Math 53 Notes on turm-liouville equations Many problems in physics, engineering, an chemistry fall in a general class of equations of the form w(x)p(x) u ] + (q(x) λ) u = w(x) on an interval a, b], plus certain bounary conitions. We ll often use the notation Lu] = w(x)p(x) u ] + q(x)u w(x) to write the turm-liouville problem as Lu] = λu. The similarity to an eigenvalue problem is not acciental. Note: every textbook seems to efine the L problem in slightly ifferent notation. You shoul be aware of this, an able to translate from one notation to another. elf-ajointness of Lu] With certain bounary conitions to be etermine soon, the L problem is self-ajoint. Define the ajoint L of an operator L implicitly by v, Lu = L v, u. trictly speaking, an operator oesn t have a uniquely etermine ajoint, because the ajoint you fin epens on your choice of inner prouct. In turm-liouville theory we ll always use the weighte L 2 inner prouct f, g = b a w(x) f (x)g(x) where f is the complex conjugate of f. When we speak of the ajoint of L, we always mean with respect to the weighte L 2 inner prouct. b Let s compute (u, v) = v, Lu Lv, u : v (u, v) = w(x) w(x)p(x) u a w(x) b = v wp u b a ] + q(x)u u ] u ] ] w(x)p(x) v + q(x)v w(x) ]] wp v. Now integrate by parts u v (u, v) = wp a u v ] + v wp u ] v b uwp. a The first term is clearly zero. The secon term will be zero if one of the following conitions hols:. The functions u, v, p, an q are perioic on the interval a, b], i.e., u(a) = u(b), u (a) = u (b), an likewise for the others. 2. u an v obey either homogeneous Neumann or homogeneous Dirichlet BCs. Note that these conitions efine a subspace of L 2 : after imposing such restrictions, we re still working in a vector space. Now, if we restrict ourselves to the subspace where the BC conitions are met, we fin (u, v) =. Referring to the efinitions of an L, we conclue that L an L are the same operators with respect to functions in that BC-restricte subspace, using the specifie inner prouct. We ll refer to L as self-ajoint, or Hermitian. trictly speaking, there s a technical istinction between Hermitian an self-ajoint: a self-ajoint operator must be boune, a concept that will be introuce in your functional analysis course. In most physics an engineering books the ifference is glosse over.
2 The spectral theorem elf-ajoint operators have two important properties: all eigenvalues are real, an eigenvectors corresponing to istinct eigenvalues are orthogonal. Theorem. Let L : be a self-ajoint operator. The eigenvalues λ of L, efine by Lu = λu are all real. Furthermore, if u an v are eigenvectors associate with istinct eigenvalues λ an µ, then v, u =. Proof. Choose any two eigenvectors v an u. Compute v, Lu Lv, u, which is known to be zero through the Hermiticity of L. v, Lu Lv, u = v, λu µv, u = λ v, u µ v, u = (λ µ ) v, u. Consier first the case where we choose ientical eigenvectors, u = v. In this case, λ = µ, an we have = (λ λ ) u, u. The inner prouct u, u cannot be zero when u =, which means λ = λ. Therefore λ is a real number. This hols for any eigenvector we choose, so all eigenvalues of L are real. Next, let u an v be ifferent eigenvectors with istinct eigenvalues, λ = µ. Because µ is real, µ = µ so we have = (λ µ) v, u. By hypothesis, λ = µ, so it must be that v, u =. Therefore, v an u are orthogonal with respect to the inner prouct,. Note that this inner prouct must be the same as use in the efinition of the ajoint. 2. Examples olve the BP u + λu = with u() = u() =. This is a L problem with w = p = an q =. The operator Lu] = u. The bounary conitions are homogeneous Dirichlet, so the operator is self-ajoint uner the inner prouct u, v = u(x)v(x). By L theory we expect real eigenvalues an orthogonal eigenvectors. Let s solve the problem. From elementary ODEs, we know that the general solution is u(x) = A cos λx + B sin λx. At this point, however, the proceure iffers somewhat from what you re use to. o far in this course, when face with a solution with unetermine coefficients such as A an B, we ve use the BCs to etermine them. That won t work here. uppose, for example, that λ = 4. Then the BC u() = gives us = A, an the BC u() = gives us B sin 2 =. The sine of 2 isn t zero, so the only possibility is B =. Our solution is therefore u(x) =, which is a rather boring solution! Can we fin an interesting solution? Yes, but only for certain values of λ. Leaving λ unetermine for the moment, look at the BCs: u() = A = u() = B sin λ =. One solution is obviously B =, but to get an interesting solution we nee to have B =. The only remaining possibility is sin λ =, which will happen iff λ = n 2 π 2, n Z. The case n = is also boring, because sin x = x. The sign of n rops out because n is square. Therefore, we can restrict ourselves to n N. 2
