Inmathematics and its applications, aSturm–Liouville problem is a second-order linearordinary differential equation of the form$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\left[p(x)\frac{\mathrm{d}y}{\mathrm{d}x}\right]+q(x)y=-\lambda w(x)y}$$for given functions${\displaystyle p(x)}$,${\displaystyle q(x)}$ and${\displaystyle w(x)}$, together with someboundary conditions at extreme values of${\displaystyle x}$. The goals of a given Sturm–Liouville problem are:

• To find the${\displaystyle \lambda }$ for which there exists a non-trivial solution to the problem. Such values${\displaystyle \lambda }$ are called theeigenvalues of the problem.
• For each eigenvalue${\displaystyle \lambda }$, to find the corresponding solution${\displaystyle y=y(x)}$ of the problem. Such functions${\displaystyle y}$ are called theeigenfunctions associated to each${\displaystyle \lambda }$.

Sturm–Liouville theory is the general study of Sturm–Liouville problems. In particular, for a "regular" Sturm–Liouville problem, it can be shown that there are an infinite number of eigenvalues each with a unique eigenfunction, and that these eigenfunctions form an orthonormal basis of a certain Hilbert space of functions.

This theory is important inapplied mathematics, where Sturm–Liouville problems occur very frequently, particularly when dealing withseparable linearpartial differential equations. For example, inquantum mechanics, the one-dimensional time-independentSchrödinger equation is a Sturm–Liouville problem.

Sturm–Liouville theory is named afterJacques Charles François Sturm (1803–1855) andJoseph Liouville (1809–1882), who developed the theory.

The main results in Sturm–Liouville theory apply to a Sturm–Liouville problem

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\left[p(x)\frac{\mathrm{d}y}{\mathrm{d}x}\right]+q(x)y=-\lambda w(x)y}$$

1

on a finite interval${\displaystyle [a,b]}$ that is "regular". The problem is said to beregular if:

• the coefficient functions${\displaystyle p,q,w}$ and the derivative${\displaystyle p^{\prime }}$ are all continuous on${\displaystyle [a,b]}$;
• ${\displaystyle p(x)>0}$ and${\displaystyle w(x)>0}$ for all${\displaystyle x\in [a,b]}$;
• the problem hasseparated boundary conditions of the form

2

The function${\displaystyle w=w(x)}$, sometimes denoted${\displaystyle r=r(x)}$, is called theweight ordensity function.

The goals of a Sturm–Liouville problem are:

• to find the eigenvalues: thoseλ for which there exists anon-trivial solution;
• for each eigenvalueλ, to find the corresponding eigenfunction${\displaystyle y=y(x)}$.

For a regular Sturm–Liouville problem, a function${\displaystyle y=y(x)}$ is called asolution if it is continuously differentiable and satisfies the equation (1) at every${\displaystyle x\in (a,b)}$. In the case of more general${\displaystyle p,q,w}$, the solutions must be understood in aweak sense.

The terms eigenvalue and eigenvector are used because the solutions correspond to theeigenvalues andeigenfunctions of aHermitiandifferential operator in an appropriateHilbert space offunctions with inner product defined using the weight function. Sturm–Liouville theory studies the existence and asymptotic behavior of the eigenvalues, the corresponding qualitative theory of the eigenfunctions and theircompleteness in the function space.

The main result of Sturm–Liouville theory states that, for any regular Sturm–Liouville problem:

• The eigenvalues${\displaystyle {\lambda }_1,{\lambda }_2,\dots }$ are real and can be numbered so that
• Corresponding to each eigenvalue${\displaystyle {\lambda }_n}$ is a unique eigenfunction${\displaystyle y_n=y_n(x)}$ (up to constant multiple), called thenthfundamental solution.
• The normalized eigenfunctions${\displaystyle y_n}$ form anorthonormal basis under thew-weighted inner product in theHilbert space${\displaystyle L^2([a,b],w(x)\mathrm{d}x)}$; that is,$${\displaystyle ⟨y_n,y_m⟩={\int }_a^b{y}_n(x)y_m(x)w(x)\mathrm{d}x={\delta }_{nm},}$$ where${\displaystyle {\delta }_{nm}}$ is theKronecker delta. Each linear combination of the eigenfunctions are uniformly convergent in the domain${\displaystyle [a,b]}$ and term by term differentiation is allowed.

