Sturm-Liouville Matrix Differential Systems with Singular Leading Coefficients

Transcription

1 Sturm-Liouville Matrix Differential Systems with Singular Leading Coefficients Iva Drimalová, Werner Kratz und Roman Simon Hilscher Preprint Series: Fakultät für Mathematik und Wirtschaftswissenschaften UNIVERSIÄ ULM

2 SURM LIOUVILLE MARIX DIFFERENIAL SYSEMS WIH SINGULAR LEADING COEFFICIEN IVA DŘÍMALOVÁ, WERNER KRAZ, AND ROMAN ŠIMON HILSCHER Abstract. In this paper we study a general even order symmetric Sturm Liouville matrix differential equation, whose leading coefficient may be singular on the whole interval under consideration. Such an equation is new in the current literature, as it is equivalent with a system of Sturm Liouville equations with different orders. We identify the so-called normal form of this equation, which allows to transform this equation into a standard controllable linear Hamiltonian system. Based on this new transformation we prove that the associated eigenvalue problem with Dirichlet boundary conditions possesses all the traditional spectral properties, such as the equality of the geometric and algebraic multiplicities of the eigenvalues, orthogonality of the eigenfunctions, the oscillation theorem and Rayleigh s principle, and the Fourier expansion theorem. We also discuss sufficient conditions, which allow to reduce a general even order symmetric Sturm Liouville matrix differential equation into the normal form. hroughout the paper we provide several examples, which illustrate our new theory. 1. Introduction Let m, n N be fixed dimensions. In this paper we study the existence theory and spectral properties of the 2n-th order matrix Sturm Liouville differential equation n L[y]t := 1 j R j t y j j = 0 SL j=0 on a given compact interval [a, b] R, where y : [a, b] R m. We shall always assume the following standing hypothesis. A1 he coefficients R j t, j {0, 1,..., n}, are real, symmetric, and piecewise continuous m m matrix-valued functions on [a, b]. Remark 1.1. In the present setting the piecewise continuity of a function ft on [a, b], i.e., f C p on [a, b], means that there is a finite partition {a = t 0 < t 1 < < t N = b} such that ft is continuous on t i 1, t i for all i {1,..., N} and the one-sided limits ft ± i of ft at the points t i exist finite. In this respect all conditions and assertions on ft regarding such a piecewise continuous function are interpreted to hold for ft + for all t [a, b and also for ft for all t a, b]. In particular, if ft 0 on [a, b], then ft α for all t [a, b] for some α > 0. In the classical theory equation SL is studied under the assumption R n t is invertible on [a, b]. 1.1 Hence, by Remark 1.1, this condition yields that det R n t α for all t [a, b] for some α > 0, and therefore Rn 1 t is also piecewise continuous on [a, b]. Condition 1.1 implies that SL can be transformed into the linear Hamiltonian system x = At x + Bt u, u = Ct x A t u, H Key words and phrases. Sturm Liouville differential equation; Linear Hamiltonian system; Generalized quasiderivative; Oscillation theory; Spectral theory; Quadratic functional; Rayleigh principle. his research was supported by the Czech Science Foundation under grant GA S and by grant MUNI/A/1154/2015 of Masaryk University. 1

3 2 IVA DŘÍMALOVÁ, WERNER KRAZ, AND ROMAN ŠIMON HILSCHER where At, Bt, Ct are piecewise continuous d d matrices on [a, b] with d = mn, such that Bt and Ct are symmetric, see [2, Section 2.7] or [6, Section 8.4] or [9] and Lemma 2.1. Hence, the known theory yields under 1.1 the results for equation SL. In some situations, e.g. for the second order scalar Sturm Liouville equations m = n = 1, the leading coefficient R 1 t is allowed to be singular at the endpoints of the interval [a, b] or sometimes at some interior point the so-called singular Sturm Liouville equations according to [1, 13, 14]. his happens for the Legendre equation, Bessel equation, and many others however, these are not covered for this paper, see [4]. In this paper we consider equation SL with the leading coefficient singular throughout the interval [a, b], so that equation SL corresponds to a system of Sturm Liouville equations with different orders, see system in Remark 2.5. Such systems do not have a counterpart in the current literature and hence, this paper can be regarded as a pioneer work in this direction. In this paper we improve the method of K. Setzer in [10], who considered the case of m = n = 2 or m = n = 3, and extend it to arbitrary matrix coefficients in SL and to an arbitrary order i.e., to m, n N, see the open problem in [10, Remark 3.2.3]. We identify a special block structure of the coefficients R 1 t,..., R n t, which allows one to transform equation SL into the linear Hamiltonian system H with d mn heorem 2.7. More precisely, we show that the full regularity in the leading coefficient R n t in 1.1 can be distributed into some or all coefficients R n t,..., R 1 t in such a way that the sum of the ranks of R n t,..., R 1 t is at least m. In this case we refer equation SL to be in the so-called normal form Definition 2.2. ypically, when the regularity assumption 1.1 is distributed only between the first two leading coefficients R n t and R n 1 t, we consider these coefficients with the block structure rn t 0 sn 1 t p n 1 t R n t =, R n 1 t =, q n 1 t r n 1 t where r n t and r n 1 t are symmetric and invertible matrices such that for all t [a, b] rank r n t + rank r n 1 t = m, s n 1 t is symmetric, q n 1 t = p n 1t, 1.3 Knowing that under these more general assumptions equation SL or the corresponding system of Sturm Liouville equations with different orders can be transformed into a controllable linear Hamiltonian system H, we can apply the existence and eigenvalue theory of these systems known in [6, 8, 9, 12] to obtain the corresponding results for equation SL. his gives us the global existence and uniqueness of solutions heorem 2.13, equality of geometric and algebraic multiplicities of eigenvalues, orthogonality of the eigenfunctions, oscillation theorem, Rayleigh s principle, and Fourier s expansion in terms of the eigenfunctions heorems 3.3 and 3.6. he general method of this paper also yields a new and interesting connection between classical Sturm Liouville differential equations with 1.1 for different values of n, see Remarks 2.5 and 2.6. For example, between the known Sturm Liouville equations SL of order two n = 1 and four n = 2 there are m 1 equations in the normal form, which can be transformed into the system H by our algorithm. Finally, in the last part of this paper heorem 4.1 we derive conditions, which guarantee that a general Sturm Liouville differential equation can be reduced to the above mentioned normal form. hroughout the paper we also present several examples, which illustrate our new approach. 2. Normal form of singular Sturm Liouville systems In this section we define the so-called normal form of equation SL and prove that equation SL in the normal form, with possibly singular leading coefficient, is equivalent with the linear Hamiltonian system H. his generalizes the standard regular case, i.e., equation SL with 1.1, which we also recall for comparison with the more general case. he following well-known result can be found in [2, Section 2.7].

