Sturm-Liouville theory
Encyclopedia
In mathematics
and its applications, a classical Sturm–Liouville equation, named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville
(1809–1882), is a real second-order linear differential equation
of the form
where y is a function of the free variable x. Here the functions p(x) > 0, q(x), and w(x) > 0 are specified at the outset. In the simplest of cases all coefficients are continuous on the finite closed interval [a,b], and p has continuous derivative. In this simplest of all cases, this function "y" is called a solution if it is continuously differentiable on (a,b) and satisfies the equation at every point in (a,b). In addition, the unknown function y is typically required to satisfy some boundary conditions at a and b. The function w(x), which is sometimes called r(x), is called the "weight" or "density" function.
The value of λ is not specified in the equation; finding the values of λ for which there exists a non-trivial solution of satisfying the boundary conditions is part of the problem called the Sturm–Liouville (S-L) problem.
Such values of λ when they exist are called the eigenvalues of the boundary value problem defined by and the prescribed set of boundary conditions. The corresponding solutions (for such a λ) are the eigenfunction
s of this problem. Under normal assumptions on the coefficient functions p(x), q(x), and w(x) above, they induce a Hermitian differential operator
in some function space
defined by boundary conditions
. The resulting theory of the existence and asymptotic behavior of the eigenvalues, the corresponding qualitative theory of the eigenfunctions and their completeness in a suitable function space
became known as Sturm–Liouville theory. This theory is important in applied mathematics, where S–L problems occur very commonly, particularly when dealing with linear partial differential equation
s that are separable
.
with separated boundary conditions of the form
the main tenet of Sturm–Liouville theory states that:
Note that, unless p(x) is continuously differentiable and q(x), w(x) are continuous, the equation has to be understood in a weak sense
.
s can be recast in the form on the left-hand side of by multiplying both sides of the equation by an appropriate integrating factor
(although the same is not true of second-order partial differential equation
s, or if y is a vector.)
can be written in Sturm–Liouville form as
The Legendre equation,
can easily be put into Sturm–Liouville form, since D(1 − x2) = −2x, so, the Legendre equation is equivalent to
Less simple is such a differential equation as
Divide throughout by x3:
Multiplying throughout by an integrating factor
of
gives
which can be easily put into Sturm–Liouville form since
so the differential equation is equivalent to
In general, given a differential equation
dividing by P(x), multiplying through by the integrating factor
and then collecting gives the Sturm–Liouville form.
can be viewed as a linear operator mapping a function u to another function Lu. We may study this linear operator in the context of functional analysis
. In fact, equation can be written as
This is precisely the eigenvalue problem; that is, we are trying to find the eigenvalues λ1, λ2, λ3, ... and the corresponding eigenvectors u1, u2, u3, ... of the L operator. The proper setting for this problem is the Hilbert space
L2([a, b],w(x) dx) with
scalar product
In this space L is defined on sufficiently smooth functions which satisfy the above boundary condition
s. Moreover, L gives rise to a self-adjoint
operator.
This can be seen formally by using integration 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 of L corresponding to different eigenvalues are orthogonal. However, this operator is unbounded and hence existence of an orthonormal basis of eigenfunctions is not evident. To overcome this problem one looks at the resolvent
where z is chosen to be some real number which is not an eigenvalue. Then, computing the resolvent amounts to solving the inhomogeneous 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
. Finally, note that
is equivalent to 
.
If the interval is unbounded, or if the coefficients have singularities at the boundary points, one calls L singular. In this case the spectrum does no longer consist 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 in quantum mechanics
, since the one-dimensional Schrödinger equation
is a special case of a S–L equation.
where the unknowns are λ and u(x). As above, we must add boundary conditions, we take for example
Observe that if k is any integer, then the function
is a solution with eigenvalue λ = −k2. We know that the solutions of a S–L problem form an orthogonal basis, and we know from Fourier series that this set of sinusoidal functions is an orthogonal basis. Since orthogonal bases are always maximal (by definition) we conclude that the S–L problem in this case has no other eigenvectors.
