Search

Semi-analytical solutions of the Schrodinger equation for a particle moving in the electrostatic field of two other particles a fixed distance apart, are derived in such a way that the resulting matrix eigenvalue equations contain real symmetric band matrices. Numerical techniques appropriate to the solution of the two simultaneous matrix eigenvalue equations are described; in particular the bisection method is used to determine precisely the significant truncation order of the matrices for a given numerical precision.

If H is a Hermitian matrix and W = AH + HA ∗ is positive definite, then A has as many eigenvalues with positive (negative) real part as H has positive (negative) eigenvalues [5]. Theorems of this type are known as inertia theorems. In this note the rank of the controllability matrix of A and W is used to derive a new inertia theorem. As an application, a result in [8] and [4] on a damping problem of the equation M x + (D + G) xdot; + Kx = 0 is extended.

The model-space effective Hamiltonian .5ferr is formulated on the basis of the unit~ry­ transformation method. The .5ferr is obtained through the similarity transformation of the original Hamiltonian. The .5ferf is proved to be Hermitian and independent of energy eigenvalue. A method is proposed which constructs a perturbative solution for the unitary operator leading to .5feff· The connection of model-space wave functions with true eigen­ functions is investigated by means of a quasi-particle picture introduced through the canonical transformation of single-particle operators.

It is a common experience that the perturbation, or even the omission, of some elements of a matrix often has negligible effect on some of the eigenvalues of the whole matrix. Here some new theorems are presented on this isolation effect in Hermitian matrices. The results are of importance in the computation of eigenvalues, particularly for tridiagonal matrices.

A minimax characterization of a subset of the real eigenvalues ω of the quadratic eigenvalue problem (ω2A + ωB + C)ξ = 0 is presented, where A, B and C are linear Hermitian operators (matrices) on and into the n-dimensional complex Hilbert space En, and A > 0. This minimax characterization extends to all the eigenvalues of a semi-overdamped system, and Duffin's minimax theory for overdamped systems is recovered as a special case.

A generalization of the Jacobi and Eberlein methods for partitioned matrices is described. Whereas the generalization of the Jacobi method is fairly straightforward and obvious, the generalization of the Eberlein method involves the use of more advanced techniques. We thereby obtain an algorithm with a very modest core store requirement that uses the backing store in a highly efficient manner. An application to non-modal damping of vibration problems is discussed.

Low-lying excitation energies are calculated by the use of the many-body quasi-degenerate Rayleigh-Schrodinger perturbation theory. An eigenvalue problem is derived for the third-order estimation of the low-lying excitation energies. The elements of the hermitian matrix appearing in this eigenvalue problem are determined up to the third order by the diagrammatic method. Some simple illustrative calculation on π-electron systems are carried out.

A hermitian dynamical matrix in terms of Cartesian displacement coordinates is derived starting from a harmonic force constant matrix expressed in terms of the Wilson's internal displacement coordinates. The method accounts both for inter- and intramolecular coupling. The frequency spectrum can be obtained from the dynamical matrix without directly solving the secular equation by the application of Dean's negative eigenvalues theorem.

the choice of metric is arbitrary. In this note we consider a comparison of metrics induced by certain matrix functions of A in terms of rapidity of convergence. A is an n-square, nonsingular, complex matrix, x and b are complex, column n-vectors, z* is the conjugate transpose of z, X(A) denotes an arbitrary eigenvalue of A, and Xm(A) and XM(A) are the minimum and maximum eigenvalues of A respectively when all X(A) are real. In this case we denote by IA the closed interval [Xm(A), XM(A)]. If R is a positive-definite Hermitian (p.d.h.) matrix define

A density-matrix approach to constrained eigenvalue problems is presented. It is shown that all of the linearly independent eigenvectors of an Hermitian matrix can be generated with the idempotency equations ($\mathit{P}$ equations) developed in previous papers of this series. In particular, the method is applied to variational calculations in ${\mathrm{H}}_{2}^{+}$ and He.

For pt. I see abstr. A10632 of 1973. The problem of l degeneracies in the reduction of irreducible representations (p, q) of SU(3) with respect to its O(3) subgroup is considered. Orthonormal states corresponding to the same l value are defined as eigenvectors of hermitian operators and, by the use of l shift operators, an algorithm is given for the calculation of the eigenvalues of thses hermitian operators. A full analysis is given of the (p, 0) representations, which contain no l degeneracies, and all eigenvalues of the third and fourth order O(3) scalar operators are calculated for the (4, 2), (5, 2) and (6, 2) representations which contain doubly l degenerate states.