4 SURM LIOUVILLE MARIX DIFFERENIAL SYSEMS 3 Lemma 2.1. Assume that 1.1 holds. hen the transformation where x i := y i 1, u i := x = x 1,..., x n, u = u 1,..., u n, n 1 j i R j t y j j i, i {1,..., n}, 2.1 j=i transforms the 2n-th order differential equation SL into an equivalent linear Hamiltonian system H, whose d d coefficients satisfy d = mn and 0 I I... 0 At = , I Bt = diag{0,..., 0, R 1 n t}, Ct = diag{r 0 t, R 1 t,..., R n 1 t}. 2.3 he symbol diag in 2.3 denotes a block diagonal matrix with the indicated entries. Note that the coefficients At, Bt, Ct in 2.2 and 2.3 have the same block structure, namely every single block is an m m matrix. Note also that the inverse of the leading coefficient R n t occurs directly in 2.3. Hence, if R n t is singular for some t [a, b], then the result in Lemma 2.1 is not applicable. he expressions in 2.1 are called the quasiderivatives y [i] of y. More precisely, with the notation 2.1 we have y [j] := x j+1, y [n+j] := u n j, j {0,..., n 1}, 2.4 with y [2n] = L[y] being the differential expression defining equation SL. We now introduce a normal form of equation SL, in which the leading coefficient is possibly singular. As it is common, we use the notation M l for the leading principal submatrix of order l of the matrix M. Moreover, we use the notation I l and 0 l for the l l identity matrix and the l l zero matrix, respectively. his will make the block structure of the occurring matrices more understandable. Definition 2.2. Equation SL is called to be in normal form, if there exist nonnegative integers k 1,..., k n such that n k j = m. 2.5 and such that the m m coefficients R 1 t,..., R n t satisfy Rn t l1 0 R n t = 0 0 m l1 Rj t ln j+1 0 R j t = 0 0 m ln j+1 j=1, R n t l1 =: r n t, 2.6, R j t ln j+1 =: sj t p j t, 2.7 q j t r j t for all j {1,..., n 1}, where l i is the partial sum of the k j s, i.e., i l i := k j, i {1,..., n}, 2.8 j=1 and where, r j t is a symmetric and invertible k n j+1 k n j+1 matrix for j {1,..., n}. In addition, according to assumption A1, the matrix s j t is symmetric and q j t = p j t for all t [a, b]. We call the n-tuple k 1,..., k n as the type of the normal form.

5 4 IVA DŘÍMALOVÁ, WERNER KRAZ, AND ROMAN ŠIMON HILSCHER Remark 2.3. A regular Sturm Liouville equation, i.e., SL with 1.1, is always in the normal form with the type m, 0,..., 0. In this paper we allow the equation SL to have an arbitrary type k 1,..., k n satisfying 2.5 and Observe that by A1 the matrices s j t, p j t, q j t, and r j t are piecewise continuous on [a, b], and so is r 1 j t by the invertibility of r j t and Remark 1.1. Finally, we note that by 2.5 and 2.8 we have l 1 = k 1 and l n = m. Next we introduce the generalized quasiderivatives of a function y : [a, b] R m. Definition 2.4. Let be given nonnegative integers k j and l j for j {1,..., n} as in Definition 2.2. Let y : [a, b] R m be divided into components y = y 1,..., y n R m, where y j R k j, j {1,..., n}. 2.9 he generalized quasiderivatives y [j] of y are defined by y j y [j] := 1. R l n j, j {0,..., n 1}, y [n] := r n t y n 1 R k 1, 2.10 y j n j and y [n+j] y [n+j 1] := + R n j t l 0 j+1 y [n j 1] R l j+1, j {1,..., n 1} In contrast with the ordinary quasiderivatives of y in 2.4, the generalized quasiderivatives y [j] in 2.10 and 2.11 have smaller dimensions. Note also that for j {1,..., n 1} y [j 1] y [j] =, y [n+j] y [n+j 1] sn j t p n j t y [n j] = q n j t r n j t y j n j+1 y n j j+1 Moreover, these generalized quasiderivatives depend not only on y, but also on the type and on the coefficients. More precisely, we have y [j] = y [j] y; k 1,..., k n j for j {0,..., n 1}, and y [n+j] = y [n+j] y; k 1,..., k j+1 ; R n t,..., R n j t for j {0,..., n 1}. If the equation SL is in the normal form, then the leading coefficient R n t is allowed to be singular the case of n 2 and k 1 < m. For n 2 the submatrix r j t represents the regular part of the coefficient R j t, which is used in order to get the total regularity of order m. Conditions 2.5 and 2.8 determine how the full regularity assumption 1.1 for the leading coefficient R n t is distributed between the other coefficients R n 1 t,..., R 1 t. Note that in 2.5 and 2.8 the values k j = 0 are allowed. In this case the assumptions in 2.6 and 2.7 have a special interpretation, where the corresponding blocks of dimension zero are missing, see Remark 2.9 and Example Note also that the condition 1.1 holds whenever k 1 = m. his is the case, for example, for n = 1 and the second order equation R 1 t y +R 0 t y = 0 with invertible R 1 t on [a, b]. Remark 2.5. For illustration of the new types of equations, which arise between the standard Sturm Liouville equations with invertible leading coefficients, we consider the situation with n = 2 and an arbitrary m N. As in 1.2 and 1.3, we write SL in the form r2 t 0 y s1 t p 1 t y + R 0 t y = 0, q 1 t r 1 t where rank r 2 t = k 1, rank r 1 t = k 2, and k 1 + k 2 = m. If we split y = y 1, y 2, where y 1 : [a, b] R k 1 and y 2 : [a, b] R k 2, then equation 2.13 is equivalent with the system