Some classical results may be established about the oscillation and non-oscillation properties of solutions to certain Sturm-Liouville problems. In particular these establish that linearly independent solutions oscillate "equally rapidly" and the conditions under which solutions oscillate more rapidly.

Consider the Sturm-Liouville problem:

${\displaystyle \frac{d}{dx}\left[p(x)\frac{dy}{dx}\right]-q(x)y=0}$

It can be shown that there are no non-trivial solutions to the above equation which have infinitely many zeros on some closed interval$(a,b)$. A proof of this result would look something like as follows:

Suppose - for contradiction - that such a non-trivial solution$u$ existed, then the set$\{x\in (a,b):u(x)=0\}$ is infinite. TheBolzano-Weierstrass Theorem tells us that this set has some limit point$c\in [a,b]$,$u$ is a continuous function we have$u(c)=0$. By TheMean Value Theorem we have that for all$h>0$ there exists some$\theta \in [0,1)$ for which$u(c+h)=u(c)+h{u}^{\prime }(c+\theta h)$ and as$c$ is a limit point of a sequence of zeros, there's some$h$ for which$u(c+h)=0$ and hence for which$u^{\prime }(c+\theta h)=0$. Applying the continuity of$u(x)$ gives us that$u^{\prime }(c)=0$, from which we obtain that$u(x)=0$ everywhere.

Sturm's Separation Theorem: If$y_1,y_2$ are linearly independent solutions to the differential equation, and if$y_1$has two consecutive zeros at$x_1$ and$x_2$, then$y_2$ equals zero somewhere on the open interval$(x_1,x_2)$. Informally this means that the zeros of each linearly independent solution fall between the zeros of the other solution.

Sturm's Fundamental Theorem: Suppose that$u$ is a solution of

$\frac{d}{dx}\left[p(x)u^{\prime }\right]-q_1(x)u=0,$

and$v$ is a solution of

$\frac{d}{dx}\left[p(x)v^{\prime }\right]-q_2(x)v=0,$

where$q_1(x)>q_2(x)$. If$x_1,x_2$ are two consecutive zeros of$u$, then$v$ is zero somewhere on the interval$(x_1,x_2)$.In particular if$v$ is zero whenever$u$ is zero, v oscillates more rapidly than$u$.

The differential equation (1) is said to be inSturm–Liouville form orself-adjoint form. All second-order linear homogenousordinary differential equations can be recast in the form on the left-hand side of (1) by multiplying both sides of the equation by an appropriateintegrating factor (although the same is not true of second-orderpartial differential equations, or ify is avector). Some examples are below.

$${\displaystyle x^2{y}^{″}+x{y}^{\prime }+\left(x^2-{\nu }^2\right)y=0}$$which can be written in Sturm–Liouville form (first by dividing through byx, then by collapsing the first two terms on the left into one term) as$${\displaystyle {\left(x{y}^{\prime }\right)}^{\prime }+\left(x-\frac{{\nu }^2}{x}\right)y=0.}$$

$${\displaystyle \left(1-x^2\right)y^{″}-2x{y}^{\prime }+\nu (\nu +1)y=0}$$which can be put into Sturm–Liouville form, since⁠d/dx⁠(1 −x²) = −2x, so the Legendre equation is equivalent to$${\displaystyle {\left(\left(1-x^2\right)y^{\prime }\right)}^{\prime }+\nu (\nu +1)y=0}$$

$${\displaystyle x^3{y}^{″}-x{y}^{\prime }+2y=0}$$

Divide throughout byx³:$${\displaystyle y^{″}-\frac{1}{x^2}{y}^{\prime }+\frac{2}{x^3}y=0}$$