The requirements of positivity on Hermitian matrices are stated as conditions on the elementary symmetric functions ${c}_{j}$ of the eigenvalues. The exact restrictions are found for matrices of arbitrary dimensionality; weaker conditions on the ${c}_{j}$ are also given which hold in every case.

1. Main theorems. Let A be a positive definite Hermitian matrix of finite order, and let A and X be its maximal and minimal eigenvalue respectively. The condition number of A is the ratio P(A) =A/X introduced by Todd [1]. Let 'G be a class of regular linear transformations. Define ATT*A T. We say that A is best conditioned with respect to 'G if P(A T) > P(A) for all TEE T. In order to investigate whether A is best conditioned we remember that

Given a hermitian (normal) matrixA with known eigenelements, we study the behavior of these elements under a hermitian perturbationH. With a symmetric 2×2 matrix, the problem is explicit (algebraic equation of 2nd degree), and we try, in the case of ann×n hermitian matrix, to obtain upper bounds which are as close as possible of exact results forn=2. The results are collected in § I. They state in a unified manner some results of Davis [2], Gavurin [4], Golub [5], Kato-Temple [7], Ortega [3], Wilkinson [8] and others. In § II, we apply this theory to produce error bounds for eigenelements computed by theQR and Jacobi methods. The given error bounds are realistic and easy to compute during the algorithms. When the separation distance between eigenvalues ofA approaches zero, the problem of computing eigenelements ofA+H is ill-conditionned with respect to the eigenvalues, the eigenvectors being orthogonal. The precision then obtained is given.

A new method is devised to include the symmetry in perturbation series, in case the unperturbed hamiltonian has a lower (or different in general) symmetry than the whole problem. We define a unitary transformation such that the transformed unperturbed hamiltonian will have the same symmetry as the interacting system. This new unperturbed hamiltonian is hermitian and has indeed the same eigenvalues as the old one. Usual perturbation expansions can be thence utilized. The method is illustrated through some simple examples. The non-uniqueness of the defined unitary transformation is discussed.

If A = D + E where D is the matrix of diagonal elements of A, then when A has some multiple or very close eigenvalues, E has certain characteristic properties. These properties are considered both for hermitian and non-hermitian A. The properties are important in connexion with several algorithms for diagonalizing matrices by similarity transformations.

The overdamped eigenvalue problem (ω2A + ωB + C)ξ = 0 with A, B, and C Hermitian operators on and into the n-dimensional complex Hilbert space En, A > 0, (η, Bη)2−4(η, Aη)(η,Cη)> 0 for η ≠ 0, is shown to be reducible to a linear selfadjoint eigenvalue problem in En × En.

In (1) Fan showed that if A is a Hermitian matrix with eigenvalues λ1 ≤ … ≤ λn then, for k ≤ n,,,where X1, … , xk run over all sets of k orthonormal (o.n.) vectors in unitary n-space V.

Using the ideas of information theory, it is pointed out that the Gaussian ensemble for random Hermitian matrices can be characterized as the ``most random'' ensemble of these matrices. Pursuing the same characterization for positive matrices, we are led in a natural way to the definition of the exponential ensemble. Transforming to a representation of eigenvectors and eigenvalues, the joint distribution function for the eigenvalues of positive Hermitian matrices is found for this ensemble. The asymptotic single‐level density formula is derived, using a semiclassical model. It is found that the density is convex from below over most of the domain of eigenvalues. Since this is similar to the exponential dependence expected for nuclear spectra, the density is examined in a region near the level taken to correspond with the lowest nuclear level. It turns out, however, that the density is concave from below near this level, and that a large number of levels are contained in this concave region. Hence the exponential ensemble does not fairly represent nuclear energy levels, at least in this region. Various changes are made in the measure on the matrix ensemble to determine to what extent the level density depends on this measure. It is seen that the level density graph retains a characteristic shape for a wide variety of measures. The relationship of the limiting behavior of the level density for positive matrices to the semicircle law is noted.

In this paper we study the distribution of eigenvalues for two sets of random Hermitian matrices and one set of random unitary matrices. The statement of the problem as well as its method of investigation go back originally to the work of Dyson [i] and I. M. Lifsic [2], [3] on the energy spectra of disordered systems, although in their probability character our sets are more similar to sets studied by Wigner [4]. Since the approaches to the sets we consider are the same, we present in detail only the most typical case. The corresponding results for the other two cases are presented without proof in the last section of the paper. §1. Statement of the problem and survey of results We shall consider as acting in iV-dimensiona l unitary space ///v, a selfadjoint operator BN (re) of the form

Let be a hermitian matrix in partitioned form. Let the eigenvalues of A, B, C be α1 ≥ … ≥ αa, β1 ≥ … ≥ βb, γ1 ≥ … ≥ γn, respectively. In this paper four classes of inequalities are proved comparing the αi. and βj with the γk. The simplest of these is: if the subscripts is, js satisfy 1 ≤ i1 < … < im ≤ α, 1 ≤ j1 < … < jm ≤ b.

