A Numerical Investigation of the c-Numerical Ranges of Differential Operators

In this paper, we consider the problem of computing the c-numerical range numerically for block differential operators, particularly these of Schrödinger type, Hain–Lüst type, and Stokes type.

generalizations of numerical range. For 1 ≤ k ≤ n, the k-numerical range of linear operator A on C n , introduced by Halmos [13] is defined as follows: which is proved to be convex by Berger [14]. For any c = (c 1 , . . . , c k ) ∈ R k , the c-numerical range of linear operator A on C n , first introduced by Marcus [15], is defined as follows: c i x * i Ax i : {x 1 , . . . , x n } is an orthonormal set .  [13]). In addition, for c 1 = c 2 = · · · = c k = 1, the c-numerical range W c (A) turns into W k (A). Indeed, W k (A) ⊆ W (A). Westwick [16] has shown that W c (A) is convex for any c ∈ R n . In addition, he gave an example which shows that, for complex vectors c ∈ C n with n ≥ 3, the range W c (A) may fail to be convex. The c-numerical range is unitary similarity invariant, W c (A) = W c (U AU * ), for any unitary matrix U . It is also transpose invariant, W c (A) = W c (A T ). Clearly, we have the basic property: for every a, b ∈ C. A review of the properties of c-numerical ranges of operator matrices may be found in [17].
In this paper, we consider how to compute the c-numerical range by finite difference methods, which reduce the problem to that of computing the c-numerical range of a (finite) matrix and block matrix.
The paper is organized as follows. In Sect. 2.1, some theoretical results are investigated dealing with the c-numerical range of operators using finite difference method. In Sect. 3, we shall applying these results to compute the c-numerical range of differential operators. We shall consider two particular special cases. In the first case, A is the Hain-Lüst operator, it is clear the underlying Hilbert space in this case is H := L 2 (0, 1) × L 2 (0, 1) and the operator is as follows: (1.6) in which where L = − d 2 dx 2 + q is a Schrödinger operator (with bounded potential q), while w, w and z are bounded multiplication operators. The domain of L is given by the following: This operator has been introduced by Hain and Lüst [18] in application to problems of magnetohydrodynamics and the problem of this type has been studied in [19][20][21][22][23]. We shall be concerned with the effects of discretization, both of the unbounded second -order differential operator L and of the bounded multiplication operators which appear in the other entries. In the last case, A is the Stokes-type system of ordinary differential equations. The underlying Hilbert space in this case is also H := L 2 (0, 1)× L 2 (0, 1) and the operator is as follows: (1.7) the domain of A is given by the following: Note that A is not closed; however, A is closable and its closure is self-adjoint.

An Approximating Discrete Operator
We shall replace the Schrödinger operator by a matrix. Suppose u ∈ D(L). Pick N + 1 points x 0 , x 1 , . . . , x N in the interval [0, 1]; for the sake of simplicity, we assume equal spacing, so that We would like to form the vectors: Moreover, if we are to use the quadrature estimates, we shall need more smoothness in u. We, therefore, observe that the set is a core of the operator L; for u ∈ C(L), Eq. (2.1) makes sense. Recall the second-order divided difference approximation of the second derivative: where T = tridiag(1, −2, 1) is the tridiagonal matrix with entries T j j = −2, T j, j±1 = 1. We also define diagonal matrices: Our matrix replacement of the Schrödinger operator is a matrix of dimension (N − 1) × (N − 1) given by the following: We shall use this technique in this paper for computing c-numerical range. Muhammad and Marlleta [24] replaced the Hain-Lüst operator and Stokes operator by matrices. To discretize the Hain-Lüst operator, suppose that ( u v ) ∈ D(A). Pick N + 1 points x 0 , x 2 , . . . , x N in the interval [0, 1]; for the sake of simplicity, we assume equal spacing, so that We would like to form the vectors: . . .
however, since we only have v ∈ L 2 (0, 1), point values of v are meaningless. Moreover, we shall need more smoothness in u and v if we are to use the quadrature estimates we need. We, therefore, observe that the set is a core of the operator A; for u v ∈ C(A), Eq. (2.6) makes sense. We also define diagonal matrices: Our matrix replacement of the Hain-Lüst operator is a matrix of dimension 2(N − 1) × 2(N − 1) given by the following: Hence, our matrix replacement of the Stokes operator is a matrix of dimension (2N − 1) × (2N − 1) given by the following: and Z N = −3 2 I N ×N . We shall use this technique in this paper for computing c -numerical range.

c-Numerical Range Inclusion
The following theorem shows that by finite difference discretization approximation, we derived that the every point of the c-numerical range of the Schrödinger operator L = − d 2 dx 2 + q(x) can be approximated to arbitrary accuracy.
Then, it is clear which we need two types of integral which are estimated by quadrature. Type 1: L 2 integrals of smooth functions Let u 1 , u 2 , . . . , u k ∈ C 4 (0, 1) with Dirichlet boundary conditions at 0 and 1. Then By a straightforward computation, the following result can be verified.

