturbations affect the spectral properties of quantum systems, making it an essential tool in mathematical physics.
Bounded operator: A bounded operator is a linear transformation between two normed spaces that maps bounded sets to bounded sets, ensuring that there exists a constant such that the operator's norm is finite. This concept is crucial for understanding the behavior of operators in functional analysis, particularly in the context of Banach and Hilbert spaces, where operators can be classified based on their continuity and stability under limits.
Compactness: Compactness in functional analysis refers to a property of operators, particularly linear operators between Banach spaces, where the operator maps bounded sets to relatively compact sets. This concept is crucial as it connects with continuity, convergence, and spectral properties of operators, allowing us to generalize finite-dimensional results to infinite-dimensional spaces.
Continuity: Continuity refers to the property of a function where small changes in the input result in small changes in the output. This concept is vital in analysis as it ensures that the behavior of functions is predictable and stable, particularly when dealing with linear operators and spaces. Understanding continuity is crucial in various contexts, such as operator norms, the behavior of adjoints, and applications within spectral theory and functional analysis.
Continuous Spectrum: A continuous spectrum refers to a set of values that are not isolated but rather form a continuous range. In functional analysis, this concept is particularly relevant for operators where the spectrum does not consist of discrete eigenvalues but spans an interval, indicating that the operator's behavior can vary smoothly over a continuum of states.
Discrete spectrum: A discrete spectrum refers to a set of isolated eigenvalues that characterize the behavior of an operator in a Hilbert space. This concept is important because it helps in understanding the nature of the operator's spectrum, particularly in distinguishing between points of accumulation and isolated points, which can influence the spectral mapping theorem and functional calculus, as well as the analysis of quantum systems such as Schrödinger operators.
Eigenfunction: An eigenfunction is a non-zero function that, when acted upon by a linear operator, results in the function being multiplied by a scalar known as the eigenvalue. This concept is fundamental in various areas of mathematics and physics, connecting to important properties of differential equations and operator theory. Eigenfunctions are essential for understanding the behavior of systems described by these equations, as they help to decompose complex problems into simpler components that can be analyzed individually.
Eigenfunctions: Eigenfunctions are special functions associated with a linear operator, such that when the operator is applied to them, the output is simply a scalar multiple of the original function. This property makes eigenfunctions crucial in understanding the behavior of differential and integral operators, as they reveal the fundamental modes of operation for these mathematical tools. They also play a vital role in spectral analysis, particularly in quantum mechanics, where eigenfunctions correspond to possible states of a system.
Eigenvalues: Eigenvalues are special numbers associated with a linear transformation represented by a matrix or an operator. They characterize how the transformation scales certain vectors, known as eigenvectors, and provide insights into the properties of the operator or matrix. Understanding eigenvalues is crucial for solving differential equations, analyzing stability in systems, and interpreting quantum mechanical systems.
Essential Spectrum: The essential spectrum of an operator consists of those values in the spectrum that are not isolated eigenvalues with finite multiplicity. It gives insight into the behavior of an operator, particularly in terms of compactness and its relation to the perturbation of operators. Understanding the essential spectrum helps to analyze stability and the nature of solutions in various mathematical contexts.
Functional Calculus: Functional calculus is a mathematical framework that allows for the extension of functions to operate on elements of a normed space, particularly in the context of operators on Hilbert or Banach spaces. This concept plays a crucial role in analyzing the properties of operators, particularly normal operators, and connects to the behavior of systems in quantum mechanics and spectral analysis of differential operators.
Hilbert Space: A Hilbert space is a complete inner product space that is a fundamental concept in functional analysis, combining the properties of normed spaces with the geometry of inner product spaces. It allows for the extension of many concepts from finite-dimensional spaces to infinite dimensions, facilitating the study of sequences and functions in a rigorous way.
Limiting Absorption Principle: The limiting absorption principle is a concept in functional analysis that deals with the behavior of resolvents of certain operators, particularly in the context of differential operators like Schrödinger operators. This principle states that the resolvent can be analytically continued to a neighborhood of the spectrum, allowing us to relate the spectral properties of the operator to its resolvent. It plays a crucial role in understanding how perturbations affect spectral properties and ensures that we can analyze the operator's spectrum even when certain assumptions about the potential may not hold.