Given the preceding, let us now solve the inhomogeneous problem
with the same boundary conditions. In this case, we must write f(x) = x in a Fourier series. The reader may check, either by integrating ∫exp(ikx)x dx or by consulting a table of Fourier transforms, that we thus obtain
This particular Fourier series is troublesome because of its poor convergence properties. It is not clear a priori whether the series converges pointwise. Because of Fourier analysis, since the Fourier coefficients are "square-summable
", the Fourier series converges in L2 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's series converges at every point of differentiability, and at jump points (the function x, considered as a periodic function, has a jump at π) converges to the average of the left and right limits (see convergence of Fourier series
).
Therefore, by using formula , we obtain that the solution is
In this case, we could have found the answer using antidifferentiation. This technique yields u = (x3 − π2x)/6, whose Fourier series agrees with the solution we found. The antidifferentiation technique is no longer useful in most cases when the differential equation is in many variables.
s can be solved with the help of S-L theory. Suppose we are interested in the modes of vibration 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 the wave equation
:
The method of separation of variables
suggests looking first for solutions of the simple form W = X(x) × Y(y) × T(t). For such a function W the partial differential equation becomes X"/X + Y"/Y = (1/c2)T"/T. Since the
three terms of this equation are functions of x,y,t separately, they must be constants. For example, the first term gives X"=
X for a constant 
. The boundary conditions ("held in a rectangular frame") are W=0 when x=0, L1 or y = 0, L2 and define the simplest possible S-L eigenvalue problems as in the example
, yielding the "normal mode solutions" for W with harmonic time dependence,
where m and n are non-zero integer
s, Amn are arbitrary constants, and
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
. This representation may require a convergent infinite sum.
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.
1. Shooting methods. These methods proceed by guessing a value of λ, 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 endpoint b with the other desired boundary condition, and finally increasing or decreasing λ as necessary to correct the original value. This strategy is not applicable for locating complex eigenvalues.
2. Finite difference method
.
3. The Spectral Parameter Power Series (SPPS) method makes use of a generalization of the following fact about second order ordinary differential equations: if y is a solution which does not vanish at any point of [a,b], then the function
is a solution of the same equation and is linearly independent from y. Further, all solutions are linear combinations of these two solutions. In the SPPS algorithm, one must begin with an arbitrary value λ0* (often λ0*=0; it does not need to be an eigenvalue) and any solution y0 of (1) with λ=λ0* which does not vanish on [a,b]. (Discussion below of ways to find appropriate y0 and λ0*.) Two sequences of functions X(n)(t), X~(n)(t) on [a,b], referred to as iterated integrals, are defined recursively as follows. First when n=0, they are taken to be identically equal to 1 on [a,b]. To obtain the next functions they are multiplied alternately by 1/(py02) and wy02 and integrated, specifically
when n>0. The resulting iterated integrals are now applied as coefficients in the following two power series in λ:
Then for any λ (real or complex), u0 and u1 are linearly independent solutions of the corresponding equation (1). (The functions p(x) and q(x) take part in this construction through their influence on the choice of y0.)
Next one chooses coefficients c0, c1 so that the combination y=c0u0 + c1u1 satisfies the first boundary condition (2). This is simple to do since X(n)(a)=0 and X~(n)(a)=0, for n>0. The values of X(n)(b) and X~(n)(b) provide the values of u0(b) and u1(b) and the derivatives u0'(b) and u1'(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 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. The representations , also have theoretical applications in Sturm–Liouville theory.
. This trick gives a solution y0 of (1) for the value λ0=0. In practice if (1) has real coefficients, the solutions based on y0 will have very small imaginary parts which must be discarded.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and its applications, a classical Sturm–Liouville equation, named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville
Joseph Liouville
- Life and work :Liouville graduated from the École Polytechnique in 1827. After some years as an assistant at various institutions including the Ecole Centrale Paris, he was appointed as professor at the École Polytechnique in 1838...