Lemma 2.2
Suppose that u 1 , u 2 , . . . , u k ∈ C N −1 , and u 1 , u 2 , . . . , u k is an eigenfunction of the Schrödinger operator L; then Type 2: The integrals Lu, u L 2 for smooth u For u 1 , u 2 , . . . , u k as before, we observe that Lu 1 , Then, by taking inner products on both sides of Eq. (2.15), we get the following:  (2.17) and thus, using Lemma 2.2 and dividing both sides of Eq. (2.17) by k i=1 u i , u i L 2 , we get the following:

Multiplying by h and adding
This implies that We introduce Definition 2.3, which is make sense for any bounded or an unbounded linear operator to bound some crucial parts in our main Theorem 2.6. Suppose that A is an unbounded linear operator A : D(A) ⊂ H → H . Consider a sesquilinear form: We define R-partial c-numerical range as follows: Definition 2.3 Let A be an unbounded operator with quadratic form a(·). Then, the set We recall some lemmas useful to prove the main Theorem 2.6.  u 1 , u 2 , . . . , u k in C N −1 , u 1 , u 2 , . . . , u k in H 1 0 (0, 1) be the piecewise linear functions given by u r (x) = u r ( j−1)

Lemma 2.4 Let A be an operator which is defined in Definition
Proof For 1 ≤ r ≤ k and let u 1 , u 1 , . . . , u k ∈ H 1 0 (0, 1), be the piecewise linear functions given by u r (x) = u r ( j−1) Rearranging gives The following theorem is our main result.

Theorem 2.6 Suppose that the coefficient q appearing in the Schrödinger operator
Proof Suppose that λ N ∈ W cR (L N ); then, there exist an orthonormal vectors u 1 , u 2 , . . . , u k in C N −1 , such that λ N = c 1 L N u 1 , u 1 + c 2 L N u 2 , u 2 + · · · + c k L N u k , u k .
We begin by constructing a piecewise linear functions u 1 , u 2 , . . . , u k in H 1 0 (0, 1), such that u r (x j ) = u r j , and u r 0 = 0 = u r N for 1 ≤ r ≤ k. It is clear u 1 , u 2 , . . . , u k / ∈ D(L), because u 1 , u 2 , . . . , u k / ∈ H 2 (0, 1). This is not a problem when using the characterization of W c (A) in Definition (2.3); we simply note that the quadratic form associated with L is the usual Dirichlet form: so, first, we need to estimate I 1 on the right-hand side of Eq. (2.19): Here, c 1 (u 1 , u 1 ) = c 1 l 0 (u 1 , u 1 ) + c 1 1 0 q|u 1 | 2 dx, where c 1 l 0 (u 1 , u 1 ) = c 1 1 0 |u 1 | 2 dx. By summation by parts and we are left to deal with N j=1 x j (2.21) Assuming that q(·) is Lipschitz, and using the Cauchy-Schwarz inequality, we get the following: this yields, in light of (2.20): By Lemma 2.5 for r = 1: By the Lipschitz property of q and using (2.22), the fourth term on the right-hand side of Eq. (2.21) satisfies the following: while the sixth term admits the bound The third term of the right-hand side of (2.21) satisfies the following: with (2.21) again providing the last step. Summing all the contributions, we obtain the following: and so We are now in a position to estimate the following: (2.32) Now, we estimate From Eqs. (2.20) to (2.23), we have the following: (2.34) Thus, we get the following: (2.36) By the same argument, the second term and the last term of the right-hand side of Eq. (2.19) satisfies the following:

Numerical Experiments on Differential Operator
In this section, we study some concrete examples and demonstrate that, in spite of the results obtained in the previous section, practical calculation of the c-numerical range is very far from being straightforward. We shall show that the discretization techniques may often yield misleading results, and that a good knowledge of existing analytical estimates of q-numerical range is often needed to understand the results. The computations were performed in MATLAB.
For our example, these yield (λ) ≥ −39 k j=1 c j To estimate Im(λ), observe that Im(λ) = This completes the estimates on W c (A).