Perturbation Theory: Perturbation theory is a mathematical approach used to find an approximate solution to a problem that cannot be solved exactly, by starting from the exact solution of a related, simpler problem and adding small changes or 'perturbations'. This technique is vital in various fields, particularly in understanding how small changes in parameters can affect the properties of operators, and it connects deeply with compact operators, spectral theory, and applications in quantum mechanics.
Point Spectrum: The point spectrum of a linear operator refers to the set of eigenvalues for which the operator fails to be invertible, meaning there exists a non-zero vector such that the operator applied to that vector equals the eigenvalue times that vector. This concept is critical in understanding the spectral properties of operators and is closely related to various types of operators, including self-adjoint and compact operators, as well as their spectral decomposition.
Residual Spectrum: The residual spectrum refers to the set of complex numbers that are part of the spectrum of a bounded linear operator but do not correspond to eigenvalues. It is crucial for understanding the behavior of operators, particularly in relation to the resolvent and spectral theory, as it highlights the subtle distinctions between different parts of the spectrum and their implications for operator properties.
Riesz Spectral Theorem: The Riesz Spectral Theorem is a fundamental result in functional analysis that characterizes the spectrum of a bounded linear operator on a Hilbert space. It establishes a connection between the spectral properties of self-adjoint operators and the geometry of the underlying space, leading to the decomposition of the space into orthogonal subspaces corresponding to the operator's spectrum. This theorem plays a crucial role in understanding various applications, particularly in quantum mechanics and the analysis of differential operators.
Self-adjoint operator: A self-adjoint operator is a linear operator on a Hilbert space that is equal to its own adjoint, meaning that the inner product satisfies $$\langle Ax, y \rangle = \langle x, Ay \rangle$$ for all vectors x and y in the space. This property ensures that the operator has real eigenvalues and orthogonal eigenvectors, which are essential features in various mathematical and physical applications.
Spectral Theorem: The spectral theorem is a fundamental result in functional analysis that characterizes the structure of linear operators on Hilbert spaces, particularly self-adjoint and normal operators. It states that these operators can be represented in terms of their eigenvalues and eigenvectors, allowing for diagonalization in an appropriate basis. This theorem connects with various concepts, such as adjoint operators, unbounded operators, and projection operators, providing insights into their spectral properties.
Spectrum: In functional analysis, the spectrum of an operator is the set of complex numbers that describes the behavior of the operator in terms of its eigenvalues and resolvent. It provides crucial information about the operator's properties, including whether it is invertible, compact, or bounded, and plays a fundamental role in understanding various types of operators across different contexts.
Spectrum: In functional analysis, the spectrum of an operator refers to the set of complex numbers that describe the possible values of the operator's eigenvalues and the behavior of its resolvent. The spectrum provides critical insights into the operator's properties, including stability, compactness, and spectral decomposition, making it a foundational concept in various areas of analysis.
Stability analysis: Stability analysis is a mathematical method used to determine the behavior of a system or operator under small perturbations or changes in parameters. It assesses whether the solutions of a differential equation or the eigenvalues of an operator remain bounded or converge to a particular state when subjected to minor disturbances. In the context of Schrödinger operators, stability analysis helps understand how the spectral properties influence the physical behavior of quantum systems, including their responses to external forces.
Time-independent Schrödinger operator: The time-independent Schrödinger operator is a differential operator that arises in quantum mechanics and describes the energy states of a quantum system without considering time evolution. This operator is key in spectral analysis as it helps identify the spectrum of energy levels and corresponding eigenfunctions, which are essential for understanding the behavior of quantum systems in stationary states.
Weyl's Criterion: Weyl's Criterion is a fundamental result in spectral theory that provides a necessary and sufficient condition for the absence of eigenvalues in the spectrum of a self-adjoint operator. This criterion is particularly important in the spectral analysis of Schrödinger operators, as it helps determine the nature of the spectrum and understand the existence of bound states and continuum states.
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