Sturm–Liouville theory

In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the formThis theory is important in applied mathematics, where Sturm–Liouville problems occur very frequently, particularly when dealing with separable linear partial differential equations.For example, in quantum mechanics, the one-dimensional time-independent Schrödinger equation is a Sturm–Liouville problem.The main results in Sturm–Liouville theory apply to a Sturm–Liouville problem on a finite intervalThe terms eigenvalue and eigenvector are used because the solutions correspond to the eigenvalues and eigenfunctions of a Hermitian differential operator in an appropriate Hilbert space of functions with inner product defined using the weight function.In this space L is defined on sufficiently smooth functions which satisfy the above regular boundary conditions.However, this operator is unbounded and hence existence of an orthonormal basis of eigenfunctions is not evident.Then, computing the resolvent amounts to solving a nonhomogeneous equation, which can be done using the variation of parameters formula.This shows that the resolvent is an integral operator with a continuous symmetric kernel (the Green's function of the problem).As a consequence of the Arzelà–Ascoli theorem, this integral operator is compact and existence of a sequence of eigenvalues αn which converge to 0 and eigenfunctions which form an orthonormal basis follows from the spectral theorem for compact operators.In this case, the spectrum no longer consists of eigenvalues alone and can contain a continuous component.Consider a general inhomogeneous second-order linear differential equationIn general, if initial conditions at some point are specified, for example y(a) = 0 and y′(a) = 0, a second order differential equation can be solved using ordinary methods and the Picard–Lindelöf theorem ensures that the differential equation has a unique solution in a neighbourhood of the point where the initial conditions have been specified.Notice that by adding a suitable known differentiable function to y, whose values at a and b satisfy the desired boundary conditions, and injecting inside the proposed differential equation, it can be assumed without loss of generality that the boundary conditions are of the form y(a) = 0 and y(b) = 0.Here, the Sturm–Liouville theory comes in play: indeed, a large class of functions f can be expanded in terms of a series of orthonormal eigenfunctions ui of the associated Liouville operator with corresponding eigenvalues λi:Since orthogonal bases are always maximal (by definition) we conclude that the Sturm–Liouville problem in this case has no other eigenvectors.The reader may check, either by integrating ∫ eikxx dx or by consulting a table of Fourier transforms, that we thus obtainSuppose we are interested in the vibrational modes of a thin membrane, held in a rectangular frame, 0 ≤ x ≤ L1, 0 ≤ y ≤ L2.The method of separation of variables suggests looking first for solutions of the simple form W = X(x) × Y(y) × T(t).The boundary conditions ("held in a rectangular frame") are W = 0 when x = 0, L1 or y = 0, L2 and define the simplest possible Sturm–Liouville eigenvalue problems as in the example, yielding the "normal mode solutions" for W with harmonic time dependence,The functions Wmn form a basis for the Hilbert space of (generalized) solutions of the wave equation; that is, an arbitrary solution W can be decomposed into a sum of these modes, which vibrate at their individual frequencies ωmn.Consider a linear second-order differential equation in one spatial dimension and first-order in time of the form:The first of these equations must be solved as a Sturm–Liouville problem in terms of the eigenfunctions Xn(x) and eigenvalues λn.[clarification needed] The spectral parameter power series (SPPS) method makes use of a generalization of the following fact about homogeneous second-order linear ordinary differential equations: if y is a solution of equation (1) that does not vanish at any point of [a,b], then the functionIn the SPPS algorithm, one must begin with an arbitrary value λ∗0 (often λ∗0 = 0; it does not need to be an eigenvalue) and any solution y0 of (1) with λ = λ∗0 which does not vanish on [a,b].Two sequences of functions X(n)(t), X̃(n)(t) on [a,b], referred to as iterated integrals, are defined recursively as follows.Then for any λ (real or complex), u0 and u1 are linearly independent solutions of the corresponding equation (1).For numerical work one may truncate this series to a finite number of terms, producing a calculable polynomial in λ whose roots are approximations of the sought-after eigenvalues.When λ = λ0, this reduces to the original construction described above for a solution linearly independent to a given one.In practice if (1) has real coefficients, the solutions based on y0 will have very small imaginary parts which must be discarded.
mathematicsordinary differential equationboundary conditionseigenvalueseigenfunctionsapplied mathematicsseparablepartial differential equationsquantum mechanicsSchrödinger equationJacques Charles François SturmJoseph Liouvilleseparated boundary conditionsnon-trivialweak senseHermitiandifferential operatorHilbert spacefunctionscompletenessorthonormal basisKronecker deltaordinary differential equationsintegrating factorvectorBessel equationLegendre equationlinear operatorfunctional analysiseigenvalueself-adjointintegration by partsunboundedresolventvariation of parametersintegral operatorGreen's functionArzelà–Ascoli theoremspectral theorem for compact operatorsPicard–Lindelöf theoremorthogonal basisFourier seriessquare-summableconvergence of Fourier seriesantidifferentiationvibrational modeswave equationseparation of variablesintegersconvergentRayleigh–Ritz methodFinite difference methodpower seriesSturm separation theoremNormal modeOscillation theorySpectral theory of ordinary differential equationsAtkinson–Mingarelli theoremBibcodeGesztesy, FritzEncyclopedia of MathematicsEMS PressTeschl, GeraldProvidenceAmerican Mathematical SocietyCambridge, MassachusettsHarvard University PressBirkhäusertopicsglossaryBanachFréchetHilbertHölderNuclearOrliczSchwartzSobolevTopological vectorBarrelledCompleteLocally convexReflexiveHahn–BanachRiesz representationClosed graphUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluAdjointBoundedCompactHilbert–SchmidtNormalTrace classTransposeUnitaryBanach algebraC*-algebraSpectrum of a C*-algebraOperator algebraGroup algebra of a locally compact groupVon Neumann algebraInvariant subspace problemMahler's conjectureHardy spaceHeat kernelIndex theoremCalculus of variationsFunctional calculusIntegral linear operatorJones polynomialTopological quantum field theoryNoncommutative geometryRiemann hypothesisDistributionGeneralized functionsApproximation propertyBalanced setChoquet theoryWeak topologyBanach–Mazur distanceTomita–Takesaki theory