(1809–1882), is a real second-order linear differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
of the form
where y is a function of the free variable x. Here the functions p(x) > 0, q(x), and w(x) > 0 are specified at the outset. In the simplest of cases all coefficients are continuous on the finite closed interval [a,b], and p has continuous derivative. In this simplest of all cases, this function "y" is called a solution if it is continuously differentiable on (a,b) and satisfies the equation at every point in (a,b). In addition, the unknown function y is typically required to satisfy some boundary conditions at a and b. The function w(x), which is sometimes called r(x), is called the "weight" or "density" function.
The value of λ is not specified in the equation; finding the values of λ for which there exists a non-trivial solution of satisfying the boundary conditions is part of the problem called the Sturm–Liouville (S-L) problem.
Such values of λ when they exist are called the eigenvalues of the boundary value problem defined by and the prescribed set of boundary conditions. The corresponding solutions (for such a λ) are the eigenfunction
Eigenfunction
In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
s of this problem. Under normal assumptions on the coefficient functions p(x), q(x), and w(x) above, they induce a Hermitian differential operator
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
in some function space
Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...
defined by boundary conditions
Boundary value problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions...
. The resulting theory of the existence and asymptotic behavior of the eigenvalues, the corresponding qualitative theory of the eigenfunctions and their completeness in a suitable function space
Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...
became known as Sturm–Liouville theory. This theory is important in applied mathematics, where S–L problems occur very commonly, particularly when dealing with linear partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s that are separable
Separation of variables
In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation....
.
Sturm–Liouville theory
Under the assumptions that the S–L problem is regular, that is, p(x), w(x) > 0, and p(x), p(x), q(x), and w(x) are continuous functions over the finite interval [a, b],with separated boundary conditions of the form
the main tenet of Sturm–Liouville theory states that:
- The eigenvalues λ1, λ2, λ3, ... of the regular Sturm–Liouville problem -- are real and can be ordered such that
- Corresponding to each eigenvalue λn is a unique (up to a normalization constant) eigenfunction yn(x) which has exactly n − 1 zeros in (a, b). The eigenfunction yn(x) is called the n-th fundamental solution satisfying the regular Sturm–Liouville problem --.
- The normalized eigenfunctions form an orthonormal basisOrthonormal basisIn mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
- in the Hilbert spaceHilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
L2([a, b],w(x) dx). Here δmn is a Kronecker delta.
Note that, unless p(x) is continuously differentiable and q(x), w(x) are continuous, the equation has to be understood in a weak sense
Weak solution
In mathematics, a weak solution to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. There are many different definitions of weak solution, appropriate for...
.
Sturm–Liouville form
The differential equation is said to be in Sturm–Liouville form or self-adjoint form. All second-order linear ordinary differential equationOrdinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s can be recast in the form on the left-hand side of by multiplying both sides of the equation by an appropriate integrating factor
Integrating factor
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus, in this case often multiplying through by an...
(although the same is not true of second-order partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s, or if y is a vector.)
Examples
The Bessel equation:
can be written in Sturm–Liouville form as
The Legendre equation,
can easily be put into Sturm–Liouville form, since D(1 − x2) = −2x, so, the Legendre equation is equivalent to
Less simple is such a differential equation as
Divide throughout by x3:
Multiplying throughout by an integrating factor
Integrating factor
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus, in this case often multiplying through by an...
of
gives
which can be easily put into Sturm–Liouville form since
so the differential equation is equivalent to
In general, given a differential equation
dividing by P(x), multiplying through by the integrating factor
and then collecting gives the Sturm–Liouville form.
Sturm–Liouville equations as self-adjoint differential operators
The map
can be viewed as a linear operator mapping a function u to another function Lu. We may study this linear operator in the context of functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
. In fact, equation can be written as
This is precisely the eigenvalue problem; that is, we are trying to find the eigenvalues λ1, λ2, λ3, ... and the corresponding eigenvectors u1, u2, u3, ... of the L operator. The proper setting for this problem is the Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
L2([a, b],w(x) dx) with
scalar product
In this space L is defined on sufficiently smooth functions which satisfy the above boundary condition
Boundary value problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions...
s. Moreover, L gives rise to a self-adjoint
Self-adjoint
In mathematics, an element x of a star-algebra is self-adjoint if x^*=x.A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation...
operator.