6 SURM LIOUVILLE MARIX DIFFERENIAL SYSEMS 5 of two Sturm Liouville differential equations with different orders four and two, i.e., with R 0 t = {τ ij t} i,j {1,2} we can write 2.13 as r2 t y 1 s1 t y 1 + p 1 t y 2 + τ11 t y 1 + τ 12 t y 2 = 0, 2.14 q 1 t y 1 + r 1 t y 2 + τ21 t y 1 + τ 22 t y 2 = More precisely there are k 1 equations of order four, and there are k 2 equations of order two. If k 1 = m, then k 2 = 0 and equation 2.13 is a standard Sturm Liouville equation SL with regular leading coefficient R 2 t. On the other hand, if k 1 = 0, then k 2 = m and 2.13 becomes the second order Sturm Liouville equation with regular leading coefficient R 1 t. he remaining m 1 different cases with k 1 1 and k 2 1 then represent a natural connection between the above mentioned standard equations SL of order two and order four with regular leading coefficients. Remark 2.6. Similarly to the previous remark, we notice that equation SL of order 2n in the normal form covers exactly m+n 1 m various equations or systems of differential equations with different orders, including the special ones with regular leading coefficients from order two to order 2n. he next part of this section will be devoted to the formulation of the main result of this paper and its proof. In the following theorem we show, how to transform a general 2n-th order Sturm Liouville equation SL in the normal form into the linear Hamiltonian system H. heorem 2.7. Assume that A1 holds and equation SL is in the normal form of type k 1,..., k n with l i defined in 2.8 according to Definition 2.2. Let y be divided as in 2.9 and let y [j] for j {0,..., 2n 1} be the generalized quasiderivatives of y according to Definition 2.4. hen the transformation x = x 1,..., xn, u = u 1,..., un n, x, u R d, d := l j, 2.16 where x j := y [j 1], u j := y [2n j], j {1,..., n}, 2.17 transforms equation SL into the linear Hamiltonian system H with d d coefficients At, Bt, Ct given by 0 ln A 1 t ln 1 A 2 t... 0 At := , A n 1 t l1 where for t [a, b] we define Bt := diag {B 1 t,..., B n t}, Ct := diag {C 1 t,..., C n t}, 2.19 A j t := r 1 j 0ln j 0 B j t := 0 r 1 j t C 1 t := R 0 t, I ln j t q j t j=1, j {1,..., n 1}, 2.20, j {1,..., n 1}, B n t := r 1 n t, 2.21 C j t := s j 1 t p j 1 t r 1 j 1 t q j 1t, j {2,..., n}. 2.22

7 6 IVA DŘÍMALOVÁ, WERNER KRAZ, AND ROMAN ŠIMON HILSCHER Moreover, the associated linear Hamiltonian system H, or more precisely the pair A, B given by is completely controllable on [a, b]. Note that by assumption A1 and Remark 1.1 the resulting d d coefficient matrices At, Bt, Ct are piecewise continuous, and Bt and Ct are symmetric on [a, b]. Also, we refer to [6, Definition 2.3.1, pg. 53] for the complete controllability or equivalently the identical normality notion. Proof of heorem 2.7. Let the quantities x, u and At, Bt, Ct be as in the theorem, and fix t [a, b]. In order to make the presentation shorter, we suppress the argument t in the calculations below. First we prove that x = Ax + Bu. Following the same dimensions as in 2.16, we denote by Ax+Bu j and Cx A u j the j-th components of the indicated vectors, j {1,..., n}. From the definition of y [n] and u n in 2.10 and 2.17 we get x n = y [n 1] = y n 1 1 while for j {1,..., n 1} we have 2.17 Ax + Bu j = A j x j+1 + B j u j = 2.12 = = Iln j r 1 j q j y [j] y j n j+1 y [j] = rn 1 [n] 2.17 y = rn 1 u n = Ax + Bu n, 2.23 Iln j r 1 j q j 0ln j 0 0 r 1 j y [j] 0ln j r 1 j y [2n j] [ y [2n j 1] sj p j y [j] + 0 q j r j y j n j+1 ] 2.12 = y [j 1] = x j Equalities 2.23 and 2.24 show that x = Ax + Bu holds. It can be seen from 2.11 that n y [2n 1] = 1 j R j y j j 1, which together with the equation SL and the choice of u 1 yields u 1 j= = y [2n 1] SL = R 0 y 2.10 [0] 2.17 = R 0 y = R 0 x 1 = Cx A u Next, for j {2,..., n} we have by 2.17 Cx A u j = C j x j A j 1u j 2.12 = s j 1 p j 1 r 1 j 1 q j 1 y [j 1] = y [2n j] 2.17 = u j, + I ln j+1, p j 1 r 1 j 1 y [2n j] + sj 1 y [j 1] + p j 1 y j 1 q j 1 y [j 1] + r j 1 y j 1 n j which together with 2.25 implies that u = Cx A u holds. Finally, from the construction of x and u in 2.16 and 2.17 via the generalized quasiderivatives of y in Definition 2.4 it follows that if xt 0, ut is a solution of H on some nondegenerate subinterval of [a, b], then xt, ut 0, 0 on [a, b]. his means that the system H is identically normal, or equivalently, completely controllable on [a, b]. he proof is complete. Remark 2.8. he components u i in 2.1 are defined through the quasiderivatives in 2.11 via a recursion. By direct calculations we can show an explicit formula for u i, namely n u i = 1 j i I ln i+1, 0 ln i+1 m l n i+1 R j t y j j i, i {1,..., n}, j=i n j