Multiplying throughout by anintegrating factor of$${\displaystyle \mu (x)=\exp \left(\int -\frac{dx}{x^2}\right)=e^{1/x},}$$gives$${\displaystyle e^{1/x}{y}^{″}-\frac{e^{1/x}}{x^2}{y}^{\prime }+\frac{2e^{1/x}}{x^3}y=0}$$which can be put into Sturm–Liouville form since$${\displaystyle \frac{d}{dx}{e}^{1/x}=-\frac{e^{1/x}}{x^2}}$$so the differential equation is equivalent to$${\displaystyle {\left(e^{1/x}{y}^{\prime }\right)}^{\prime }+\frac{2e^{1/x}}{x^3}y=0.}$$

$${\displaystyle P(x)y^{″}+Q(x)y^{\prime }+R(x)y=0}$$

Multiplying through by the integrating factor$${\displaystyle \mu (x)=\frac{1}{P(x)}\exp \left(\int \frac{Q(x)}{P(x)}dx\right),}$$and then collecting gives the Sturm–Liouville form:$${\displaystyle \frac{d}{dx}\left(\mu (x)P(x)y^{\prime }\right)+\mu (x)R(x)y=0,}$$or, explicitly:$${\displaystyle \frac{d}{dx}\left(\exp \left(\int \frac{Q(x)}{P(x)}dx\right)y^{\prime }\right)+\frac{R(x)}{P(x)}\exp \left(\int \frac{Q(x)}{P(x)}dx\right)y=0.}$$

The mapping defined by:$${\displaystyle Lu=-\frac{1}{w(x)}\left(\frac{d}{dx}\left[p(x)\frac{du}{dx}\right]+q(x)u\right)}$$can be viewed as alinear operatorL mapping a functionu to another functionLu, and it can be studied in the context offunctional analysis. In fact, equation (1) can be written as$${\displaystyle Lu=\lambda u.}$$

This is precisely theeigenvalue problem; that is, one seeks eigenvaluesλ₁,λ₂,λ₃,... and the corresponding eigenvectorsu₁,u₂,u₃,... of theL operator. The proper setting for this problem is theHilbert space${\displaystyle L^2([a,b],w(x)dx)}$ with scalar product$${\displaystyle ⟨f,g⟩={\int }_a^b\overline{f(x)}g(x)w(x)dx.}$$

In this spaceL is defined on sufficiently smooth functions which satisfy the above regular boundary conditions. Moreover,L is aself-adjoint operator:$${\displaystyle ⟨Lf,g⟩=⟨f,Lg⟩.}$$

This can be seen formally by usingintegration by parts twice, where the boundary terms vanish by virtue of the boundary conditions. It then follows that the eigenvalues of a Sturm–Liouville operator are real and that eigenfunctions ofL corresponding to different eigenvalues are orthogonal. However, this operator isunbounded and hence existence of an orthonormal basis of eigenfunctions is not evident. To overcome this problem, one looks at theresolvent$${\displaystyle {\left(L-z\right)}^{-1},z\in \mathbb{R},}$$wherez is not an eigenvalue. Then, computing the resolvent amounts to solving a nonhomogeneous equation, which can be done using thevariation of parameters formula. This shows that the resolvent is anintegral operator with a continuous symmetric kernel (theGreen's function of the problem). As a consequence of theArzelà–Ascoli theorem, this integral operator is compact and existence of a sequence of eigenvaluesα_n which converge to 0 and eigenfunctions which form an orthonormal basis follows from thespectral theorem for compact operators. Finally, note that$${\displaystyle {\left(L-z\right)}^{-1}u=\alpha u,Lu=\left(z+{\alpha }^{-1}\right)u,}$$are equivalent, so we may take${\displaystyle \lambda =z+{\alpha }^{-1}}$ with the same eigenfunctions.