The kernel of the integral equation for the nonrelativistic scattering of a spinless particle by a potential —λV can easily be symmetrized, if V has a definite sign. Under suitable conditions for the potential the symmetrized kernel is square integrable and generates a completely continuous transformation of the space L2 that has a pure point spectrum.The scattering kernel is a two‐valued function of the energy s. For negative real values of s in the first Riemann sheet it is symmetric (Hermitian) and can be spectral decomposed in the usual way. Eigenvalues and eigenelements can be analytically continued, at least into the first s sheet and even into the finite second s sheet, if the potential decreases faster than any exponential function for r → ∞. Eigenelements at two complex conjugate points s, s* form a complete biorthogonal system of L2. The original and the resolvent kernel can be expressed in terms of that system and the eigenvalues, for s in the first and eventually finite second sheet.If the pot...

The set of integral equations describing the molecular scattering process developed in the previous papers of this series is reformulated as a set of differential equations, in which the coupling is associated with a Hermitian matrix. Exact eigenvectors and eigenvalues of this matrix are developed. These eigenvectors may be used as the basis of a numerical solution of the equations. They also lead to an approximate solution which is valid when the difference of the wavenumbers in the entrance and exit channels is small. The approximate solution may be considered as the lowest‐order term in a series development.

The eigenvalues λi of an Hermitian matrix A of order n satisfy the inequality ∑ i = 1n | λi | ≤ n1/2N, where N is the Frobenius norm of A. Let n and ν be, respectively, the numbers of atoms and bonds in the conjugated system of a hydrocarbon, and let E be the corresponding Huckel matrix. Then the Huckel π‐electron energy is nα + eβ, where [2ν + n(n − 1)| detE |2/n]1/2 ≤ e ≤ (2nν)1/2. Analogous bounds are obtained for π‐electron energies calculated by Wheland's method. An approximation for the Huckel π‐electron delocalization energy (DE) in the closed‐shell ground state of a hydrocarbon is DE / β = an1/2 − n + r, where a ≈ 1.30, and r = 0 or 1, according as n is even or odd.

The aim of this paper is to determine the location of eigenvalues of a general complex matrix without explicitly computing them. Essentially, the method consists in finding a Hermitian matrixH which has the same inertia as the given matrixM, and determining the inertia ofH. H is found as the solution of the Liapunov equation $$H\tilde M + \tilde M^* H = 2I$$ , where $$\tilde M$$ is a matrix in upper Hessenberg form which is similar toM. The method remains applicable even if the solution of the Liapunov equation is affected by rounding errors, as long as a given criterion is satisfied. A programmed version of this method has been run on our computer with satisfactory results.

Many existing algorithms for obtaining the eigenvalues and eigenvectors of matrices would make poor use of such a powerful parallel computer as the ILLIAC IV. In this paper, Jacobi's algorithm for real symmetric or complex Hermitian matrices, and a Jacobi-like algorithm for real nonsymmetric matrices developed by P. J. Eberlein, are modified so as to achieve maximum efficiency for the parallel computations. 1. Introduction. With the advent of parallel computers, the study of compu- tationally massive problems became economically possible. Such problems include, for example, solution of sets of partial differential equations over sizable grids, and multiplication, inversion, or determination of eigenvalues and eigenvectors of large matrices. An example of a parallel computer is the ILLIAC IV.* This computer is es- sentially an array of coupled arithmetic units driven by instructions from a common control unit. Each of the arithmetic units, called processing elements (PE's), have 2048 words of 64-bit memory with an access time under 420 nanoseconds. Each PE is capable of 64-bit floating-point multiplication in about 550 nanoseconds. Two 32-bit floating-point operations may be performed in each PE in approximately the same times. The PE instruction set is similar to that of conventional machines with two exceptions. First, the PE's are capable of communicating data to four neigh- boring PE's by means of routing instructions. Second, the PE's are able to set their own mode registers to effectively disable or enable themselves. For a more detailed

A Hermitian matrix which occurs in the theory of the quantum-mechanical inverse-scattering problem has apparently contradictory properties. It has a well-behaved inverse in spite of having zero as one of its eigenvalues. The properties of the matrix are investigated and the relevance of the results to the theory are discussed.