8 SURM LIOUVILLE MARIX DIFFERENIAL SYSEMS 7 compare with the expression for u i in 2.1. Remark 2.9. If some dimensions k j = 0, then the corresponding rows and columns in x, u and At, Bt, Ct in are absent. In this respect we obtain the statement in Lemma 2.1 as a special case of heorem 2.7. Indeed, for k 1 = m and k 2 = = k n = 0 we have l 1 = k 1 = m, so that the coefficient A j t, B j t, C j in reduce to A j t I l1 = I m and B j t 0 l1 = 0 m for j {1,..., n 1}, B n t = rn 1 t = Rn 1 t, C 1 t = R 0 t, and C j t = s j 1 t = R j 1 t for j {2,..., n}. We illustrate the above situation in the following three examples, being partly a continuation of Remark 2.5. Example Consider the fourth order equation 2.13 in the normal form, i.e., equation SL with n = 2 and m 2. Let k 1 and k 2 be positive integers with k 1 + k 2 = m. We write y = y1, y2, where y 1 : [a, b] R k 1 and y 2 : [a, b] R k 2. hen l 1 = k 1, l 2 = k 1 + k 2 = m, and d = l 1 + l 2 = 2k 1 + k 2. By heorem 2.7, the transformation suppressing the argument t x = y1 y 2 y 1, u = r k2 y1 y 2 s1 p 1 + q 1 r 1 r 2 y 1 y1 brings equation 2.13 into the linear Hamiltonian system H, where 0 k1 0 I k1 0 k1 0 0 A = 0 0 k2 r1 1 q 1, B = 0 r 1 R , C = k1 0 0 r2 1 0 s 1 p 1 r1 1. q 1 Example Consider the sixth order equation SL in the normal form, i.e., equation SL with n = 3 and m 2. Let k 1 and k 3 be positive integers, k 2 = 0, and k 1 + k 3 = m. We write y = y1, y3, where y 1 : [a, b] R k 1 and y 3 : [a, b] R k 3. Equation SL has the form suppressing the argument t r3 0 y1 s2 0 y1 s1 p 1 y1 + + R 0 y = y y 3 q 1 r 1 In this case l 1 = l 2 = k 1, l 3 = k 1 + k 3 = m, and d = l 1 + l 2 + l 3 = 3k 1 + k 3. hen y1 r3 0 y1 s2 0 y1 s1 p 1 y + 3 x = y 1, u = 0 0 k3 y k3 y 3 q 1 r 1 r 3 y 1 + s 2 y 1 y 1 r 3 y 1 transforms equation 2.26 into the linear Hamiltonian system H, where 0 k1 0 I k1 0 0 k A = 0 0 k3 r1 1 q k1 I, B = 0 r k k1 0, k r R 0 C = s 1 p 1 r1 1 q s 2 y 3 y 2 y1 Note that the rows and columns in the above expressions for x, u and A, B, C corresponding to k 2 = 0 are missing, as we commented in Remark 2.9. y 3,

9 8 IVA DŘÍMALOVÁ, WERNER KRAZ, AND ROMAN ŠIMON HILSCHER Example Consider the sixth order equation SL in the normal form, i.e., equation SL with n = 3 and m 2. Let k 1 and k 2 be positive integers, k 3 = 0, and k 1 + k 2 = m. We write y = y1, y2, where y 1 : [a, b] R k 1 and y 2 : [a, b] R k 2. Equation SL has the form suppressing the argument t r3 0 y1 s2 p 2 y1 + R 1 y + R 0 y = 0, y 2 q 2 r 2 y 2 where R 1 = s 1. In this case l 1 = k 1, l 2 = l 3 = k 1 + k 2 = m, and d = l 1 + l 2 + l 3 = 3k 1 + 2k 2. hen the transformation y1 r3 0 y1 s2 p 2 y1 y 2 + R 1 y x = y 0 0 y 2 q 2 r 2 y 2 1, u = s2 p 2 y1 r3 0 y1 y 2, q 2 r 2 y y 2 y 1 r 3 y 1 brings equation 2.27 into the linear Hamiltonian system H, where 0 k1 0 I k k k2 0 I k k A = k1 0 I k1, B = k1 0 0, k2 r2 1 q r k r3 1 R C = 0 R s 2 p 2 r2 1 q 2 Again, the rows and columns corresponding to k 3 = 0 in the expressions for x, u and A, B, C are missing. One application of heorem 2.7 leads to the following existence and uniqueness result for solutions of equation SL in the normal form. heorem Assume that A1 holds and that equation SL is in the normal form according to Definition 2.2. hen for any initial point t 0 [a, b] and any initial values y [j] t 0 with j {0, 1,..., 2n 1} the associated initial value problem for SL possesses a unique solution yt, which is defined on the whole interval [a, b]. Proof. he statement follows from heorem 2.7 and a standard existence and uniqueness result for linear differential systems. 3. Application to eigenvalue problems Consider the 2n-th order Sturm Liouville differential equation n L[y]t = 1 j R j t y j j = λ W t y, t [a, b], SLλ j=0 where y : [a, b] R m and λ R is the spectral parameter. Similarly as before, see A1, we assume the following standing hypothesis. A1 he coefficients R j t, j {0, 1,..., n}, and W t are real, symmetric, and piecewise continuous m m matrix-valued functions on [a, b].