If the interval is unbounded, or if the coefficients have singularities at the boundary points, one callsL singular. In this case, the spectrum no longer consists of eigenvalues alone and can contain a continuous component. There is still an associated eigenfunction expansion (similar to Fourier series versus Fourier transform). This is important inquantum mechanics, since the one-dimensional time-independentSchrödinger equation is a special case of a Sturm–Liouville equation.

Consider a general inhomogeneous second-order linear differential equation$${\displaystyle P(x)y^{″}+Q(x)y^{\prime }+R(x)y=f(x)}$$ for given functions${\displaystyle P(x),Q(x),R(x),f(x)}$. As before, this can be reduced to the Sturm–Liouville form${\displaystyle Ly=f}$: writing a general Sturm–Liouville operator as:$${\displaystyle Lu=\frac{p}{w(x)}{u}^{″}+\frac{p^{\prime }}{w(x)}{u}^{\prime }+\frac{q}{w(x)}u,}$$one solves the system:$${\displaystyle p=Pw,p^{\prime }=Qw,q=Rw.}$$

It suffices to solve the first two equations, which amounts to solving(Pw)′ =Qw, or$${\displaystyle w^{\prime }=\frac{Q-P^{\prime }}{P}w:=\alpha w.}$$

A solution is:

$${\displaystyle w=\exp \left(\int \alpha dx\right),p=P\exp \left(\int \alpha dx\right),q=R\exp \left(\int \alpha dx\right).}$$

Given this transformation, one is left to solve:$${\displaystyle Ly=f.}$$

In general, if initial conditions at some point are specified, for exampley(a) = 0 andy′(a) = 0, a second order differential equation can be solved using ordinary methods and thePicard–Lindelöf theorem ensures that the differential equation has a unique solution in a neighbourhood of the point where the initial conditions have been specified.

But if in place of specifying initial values at asingle point, it is desired to specify values attwo different points (so-called boundary values), e.g.y(a) = 0 andy(b) = 1, the problem turns out to be much more difficult. Notice that by adding a suitable known differentiable function toy, whose values ata andb 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 formy(a) = 0 andy(b) = 0.

Here, the Sturm–Liouville theory comes in play: indeed, a large class of functionsf can be expanded in terms of a series of orthonormal eigenfunctionsu_i of the associated Liouville operator with corresponding eigenvaluesλ_i:$${\displaystyle f(x)=\sum _i{\alpha }_i{u}_i(x),{\alpha }_i\in \mathbb{R}.}$$

Then a solution to the proposed equation is evidently:$${\displaystyle y=\sum _i\frac{{\alpha }_i}{{\lambda }_i}{u}_i.}$$

This solution will be valid only over the open intervala <x <b, and may fail at the boundaries.

Consider the Sturm–Liouville problem:

$${\displaystyle Lu=-\frac{d^{2}u}{d{x}^2}=\lambda u}$$

4

for the unknowns areλ andu(x). For boundary conditions, we take for example:$${\displaystyle u(0)=u(\pi )=0.}$$

Observe that ifk is any integer, then the function$${\displaystyle u_k(x)=\sin kx}$$is a solution with eigenvalueλ =k². We know that the solutions of a Sturm–Liouville problem form anorthogonal basis, and we know fromFourier series that this set of sinusoidal functions is an orthogonal basis. Since orthogonal bases are always maximal (by definition) we conclude that the Sturm–Liouville problem in this case has no other eigenvectors.