These special choices of λ for which solutions are interesting, i.e., not ientically zero, are calle the eigenvalues of the problem. The corresponing solutions are calle the eigenfunctions. The eigenvalues λ n = n 2 π 2 are clearly real, as expecte. What about orthogonality? Compute the inner prouct When m = n we have which is zero. When m = n, we have u m, u n = sin mπx sin nπx. u m, u n = cos (m n) πx cos (m + n) πx] 2 = ] 2 (m n)π sin(m n)πx + sin(m + n)πx (m + n)π Therefore, u m, u m = sin 2 mπx = 2. u m, u n = δ mn 2. The eigenfunctions form an orthogonal set, as expecte. 3 turm-liouville theory in multiple spatial imensions Were L theory limite to D problems, it wouln t be very useful for real-worl problems. Remarkably, the generalization to multiple spatial imensions is very simple. 3. Review of vector calculus In multiple imensions, there are several interesting combinations of partial erivatives. The graient operator acts on a real-value (or generally speaking, scalar-value) function to prouce a vector-value function. In Cartesian coorinates, it takes the form u = i u x + j u y + k u z. Geometrically, the graient is a vector oriente along the irection of greatest increase of u, having magnitue equal to the rate of change of u with istance along that irection. Another operator of interest is the ivergence,, which acts on a vector-value function an returns a scalarvalue function. In Cartesian coorinates, the ivergence is Notice that if you imagine the symbol to be a vector F = F x x + F y y + F z z. = i x + j y + k z, then F using the usual vector ot prouct gives you the formula for the ivergence in Cartesian coorinates. The ot-prouct notation can help you remember (a) that the ivergence returns a scalar, just like the ot prouct, an (b) the form of the operator in Cartesian coorinates. Don t take this ot-prouct notation too seriously: it gives you the correct formula only in Cartesian coorinates. 3
What if I compose operators? The graient acts on a scalar an prouces a vector, an the ivergence acts on a vector an prouces a scalar, so both of the following compositions make sense: ( u) ( F). The secon compose operator will arise occasionally. The first, however, is ubiquitous; we ll concentrate on it. It s common enough to have it s own name: the Laplacian operator. It also gets its own symbols: u an 2 u. The first symbol is more common among mathematicians, the secon more common among engineers an physicists. Unfortunately, in speaking I ll switch between them more-or-less at ranom. Using the efinitions above, we fin that in Cartesian coorinates u = 2 u x 2 + 2 u y 2 + 2 u z 2 You can see why 2 u is popular notation: if you think of 2 = ( ) you get the right formula... again, only in Cartesian coorinates. Now, if I have a scalar-value function v(x), we can also form the operator v u]. The quantity insie the brackets is a vector, so the ivergence operation makes sense. This raises the question: can we simplify v u] using something like a prouct rule. Yes: a simple calculation shows that where u is the Laplacian. v u] = v u + v u What about integration? There s a generalization of the funamental theorem of calculus to multiple imensions, calle the generalize tokes theorem (see, e.g., Ruin or pivak). For present purposes, a special case will suffice: the Gauss ivergence theorem, which relates an integral over a close surface to an integral v over the volume enclose by the surface: F = F n. With the ivergence theorem an the prouct rule together, it follows that or v u + v u] = v u = vn u vn u, v u. This might look familiar: it s a form of integration by parts! With this, we can compute v u u v] = vn u un v]. These last two equations are calle Green s first an secon ientities, respectively. Finally, the prouct rule vc u] = c v u + v c u] lets us erive a generalization of Green s first ientity, v c u] = c u v + cvn u, from which we can generalize Green s secon ientity, v c u] u c v]] = cn v u u v]. 4
3.2 elf-ajointness an the spectral theorem in multiple spatial imensions The generalize Green s ientities are all that s neee to exten turm-liouville theory to multiple spatial imensions. Define a ifferential operator an the L 2 inner prouct Lu] = w(x)p(x) u] + q(x)u w(x) v, u = w(x)v (x)u(x). Form v, Lu Lv, u an apply the generalize Green s secon ientity to fin v, Lu Lv, u = cn v u u v ]. This will be zero, an thus L Hermitian, if u an v satisfy homogeneous Neumann or homogeneous Dirichlet bounary conitions, where homogeneous Neumann bounary conitions in multiple imensions mean We ll often write this as n u = or u n =. n u =. The spectral theorem follows immeiately from Hermiticity. 5