10 SURM LIOUVILLE MARIX DIFFERENIAL SYSEMS 9 We assume within this section that equation SL λ is in the normal form of type k 1,..., k n according to Definition 2.2. Moreover, we suppose that the blocks r j t and the matrix W t are positive definite not only invertible, i.e., r j t > 0 for all t [a, b] and j {1,..., n}, 3.1 W t > 0 for almost every t [a, b]. 3.2 By heorem 2.7 we deduce that in this case equation SL λ is equivalent to the linear Hamiltonian system x = At x + Bt u, u = Ct x A t u λ V t x, t [a, b], H λ where At, Bt, Ct are given in 2.18 and 2.19 and Bt 0, t [a, b], 3.3 V t := diag{w t, 0 ln 1,..., 0 l1 } 0, t [a, b]. 3.4 Remark 3.1. Note that the Hamiltonian system H λ satisfies the common assumptions, so that e.g. the theory in [6,8,11,12] can be applied. More precisely, the matrices At, Bt, Ct, and V t are piecewise continuous, Bt, Ct, and V t are symmetric, the Legendre condition 3.3 holds, and by 3.4 the weight matrix V t is positive semidefinite for all t [a, b]. Moreover, as in heorem 2.7 the pair A, B is completely controllable or equivalently the system H λ is identically normal on [a, b], and the triple A, B, V is strongly observable on [a, b], see [6, Definition 3.5.1, pg. 107], under the assumptions A1, 3.1, and 3.2. his implies that focal points of conjoined bases of the system H λ are defined in the classical way, that is, a point t 0 a, b] is a focal point of a conjoined basis X, U of H λ if Xt 0 is singular and then def Xt 0 = dim Ker Xt 0 is its multiplicity, see [12, Definition 1.1]. We recall that a solution X, U of H λ is a conjoined basis if X t Ut is symmetric and rankx t, U t = d for some and hence for all t [a, b]. Based on the equivalence of the equation SL λ in the normal form and the system H λ provided by heorem 2.7, we are now able to show that equation SL λ, together with appropriate boundary conditions, possesses all the traditional spectral properties of self-adjoint eigenvalue problems. his includes the equality of the geometric and algebraic multiplicities of the eigenvalues, orthogonality of eigenfunctions, the oscillation theorem and Rayleigh s principle, and the Fourier expansion theorem. In order to derive these facts, we make some preparatory considerations. he quadratic functional associated with equation SL λ is given by b n Gy, λ := y j t b Rj t y j t dt λ y t W t yt dt = a b a j=0 n y [j 1] t Rj t l n j+1 j=1 a y [j 1] t dt + b a y t [R 0 t λ W t] yt dt. We say that the functional G, λ is positive definite and we write G, λ > 0, if Gy, λ > 0 for all y A with yt 0, where A is the admissible set for the functional G, λ, i.e., A := { y = y 1,..., y n, y j C n j+1 p, y [j 1] a = 0 = y [j 1] b, j {1,..., n} }. he following positivity result is proven in the same way as [10, heorem 2.2.5]. his proof works with Ehrling s lemma along the same lines as reference [7]. Note that if W t is piecewise continuous on [a, b] and W t > 0 for all t [a, b], then this result is contained in [8, Lemma 2.10]. Lemma 3.2. Assume that A1, 3.1, and 3.2 hold and let equation SL λ be in the normal form according to Definition 2.2. hen there exists ν < 0 such that G, ν > 0.

11 10 IVA DŘÍMALOVÁ, WERNER KRAZ, AND ROMAN ŠIMON HILSCHER Consider now the eigenvalue problem with Dirichlet boundary conditions for the equation SL λ, i.e., SL λ, y [j] a = 0 = y [j] b, j {0,..., n 1}, 3.5 and λ R. We may consider real λ only, since the eigenvalue problem 3.5 is self-adjoint. First we assume that 3.2 holds and define the eigenvalues and eigenfunctions of 3.5 in the traditional way. A number λ 0 R is an eigenvalue of 3.5 if it has a nontrivial solution y. Such a solution is called an eigenfunction for λ 0 and the dimension of eigenfunctions for λ 0 is called the geometric multiplicity of λ 0. Note that all eigenfunctions of 3.5 belong to the admissible set A. he corresponding inner product is given by y, ỹ W := b a y t W t ỹt dt. 3.6 Let the coefficients At, Bt, Ct, V t of H λ be given by 2.18 and 2.19 in heorem 2.7 and 3.4. he principal solution ˆXλ, Ûλ of H λ is defined as the solution given by the initial conditions ˆXa, λ = 0 and Ûa, λ = I. Clearly, the principal solution is a conjoined basis of H λ. If λ 0 C is an eigenvalue of 3.5, then the number def ˆXb, λ 0 = dim Ker ˆXb, λ 0 defines the algebraic multiplicity of λ 0. For λ R we set n 1 λ := the number of focal points of ˆXλ, Ûλ in a, b], 3.7 n 2 λ := the number of eigenvalues of 3.5 in, λ], 3.8 where the focal points and the eigenvalues are counted including their geometric or algebraic multiplicities. his definition is correct according to the result presented below. heorem 3.3. Assume that A1, 3.1, and 3.2 hold and let equation SL λ be in the normal form according to Definition 2.2. hen the following statements hold. i he eigenvalue problem 3.5 has infinitely many eigenvalues, which are all real, isolated, bounded from below, unbounded from above with the only accumulation point at +, and their geometric and algebraic multiplicities coincide. he multiplicity of each eigenvalue is at most d, which is given in ii he eigenfunctions corresponding to different eigenvalues are orthogonal with respect to the inner product 3.6. iii he oscillation theorem holds: n 1 λ = n 2 λ for all λ R. 3.9 iv he Rayleigh principle holds, i.e., if < λ 1 λ 2 λ j... are the eigenvalues of 3.5 with the corresponding orthonormal eigenfunctions ỹ 1, ỹ 2,..., ỹ j,..., then for every j N {0} λ j+1 = min { Gy, 0 y, y W, y A, yt 0, and y ỹ 1,..., ỹ j v Every function y A has the Fourier expansion y = c j ỹ j, where c j := y, ỹ j W for all j N. j=1 } Proof. By heorem 2.7, the eigenvalue problem 3.5 is equivalent with the linear Hamiltonian eigenvalue problem H λ, λ R, xa = 0 = xb, 3.11 whose coefficients are given by heorem 2.7 and 3.4 and where the Legendre condition 3.3 is satisfied by 3.1. We note that the system H λ in 3.11 satisfies for all λ C the identical