Given the preceding, let us now solve the inhomogeneous problem$${\displaystyle Ly=x,x\in (0,\pi )}$$with the same boundary conditions${\displaystyle y(0)=y(\pi )=0}$. In this case, we must expandf(x) =x as a Fourier series. The reader may check, either by integrating∫e^ikxxdx or by consulting a table of Fourier transforms, that we thus obtain$${\displaystyle Ly=\sum _{k=1}^{\mathrm{\infty }}-2\frac{{\left(-1\right)}^k}{k}\sin kx.}$$

This particular Fourier series is troublesome because of its poor convergence properties. It is not cleara priori whether the series converges pointwise. Because of Fourier analysis, since the Fourier coefficients are "square-summable", the Fourier series converges inL² which is all we need for this particular theory to function. We mention for the interested reader that in this case we may rely on a result which says that Fourier series converge at every point of differentiability, and at jump points (the functionx, considered as a periodic function, has a jump at π) converges to the average of the left and right limits (seeconvergence of Fourier series).

Therefore, by using formula (4), we obtain the solution:$${\displaystyle y=\sum _{k=1}^{\mathrm{\infty }}2\frac{(-1{)}^k}{k^3}\sin kx=\frac{1}{6}(x^3-{\pi }^{2}x).}$$

In this case, we could have found the answer usingantidifferentiation, but this is no longer useful in most cases when the differential equation is in many variables.

Certainpartial differential equations can be solved with the help of Sturm–Liouville theory. Suppose we are interested in thevibrational modes of a thin membrane, held in a rectangular frame,0 ≤x ≤L₁,0 ≤y ≤L₂. The equation of motion for the vertical membrane's displacement,W(x,y,t) is given by thewave equation:$${\displaystyle \frac{{\mathrm{\partial }}^{2}W}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}W}{\mathrm{\partial }{y}^2}=\frac{1}{c^2}\frac{{\mathrm{\partial }}^{2}W}{\mathrm{\partial }{t}^2}.}$$

The method ofseparation of variables suggests looking first for solutions of the simple formW =X(x) ×Y(y) ×T(t). For such a functionW the partial differential equation becomes⁠X″/X⁠ +⁠Y″/Y⁠ =⁠1/c²⁠⁠T″/T⁠. Since the three terms of this equation are functions ofx,y,t separately, they must be constants. For example, the first term givesX″ =λX for a constant λ. The boundary conditions ("held in a rectangular frame") areW = 0 whenx = 0,L₁ ory = 0,L₂ and define the simplest possible Sturm–Liouville eigenvalue problems as in the example, yielding the "normal mode solutions" forW with harmonic time dependence,$${\displaystyle W_{mn}(x,y,t)=A_{mn}\sin \left(\frac{m\pi x}{L_1}\right)\sin \left(\frac{n\pi y}{L_2}\right)\cos \left({\omega }_{mn}t\right)}$$wherem andn are non-zerointegers,A_mn are arbitrary constants, and$${\displaystyle {\omega }_{mn}^2=c^2\left(\frac{m^2{\pi }^2}{L_1^2}+\frac{n^2{\pi }^2}{L_2^2}\right).}$$

The functionsW_mn form a basis for theHilbert space of (generalized) solutions of the wave equation; that is, an arbitrary solutionW can be decomposed into a sum of these modes, which vibrate at their individual frequenciesω_mn. This representation may require aconvergent infinite sum.

Consider a linear second-order differential equation in one spatial dimension and first-order in time of the form:$${\displaystyle f(x)\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+g(x)\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+h(x)u=\frac{\mathrm{\partial }u}{\mathrm{\partial }t}+k(t)u,}$$$${\displaystyle u(a,t)=u(b,t)=0,u(x,0)=s(x).}$$

Separating variables, we assume that$${\displaystyle u(x,t)=X(x)T(t).}$$ Then our above partial differential equation may be written as:$${\displaystyle \frac{\hat{L}X(x)}{X(x)}=\frac{\hat{M}T(t)}{T(t)}}$$where$${\displaystyle \hat{L}=f(x)\frac{d^2}{d{x}^2}+g(x)\frac{d}{dx}+h(x),\hat{M}=\frac{d}{dt}+k(t).}$$