This can be seen formally by using integration by parts
Integration by parts
In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...
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 of L corresponding to different eigenvalues are orthogonal. However, this operator is unbounded and hence existence of an orthonormal basis of eigenfunctions is not evident. To overcome this problem one looks at the resolvent
where z is chosen to be some real number which is not an eigenvalue. Then, computing the resolvent amounts to solving the inhomogeneous 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
Green's function
In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...
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
Compact operator on Hilbert space
In functional analysis, compact operators on Hilbert spaces are a direct extension of matrices: in the Hilbert spaces, they are precisely the closure of finite-rank operators in the uniform operator topology. As such, results from matrix theory can sometimes be extended to compact operators using...
. Finally, note that
is equivalent to .If the interval is unbounded, or if the coefficients have singularities at the boundary points, one calls L singular. In this case the spectrum does no longer consist 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 in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, since the one-dimensional Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
is a special case of a S–L equation.
Example
We wish to find a function u(x) which solves the following Sturm–Liouville problem:where the unknowns are λ and u(x). As above, we must add boundary conditions, we take for example
Observe that if k is any integer, then the function
is a solution with eigenvalue λ = −k2. We know that the solutions of a S–L problem form an orthogonal basis, and we know from Fourier series that this set of sinusoidal functions is an orthogonal basis. Since orthogonal bases are always maximal (by definition) we conclude that the S–L problem in this case has no other eigenvectors.
Given the preceding, let us now solve the inhomogeneous problem
with the same boundary conditions. In this case, we must write f(x) = x in a Fourier series. The reader may check, either by integrating ∫exp(ikx)x dx or by consulting a table of Fourier transforms, that we thus obtain
This particular Fourier series is troublesome because of its poor convergence properties. It is not clear a priori whether the series converges pointwise. Because of Fourier analysis, since the Fourier coefficients are "square-summable
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
", the Fourier series converges in L2 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's series converges at every point of differentiability, and at jump points (the function x, considered as a periodic function, has a jump at π) converges to the average of the left and right limits (see convergence of Fourier series
Convergence of Fourier series
In mathematics, the question of whether the Fourier series of a periodic function converges to the given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics...
).
Therefore, by using formula , we obtain that the solution is
In this case, we could have found the answer using antidifferentiation. This technique yields u = (x3 − π2x)/6, whose Fourier series agrees with the solution we found. The antidifferentiation technique is no longer useful in most cases when the differential equation is in many variables.
Application to normal modes
Certain partial differential equationPartial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s can be solved with the help of S-L theory. Suppose we are interested in the modes of vibration 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 the wave equation
Wave equation
The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...
:
The method of separation of variables
Separation of variables
In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation....
suggests looking first for solutions of the simple form W = X(x) × Y(y) × T(t). For such a function W the partial differential equation becomes X"/X + Y"/Y = (1/c2)T"/T. Since the
three terms of this equation are functions of x,y,t separately, they must be constants. For example, the first term gives X"=
X for a constant . The boundary conditions ("held in a rectangular frame") are W=0 when x=0, L1 or y = 0, L2 and define the simplest possible S-L eigenvalue problems as in the exampleExample
An example is a demonstration with the aim of informing others of how a task should be performed. It is often abbreviated to e.g.Example may also refer to:* Example , a British musician...
, yielding the "normal mode solutions" for W with harmonic time dependence,
where m and n are non-zero integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s, Amn are arbitrary constants, and
The functions Wmn form a basis for the Hilbert Space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
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
. This representation may require a convergent infinite sum.Representation of Solutions and Numerical Calculation
The Sturm-Liouville differential equation (1) with boundary conditions may be solved in practice by a variety of numerical methods. In difficult cases, onemay 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.