12 SURM LIOUVILLE MARIX DIFFERENIAL SYSEMS 11 normality condition. Since under 3.2 the weight matrix V t in H λ is only positive semidefinite, we need to consider the theory of finite eigenvalues for problem Following [8, 12], a number λ 0 R is called a finite eigenvalue of 3.11 if θλ 0 := rb rank ˆXb, λ 0 1, 3.12 where rt := max λ R rank ˆXt, λ for t [a, b]. he number θλ 0 is then its algebraic multiplicity. he finite eigenfunctions x, u for λ 0 are defined by the property V t xt 0 on [a, b]. he dimension of the space of all functions V t xt, where x, u are finite eigenfunctions for λ 0, is then called the geometric multiplicity of λ 0. From [8, Remark 2.7] we know that the finite eigenvalues of 3.11 are real, isolated, bounded from below, with the same algebraic and geometric multiplicities, and that the finite eigenfunctions corresponding to different finite eigenvalues are orthogonal with respect to the inner product b x, u, x, ũ V := x t V t xt dt = a Moreover, by [8, heorem B.5] there exists l N {0} such that b a y t W t ỹt dt = y, ỹ W. n 1 λ = n 2 λ + l for all λ R, 3.13 where n 1 λ is the number of focal points of the principal solution ˆXλ, Ûλ of H λ in a, b] and n 2 λ is the number of finite eigenvalues of 3.11 in the interval, λ], all counted including their multiplicities. Equality 3.13 yields that the only possible accumulation point of finite eigenvalues is +, since n 1 λ is finite for all λ R by [5, heorem 3]. It is our first claim to prove that λ 0 is a finite eigenvalue of 3.11 with algebraic multiplicity θλ 0 if and only if it is an eigenvalue of 3.5 with the same algebraic multiplicity. By using the positivity assumptions 3.1 and 3.2 and Lemma 3.2, we know that the functional G, ν is positive definite for some ν < 0. his implies that l = 0 in 3.13, because in this case the principal solution ˆXν, Ûν has no focal points in a, b], by [5, heorem 1]. he identical normality of system H λ now implies that the matrix ˆXb, ν is invertible, so that the above defined value rb = d. In turn, from 3.12 we have θλ 0 = d rank ˆXb, λ 0 = def ˆXb, λ 0, which proves the assertion. Now if x, u is a finite eigenfunction for the finite eigenvalue λ 0, then by using the structure of V t in 3.4 and the structure of xt in 2.17 we obtain that V t xt 0 if and only if W t yt 0, where yt := y1 t,..., yn t as before. But since W t is assumed to be invertible, we have that the finite eigenfunctions of 3.11 for λ 0 and the eigenfunctions of 3.5 for λ 0 are in one-to-one correspondence with the same geometric and hence algebraic multiplicities. All the statements in parts i iii, except for the existence of infinitely many eigenvalues of 3.5, now follow from the above properties of finite eigenvalues of 3.11, compare also with [8, heorems 2.9 and A.2] or also with [12, Corollary 1.7 and Proposition 3.2]. he Rayleigh principle in part iv and the expansion theorem in part v follow from the same formulas for finite eigenvalues of 3.11 in [8, heorems 1.1 and 4.3]. Finally, since dim A =, the Rayleigh principle in 3.10 implies that 3.5 has infinitely many eigenvalues, which are therefore unbounded from above, compare also with [8, Corollary 4.1i]. he proof is complete. Remark 3.4. Similarly as in Remark 2.6, if k 1 = m, then heorem 3.3 yields the results for the classical 2n-th order eigenvalue problem SL λ, λ R, y j a = 0 = y j b, j {0,..., n 1} 3.14 with R n t = r n t > 0 on [a, b] as in 3.1. he eigenvalues of 3.14 then have the multiplicity at most d = mn. On the other hand, when k n = m, i.e., k j = 0 for all j {1,..., n 1}, then

13 12 IVA DŘÍMALOVÁ, WERNER KRAZ, AND ROMAN ŠIMON HILSCHER heorem 3.3 yields the results for the classical second order eigenvalue problem R 1 t y + R 0 t y = λ W t y, t [a, b], λ R, ya = 0 = yb 3.15 with R 1 t = r 1 t > 0 on [a, b]. he eigenvalues of 3.15 have the multiplicity at most m. A similar situation occurs when some other k j = 0. herefore, we can interpret heorem 3.3 as a connection between the two classical eigenvalue problems 3.14 and Remark 3.5. Note that in the present work we consider Dirichlet boundary conditions only, i.e., xa = 0 = xb in the eigenvalue problem 3.11, or y [j] a = 0 = y [j] b for j {0,..., n 1} in terms of y in the eigenvalue problem 3.5. By 2.10 of Definition 2.4, these boundary conditions depend on the type k 1,..., k n of the normal form only, but not on the coefficients R 1 t,..., R n t. More general boundary conditions, in particular separated boundary conditions for the Hamiltonian system H λ, can also be treated in principle, see e.g. [3]. But such more general boundary conditions involve also the values ua and ub defined by 2.16 and Hence, they depend on the coefficients R 1 t,..., R n t too, so that they become practically untreatable. A closer look at the proof of heorem 3.3 and the quoted references shows that the statement can be easily generalized to the weaker assumption W t 0, t [a, b], 3.16 instead of 3.2. In this case the finite eigenvalues of 3.5, defined by the property W t yt 0 on [a, b] or equivalently by condition 3.12, must be used. heorem 3.6. Assume that A1, 3.1, and 3.16 hold and let equation SL λ be in the normal form according to Definition 2.2. Furthermore, assume that the functional G, ν is positive definite for some ν < 0. hen the following statements hold. i he eigenvalue problem 3.5 has exactly p := dim{w t yt, y A} finite eigenvalues, p N {0} { }, which are all real, isolated, bounded from below, their only possible accumulation point is +, and their geometric and algebraic multiplicities coincide. he multiplicity of each finite eigenvalue is at most d, which is given in ii he finite eigenfunctions corresponding to different finite eigenvalues are orthogonal with respect to the semi-inner product 3.6. iii he oscillation theorem in 3.9 holds, where n 1 λ is defined in 3.7 and n 2 λ denotes the number of finite eigenvalues of 3.5 in the interval, λ]. iv he Rayleigh principle holds, i.e., if < λ 1 λ 2 λ p are the finite eigenvalues of 3.5 with the corresponding orthonormal finite eigenfunctions ỹ 1, ỹ 2,..., ỹ p, then for every j {0, 1,..., p 1} λ j+1 = min { Gy, 0 y, y W, y A, W t yt 0, and y ỹ 1,..., ỹ j v Every function y A has the Fourier expansion y = p c j ỹ j, j=1 where c j := y, ỹ j W for all j {1, 2,..., p}. Proof. he results follow from the corresponding statements regarding the finite eigenvalues and finite eigenfunctions for the problem 3.11 with 3.4, in particular, from Corollary 4.1 ii, Remark 2.7, and heorems 1.1, 2.9, 4.3, and A.2 in the reference [8]. We note that the number p in heorem 3.6 satisfies p 1, as long as W t 0 on [a, b]. }.