Since, by definition,L̂ andX(x) are independent of timet andM̂ andT(t) are independent of positionx, then both sides of the above equation must be equal to a constant:$${\displaystyle \hat{L}X(x)=\lambda X(x),X(a)=X(b)=0,\hat{M}T(t)=\lambda T(t).}$$

The first of these equations must be solved as a Sturm–Liouville problem in terms of the eigenfunctionsX_n(x) and eigenvaluesλ_n. The second of these equations can be analytically solved once the eigenvalues are known.

$${\displaystyle \frac{d}{dt}{T}_n(t)=({\lambda }_n-k(t))T_n(t)}$$$${\displaystyle T_n(t)=a_n\exp \left({\lambda }_{n}t-{\int }_0^{t}k(\tau )d\tau \right)}$$$${\displaystyle u(x,t)=\sum _n{a}_n{X}_n(x)\exp \left({\lambda }_{n}t-{\int }_0^{t}k(\tau )d\tau \right)}$$$${\displaystyle a_n=\frac{⟨X_n(x),s(x)⟩}{⟨X_n(x),X_n(x)⟩}}$$

where$${\displaystyle ⟨y(x),z(x)⟩={\int }_a^{b}y(x)z(x)w(x)dx,}$$$${\displaystyle w(x)=\frac{\exp \left(\int \frac{g(x)}{f(x)}dx\right)}{f(x)}.}$$

The Sturm–Liouville differential equation (1) with boundary conditions may be solved analytically, which can be exact or provide an approximation, by theRayleigh–Ritz method, or by thematrix-variational method of Gerck et al.^[1][2][3]

Numerically, a variety of methods are also available. In difficult cases, one may need to carry out the intermediate calculations to several hundred decimal places of accuracy in order to obtain the eigenvalues correctly to a few decimal places.

• Shooting methods^[4][5]
• Finite difference method
• Spectral parameterpower series method^[6]

Shooting methods proceed by guessing a value ofλ, solving an initial value problem defined by the boundary conditions at one endpoint, say,a, of the interval[a,b], comparing the value this solution takes at the other endpointb with the other desired boundary condition, and finally increasing or decreasingλ as necessary to correct the original value. This strategy is not applicable for locating complex eigenvalues.^{[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: ify is a solution of equation (1) that does not vanish at any point of[a,b], then the function$${\displaystyle y(x){\int }_a^x\frac{dt}{p(t)y(t{)}^2}}$$is a solution of the same equation and is linearly independent fromy. Further, all solutions are linear combinations of these two solutions. In the SPPS algorithm, one must begin with an arbitrary valueλ^∗
₀ (oftenλ^∗
₀ = 0; it does not need to be an eigenvalue) and any solutiony₀ of (1) withλ =λ^∗
₀ which does not vanish on[a,b]. (Discussionbelow of ways to find appropriatey₀ andλ^∗
₀.) Two sequences of functionsX⁽ⁿ⁾(t),X̃⁽ⁿ⁾(t) on[a,b], referred to asiterated integrals, are defined recursively as follows. First whenn = 0, they are taken to be identically equal to 1 on[a,b]. To obtain the next functions they are multiplied alternately by⁠1/py²
₀⁠ andwy²
₀ and integrated, specifically, forn > 0:

5

6

The resulting iterated integrals are now applied as coefficients in the following two power series in λ:$${\displaystyle u_0=y_0\sum _{k=0}^{\mathrm{\infty }}{\left(\lambda -{\lambda }_0^{\ast }\right)}^k{\tilde{X}}^{(2k)},}$$$${\displaystyle u_1=y_0\sum _{k=0}^{\mathrm{\infty }}{\left(\lambda -{\lambda }_0^{\ast }\right)}^k{X}^{(2k+1)}.}$$Then for anyλ (real or complex),u₀ andu₁ are linearly independent solutions of the corresponding equation (1). (The functionsp(x) andq(x) take part in this construction through their influence on the choice ofy₀.)