1. Shooting methods. These methods proceed by guessing a value of λ, 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 endpoint b with the other desired boundary condition, and finally increasing or decreasing λ as necessary to correct the original value. This strategy is not applicable for locating complex eigenvalues.
2. Finite difference method
Finite difference method
In mathematics, finite-difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.- Derivation from Taylor's polynomial :...
.
3. The Spectral Parameter Power Series (SPPS) method makes use of a generalization of the following fact about second order ordinary differential equations: if y is a solution which does not vanish at any point of [a,b], then the function
is a solution of the same equation and is linearly independent from y. Further, all solutions are linear combinations of these two solutions. In the SPPS algorithm, one must begin with an arbitrary value λ0* (often λ0*=0; it does not need to be an eigenvalue) and any solution y0 of (1) with λ=λ0* which does not vanish on [a,b]. (Discussion below of ways to find appropriate y0 and λ0*.) Two sequences of functions X(n)(t), X~(n)(t) on [a,b], referred to as iterated integrals, are defined recursively as follows. First when n=0, they are taken to be identically equal to 1 on [a,b]. To obtain the next functions they are multiplied alternately by 1/(py02) and wy02 and integrated, specifically
when n>0. The resulting iterated integrals are now applied as coefficients in the following two power series in λ:
-
and
.
Then for any λ (real or complex), u0 and u1 are linearly independent solutions of the corresponding equation (1). (The functions p(x) and q(x) take part in this construction through their influence on the choice of y0.)
Next one chooses coefficients c0, c1 so that the combination y=c0u0 + c1u1 satisfies the first boundary condition (2). This is simple to do since X(n)(a)=0 and X~(n)(a)=0, for n>0. The values of X(n)(b) and X~(n)(b) provide the values of u0(b) and u1(b) and the derivatives u0'(b) and u1'(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 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. The representations , also have theoretical applications in Sturm–Liouville theory.
Construction of a nonvanishing solution
The SPPS method can be itself used to find a starting solution y0. Consider the equation (p'y)'=μqy; i.e., q, w, and λ are replaced in (1) by 0, -q, and μ respectively. Then the constant function 1 is a nonvanishing solution corresponding to the eigenvalue μ0=0. While there is no guarantee that u0 or u1 will not vanish, the complex function y0=u0+iu1 will never vanish because two linearly independent solutions of a regular S-L equation cannot vanish simultaneously as a consequence of the Sturm separation theoremSturm separation theorem
In mathematics, in the field of ordinary differential equations, Sturm separation theorem, named after Jacques Charles François Sturm, describes the location of roots of homogeneous second order linear differential equations...
. This trick gives a solution y0 of (1) for the value λ0=0. In practice if (1) has real coefficients, the solutions based on y0 will have very small imaginary parts which must be discarded.
See also
- Normal modeNormal modeA normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies...
- Oscillation theory
- Self-adjointSelf-adjointIn mathematics, an element x of a star-algebra is self-adjoint if x^*=x.A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation...
- Variation of Parameters
- Spectral theory of ordinary differential equationsSpectral theory of ordinary differential equationsIn mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation...
Further reading
- P. Hartman, Ordinary Differential Equations, SIAM, Philadelphia, 2002 (2nd edition). ISBN 978-0-898715-10-1
- A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2003 (2nd edition). ISBN 1-58488-297-2
- G. TeschlGerald TeschlGerald Teschl is an Austrian mathematical physicist and Professor of Mathematics.He is working in the area of mathematical physics; in particular direct and inverse spectral theory with application to completely integrable partial differential equations .-Career:After studying physics at the Graz...
, Ordinary Differential Equations and Dynamical Systems, http://www.mat.univie.ac.at/~gerald/ftp/book-ode/ (Chapter 5) - G. Teschl, Mathematical Methods in Quantum Mechanics and Applications to Schrödinger Operators, http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/ (see Chapter 9 for singular S-L operators and connections with quantum mechanics)
- A. Zettl, Sturm–Liouville Theory, American Mathematical Society, 2005. ISBN 0-8218-3905-5.