14 SURM LIOUVILLE MARIX DIFFERENIAL SYSEMS ransformation to normal form Once we know how the equation SL in the normal form can be transformed into the linear Hamiltonian system H, it is natural to ask when a general equation SL can be transformed to some other equation n L[ỹ]t := 1 j Rj t ỹ j j = 0 SL j=0 in the normal form according to Definition 2.2. In this section we give sufficient conditions on the coefficients R 1 t,..., R n t, which guarantee that such a transformation exists. Moreover, we show that these conditions or at least some of them are optimal in some sense. In particular, we show in Remark 4.4 and Example 4.5 that when these conditions are violated, then the equation SL cannot be transformed into a linear Hamiltonian system. he main result of this section heorem 4.1 below is formulated in terms of the following subspaces K j t and I j t, which are associated with the fundamental spaces Ker R j t and Im R j t. For t [a, b] and j {1,..., n} we define the subspaces K j t := Ker R n t Ker R n j+1 t, I j t := Im R n t + + Im R n j+1 t, which satisfy the orthogonality relation {I j t} = K j t on [a, b], by the symmetry of the coefficients. hen we consider the main hypotheses or conditions: A2 K j t K j, and hence I j t I j, is constant on [a, b] with l j := m dim K j = dim I j for j {1,..., n}, A3 K n = {0}, and hence I n = R m, { } A4 Rn j t K j Ij = {0} on [a, b] for j {1,..., n 1}, A5 R n t 0, and R n j t 0 on K j on [a, b] for j {1,..., n 1}. We note that A3 implies condition A2 with j = n. heorem 4.1. Assume that A1 A4 hold. hen there exists a constant orthogonal matrix such that the transformation y = ỹ transforms equation SL into equation SL in the normal form of the type k 1,..., k n according to Definition 2.2, where R j t := R j t = rj t 0 0 0, t [a, b], j {0, 1,..., n}, 4.1 k 1 := l 1, k j := l j l j 1, j {2,..., n}. 4.2 Moreover, the matrices r j t from 4.1 satisfy 3.1, i.e., r j t > 0 for all t [a, b] and j {1,..., n}, if and only if, in addition to A1 A4, condition A5 holds. he proof of heorem 4.1 is displayed below, after we discuss its consequences. A combination of heorem 4.1 with heorems 2.7 and 3.3, 3.6 yields, respectively, the following results. Corollary 4.2. Assume that A1 A4 hold. hen the equation SL can be transformed into a linear Hamiltonian system H of size 2d, where d := n j=1 l j with l j given in A2. Moreover, the system H is completely controllable on [a, b], and the matrix Bt given by 2.19 and 2.21 satisfies the Legendre condition 3.3 if and only if, in addition to A1 A4, condition A5 holds. Corollary 4.3. Assume that A1, A2 A5, and 3.2 hold. hen the eigenvalue problem SL λ, y [j] a = 0 = y [j] b, j {0,..., n 1}, 4.3 with k j and as in heorem 4.1, possesses the spectral properties in heorem 3.3, i.e., assertions i to v of heorem 3.3 hold. Moreover, if 3.16 is satisfied instead of 3.2 and if the

15 14 IVA DŘÍMALOVÁ, WERNER KRAZ, AND ROMAN ŠIMON HILSCHER functional G, ν is positive definite for some ν < 0, then 4.3 possesses the spectral properties in heorem 3.6. We are now ready to present the proof of heorem 4.1. We recall that the numbers l j and k j are defined in A2 and 4.2, respectively. Proof of heorem 4.1. Assume that A1 A4 hold. We divide the proof into several steps. Step 1. From A2 with j = 1 we know that the space K 1 = Ker R n t is constant on [a, b]. hen there exists a constant orthogonal matrix 1 such that rn 1 t 0 }l1 =k 1 R n t 1 = m l1 where r n t is an invertible l 1 l 1 matrix and dim Ker R n t = m l 1 = dim K 1 on [a, b]. Moreover, r n t > 0 on [a, b] if and only if R n t 0 on [a, b]. Step 2. Let sn 1 1 t ˆp n 1 t }l1 R n 1 t 1 = with ˆq ˆq n 1 t ˆr n 1 t n 1t = ˆp n 1 t. 4.5 If d Ker 1 R n t 1, then d = 0ˆd, 1 R n 1 t 1 d = ˆpn 1 t ˆd ˆr n 1 t ˆd. Moreover, if 1 R n 1 t 1 d Im 1 R n t 1, i.e., if ˆr n 1 t ˆd = 0, then ˆp n 1 t ˆd = 0 by assumption A4 with j = 1. herefore, assumption A4 yields that Ker ˆr n 1 t Ker ˆp n 1 t on [a, b]. 4.6 Using this we obtain that Ker 1 R n t 1 Ker ˆpn 1 t 1 R n 1 t 1 = {0} Ker ˆr n 1 t 4.6 = {0} Ker ˆr n 1 t is constant on [a, b], where {0} is the zero space in R l 1. Step 3. Let ˆ 2 be a constant orthogonal m l 1 m l 1 matrix such that ˆ 2 ˆr n 1 t ˆ 2 = rn 1 t m l1 k 2 }k2 where r n 1 t is an invertible k 2 k 2 matrix and dim Ker ˆr n 1 t = m l 1 k 2 = dim Ker R n t Ker R n 1 t = dim K 2. hen by 4.6 we get ˆp n 1 t ˆ 2 = p n 1 t, 0 for some l 1 k 2 matrix p n 1 t. 4.7 Define the m m matrix 2 := diag{i l1, ˆ 2 }. hen 2 is orthogonal, rn 2 1 t 0 }l1 =k 1 R n t 1 2 = by 4.4, while by using 4.5, 4.7, and the symmetry of R n 1 t s n 1t p n 1 t R n 1 t 1 2 = q n 1 t r n 1 t 0 }l 1=k 1 }k 2 with q n 1 t := p n 1t m l2 Moreover, r n 1 t > 0 on [a, b] if and only if R n 1 t 0 on K 1 = Ker R n t on [a, b].