Next one chooses coefficientsc₀ andc₁ so that the combinationy =c₀u₀ +c₁u₁ satisfies the first boundary condition (2). This is simple to do sinceX⁽ⁿ⁾(a) = 0 andX̃⁽ⁿ⁾(a) = 0, forn > 0. The values ofX⁽ⁿ⁾(b) andX̃⁽ⁿ⁾(b) provide the values ofu₀(b) andu₁(b) and the derivativesu′₀(b) andu′₀(b), so the second boundary condition (3) becomes an equation in a power series in λ. For numerical work one may truncate this series to a finite number of terms, producing a calculable polynomial inλ whose roots are approximations of the sought-after eigenvalues.

Whenλ =λ₀, this reduces to the original construction described above for a solution linearly independent to a given one. The representations (5) and (6) also have theoretical applications in Sturm–Liouville theory.^[6]

The SPPS method can, itself, be used to find a starting solutiony₀. Consider the equation(py′)′ =μqy; i.e.,q,w, andλ are replaced in (1) by 0,−q, andμ respectively. Then the constant function 1 is a nonvanishing solution corresponding to the eigenvalueμ₀ = 0. While there is no guarantee thatu₀ oru₁ will not vanish, the complex functiony₀ =u₀ +iu₁ will never vanish because two linearly-independent solutions of a regular Sturm–Liouville equation cannot vanish simultaneously as a consequence of theSturm separation theorem. This trick gives a solutiony₀ of (1) for the valueλ₀ = 0. In practice if (1) has real coefficients, the solutions based ony₀ will have very small imaginary parts which must be discarded.

• Normal mode
• Oscillation theory
• Self-adjoint
• Variation of parameters
• Spectral theory of ordinary differential equations
• Atkinson–Mingarelli theorem

• Courant, Richard;Hilbert, David (1953). "Part V: Vibration and Eigenvalue Problems".Methods of Mathematical Physics. Vol. 1. Weinheim: Wiley-VCH.
• Gesztesy, Fritz; Nichols, Roger; Zinchenko, Maxim (2024-09-24).Sturm–Liouville Operators, Their Spectral Theory, and Some Applications. Providence, Rhode Island: American Mathematical Society.ISBN 978-1-4704-7666-3.
• "Sturm–Liouville theory",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Hartman, Philip (2002).Ordinary Differential Equations (2 ed.). Philadelphia:SIAM.ISBN 978-0-89871-510-1.
• Polyanin, A. D. & Zaitsev, V. F. (2003).Handbook of Exact Solutions for Ordinary Differential Equations (2 ed.). Boca Raton: Chapman & Hall/CRC Press.ISBN 1-58488-297-2.
• Teschl, Gerald (2012).Ordinary Differential Equations and Dynamical Systems.Providence:American Mathematical Society.ISBN 978-0-8218-8328-0. (Chapter 5)
• Teschl, Gerald (2009).Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators.Providence:American Mathematical Society.ISBN 978-0-8218-4660-5. (see Chapter 9 for singular Sturm–Liouville operators and connections with quantum mechanics)
• Zettl, Anton (2005).Sturm–Liouville Theory.Providence:American Mathematical Society.ISBN 0-8218-3905-5.
• Birkhoff, Garrett (1973).A source book in classical analysis.Cambridge, Massachusetts:Harvard University Press.ISBN 0-674-82245-5. (See Chapter 8, part B, for excerpts from the works of Sturm and Liouville and commentary on them.)
• Kravchenko, Vladislav (2020).Direct and Inverse Sturm-Liouville Problems: A Method of Solution. Cham:Birkhäuser.ISBN 978-3-030-47848-3.

• Ordinary differential equations
• Operator theory
• Spectral theory
• Partial differential equations
• Boundary value problems

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