Math 2.0

Ordinary differential equations Definition

Schrödinger operator

Builds on LaplacianOrdinary differential equation

Definition

Let I⊆ℝI \subseteq \R be an open interval and let V ⁣:I→ℝV \colon I \to \R be locally integrable. The one-dimensional Schrödinger operator with potential VV is the differential expression Hy=−y′′+Vy,H y = -y'' + V y, acting on functions y ⁣:I→ℂy \colon I \to \C such that yy and y′y' are locally absolutely continuous. More generally, for an open set Ω⊆ℝd\Omega \subseteq \R^d and a real function VV on Ω\Omega, the Schrödinger operator is H=−Δ+VH = -\Delta + V, where Δ\Delta is the Laplacian. The equation Hy=λyH y = \lambda y with λ∈ℂ\lambda \in \C is the time-independent Schrödinger equation.

Remarks

Example: for I=ℝI = \R and V(x)=x2V(x) = x^2, the function y(x)=e−x2/2y(x) = e^{-x^2/2} has y′′=(x2−1)e−x2/2y'' = (x^2 - 1) e^{-x^2/2}. So Hy=−y′′+x2y=yH y = -y'' + x^2 y = y, and yy solves the Schrödinger equation with λ=1\lambda = 1. Non-example: −y′′+ixy-y'' + i x y is not a Schrödinger operator in this sense, because the potential ixi x is not real. In units where Planck's constant and the mass are normalised, HH is the energy operator of a quantum particle. To get a self-adjoint operator on L2(I)L^2(I) one must also fix boundary conditions at the ends of II when they are needed.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptOrdinary differential equation, concept, no evidence levelOrdinary differential…conceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptLaplacian, concept, no evidence levelLaplacianconceptSchrödinger operator, definition, no evidence levelSchrödinger operatordefinition
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  1. Euclidean space object
  2. Function concept
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  7. Differentiable function of several variables concept
  8. Partial derivative concept
  9. Vector field concept
  10. Gradient concept
  11. Divergence concept
  12. Laplacian concept
  13. Schrödinger operator definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Schrödinger_equation
About this page
TagW6QS1ME39J
URIm2://W6QS1ME39J
Kinddefinition
WritingW0 stub
MSC34L40, 35J10, 81Q10
DefinesSchrödinger operator
UsesLaplacian, Ordinary differential equation
Changed2 November 2026

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