16 SURM LIOUVILLE MARIX DIFFERENIAL SYSEMS 15 Step 4. Similarly as in 4.5, let 2 1 R n 2 t 1 2 = sn 2 t ˆq n 2 t If d Ker 2 1 R n t 1 2 Ker 2 1 R n 1 t 1 2, then 0ˆd d =, 2 1 R n 2 t 1 2 d = ˆp n 2 t }l2 with ˆq ˆr n 2 t n 2t = ˆp n 2 t ˆpn 2 t ˆd ˆr n 2 t ˆd Moreover, if 2 1 R n 2 t 1 2 d Im 2 1 R n t Im 2 1 R n 1 t 1 2, i.e., if ˆr n 2 t ˆd = 0, then ˆp n 1 t ˆd = 0 follows by assumption A4 with j = 2. herefore, Using this we obtain that Ker ˆr n 2 t Ker ˆp n 2 t on [a, b] Ker 2 1 R n t 1 2 Ker 2 1 R n 1 t 1 2 Ker 2 1 R n 2 t 1 2 ˆpn 2 t 4.11 = {0} Ker = {0} Ker ˆr ˆr n 2 t n 2 t is constant on [a, b], where {0} is the zero space in R l 2. Step 5. Let ˆ 3 be a constant orthogonal m l 2 m l 2 matrix such that where r n 2 t is an invertible k 3 k 3 matrix and ˆ 3 ˆr n 2 t ˆ 3 = rn 2 t m l2 k 3 }k dim Ker ˆr n 2 t = m l 2 k 3 = dim Ker R n t Ker R n 1 t Ker R n 2 t = dim K 3. hen by 4.11 we obtain ˆp n 2 t ˆ 3 = p n 2 t, 0 for some l 2 k 3 matrix p n 2 t Define the m m matrix 3 := diag{i l2, ˆ 3 }. hen 3 is orthogonal and rn t 0 }l1 =k 1 R n t = 0 0 by 4.8. Similarly, from 4.9 we obtain R n 1 t = s n 1t p n 1 t 0 q n 1 t r n 1 t m l2. }l 1=k 1 }k 2 while from 4.10, 4.12, 4.13, and the symmetry of R n 2 t we get s n 2t p n 2 t R n 2 t = q n 2 t r n 2 t 0 }l 2 }k 3 with q n 2 t := p n 2t m l3 Moreover, r n 2 t > 0 on [a, b] if and only if R n 2 t 0 on K 2 = Ker R n t Ker R n 1 t on [a, b]. Step 6. Now the procedure is clear and it leads to the normal form for the equation SL, in which the coefficients R j t are defined by 4.1 with := n being constant and orthogonal. Indeed, the above procedure ends with s1 t p 1 t }ln 1 R 1 t = with q 1 t = p 1 t, q 1 t r 1 t

17 16 IVA DŘÍMALOVÁ, WERNER KRAZ, AND ROMAN ŠIMON HILSCHER where r 1 t is an invertible k n k n matrix. Here we use the assumption A3, i.e., l n = m or equivalently m l n 1 k n = 0, to finish the above algorithm. Moreover, we have r 1 t > 0 on [a, b] if and only if R 1 t 0 on K n 1 = Ker R n t Ker R 2 t on [a, b]. he proof is complete. Let us make some comments on the assumptions of heorem 4.1. Remark 4.4. i he constancy of the kernels in A2 is crucial for the present method. It implies that the type k 1,..., k n of the transformed equation SL does not change on [a, b], so that the associated Hamiltonian system H according to heorem 2.7 is of constant size 2d 2d. Moreover, it is important that the transformation matrix is then constant on [a, b], too. ii Note that assumption A3 is equivalent with l n = m. If the assumption A3 is violated, then there exists c R m, c 0, such that R j t c = 0 for all t [a, b] and j {1,..., n}. hus, c L[y]t = n 1 j c R j t y j j = c R 0 t y, j=0 t [a, b]. Consequently, if we consider the equation SL λ with 3.2 and the initial value yt 0 = W t 0 c at some t 0 [a, b], then we get c R 0 t 0 W t 0 c = c L[y]t 0 = λ c W t 0 yt 0 = λ W t 0 c 2, with W t 0 c 0. Hence, there is no solution to this initial value problem for large λ, and there cannot be an equivalence of SL λ with a Hamiltonian system H λ or any kind of spectral theory. iii he violation of the assumption A4 leads to a degenerate differential system, as we show in the simple Example 4.5 below. iv Of course, the nonnegativity or positivity assumptions 3.3 and 3.2 or 3.16 are needed for the spectral theory in heorems 3.3 and 3.6. Hence, the assumption A5 is essentially required for Corollary 4.3. Example 4.5. Let m = n = 2 and 1 0 R 2 t :=, R t := 0 1, R t := on [a, b]. hen K 1 = Ker R 2 t = {0} R and K 2 = Ker R 2 t Ker R 1 t = {0}, so that assumptions A1 A3 hold. But { R1 t K 1 } I1 = { R 1 t Ker R 2 t } Im R 2 t = R {0} is not the zero space in R 2, so that A4 is violated. In this case, for y = y 1, y 2 the differential system SL becomes y 4 1 y 2 + y 1 = 0, y 1 + y 2 = Hence, y 2 = y 1, which upon inserting to the first equation in 4.14 yields y 1 = 0. herefore, y 2 = 0 = y 1, so that the system SL is completely degenerate. Acknowledgements he third author is grateful to the University of Ulm for hospitality provided while conducting a part of this research project.

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