The minimax theorem for the eigenvalues of Hermitian operators is reviewed and its inverse (maxmini) theorem is derived. The connection between the Hylleraas-Undheim theorem (HUT) and the minimax theorem is clarified, and an extension of the HUT to compound resonance states is obtained. Thus, rigorous upper bounds on the resonance energies may be obtained without the explicit use of the exact open-channel projection operators. The exchange and rearrangement problems are also considered.

Let C = (Aij)1≤i, j≤t be a hermitian matrix in partitioned form; here Aij, is an ni × nj. block. The purpose of this paper is to obtain inequalities linking the eigenvalues of C to those of the main diagonal blocks A11, …, Att of C.These inequalities include, as special cases, inequalities due to N. Aronszajn and A. Hoffman.

A generalization of quantum mechanics different from the theory proposed recently by Phipps is considered. Difficulties connected with the commutation relations of the angular momentum components defined with the help of the operators pi of the fundamental equations of Phipps are presented as new arguments in favor of the choice of Phipps of the Hermitian operators to be used in his generalized theory. The application of the equations proposed in the present paper to the motion of a particle in a central Coulombic field of forces is considered. In the Dirac theory the spectrum is different from the traditional one. As the Heisenberg postulate is now obeyed, this result shows that an unusual spectrum is not a consequence of the violation of the postulate.

An operator A† that satisfies [H, A†] = h/ωA† converts a stationary‐state eigenfunction of the Hamiltonian H into another eigenfunction with energy eigenvalue increased by h/ω. Such operators describe collective excitations of many‐particle systems, and their properties can be used to construct an intrinsic Hamiltonian that is dynamically independent of the collective degrees of freedom, without introducing subsidiary conditions. The procedure developed by Lipkin, valid when h/ω is real and positive, is extended to make possible the construction of an intrinsic Hamiltonian when h/ω vanishes and A† is Hermitian, and also when h/ω is complex. The nuclear cranking model is shown to be a special case of the proposed general method for vanishing h/ω in which effective moments of inertia occur as eigenvalues of linear equations. Several examples are worked out in detail, all dealing with an interacting phonon‐electron system in the random‐phase approximation. Results derived are the explicit screened Coulomb in...

Using an independent derivation by Kohn, the full meaning of Kato's formulas for upper and lower bounds to eigenvalues of a Hermitian operator is shown. These bounds are the best possible when the only information available on a particular eigenvalue problem is a suitable trial function and an estimate of the neighboring eigenvalues to the one in question. This was asserted by Kato but not proved. A comparison is made of Kato's bounds with those derived in papers by Stevenson and Crawford and by Cohen and Feldmann. Under the conditions which result in Kato's bounds it is shown that the Stevenson–Crawford and Cohen–Feldmann bounds reduce to those of Kato. When more information is available these bounds are an improvement upon Kato's. This makes more precise the recent work of Walmsley and Cohen–Feldmann, whose results appear to prove in general the greater accuracy of the Stevenson–Crawford and Cohen–Feldmann bounds over those of Kato. A general discussion of all three sets of bounds is given in terms of the parameter λ appearing in the Stevenson–Crawford formulation.

The solution to the optimization of performance indices of array antennas such as directive gain, efficiency index, and signal-to-noise ratio, do not provide information regarding the sidelobe region of the radiation pattern. It is shown that, with proper constraints on the sidelobes, a given performance index can be optimized to give a radiation pattern with desired sidelobe levels. As most of the performance indices of an array antenna can be expressed as a ratio of two Hermitian quadratic forms, an eigenvalue method is used for the constrained optimization. This method gives explicit expressions for the excitation vector and constrained values of the performance indices. An iterative technique is used to ensure that the specified field values occur at the sidelobe peak positions. The element excitations obtained by this technique for maximum gain and uniform sidelobe level are similar to these obtained by the Dolph-Chebyshev technique.

A new proof of a theorem byWielandt for inclusion of eigenvalues of Hermitian matrices is given. Relations between that inclusion method and the Mohr stress diagram are investigated.

Let 3C be hilbert space (any dimensionality, real or complex scalars). Let P be a hermitian projection. Let A be any hermitian operator. The compression of A to P3C [3] is PAP, considered as an operator on P3R. Compressions of completely continuous positive operators are of interest in connection with estimating eigenvalues: the Fischer-Courant nminimax theorem [5, p. 2351 says the kth highest eigenvalue of PAP is not greater than that of A. Compressions enter in the study of more general mappings of operators, often via Nalmark's theorem [6]. Especially in the first connection, the case where PJC is finitedimensional is interesting. But in some problems a finite-dimensional subspace may be known, not via the operator P, but via an arbitrary set of vectors which span it; if they are not orthonormal, one would rather not have to find P. This suggests that the following elementary formulas may be worth pointing out. I suppose that at least Formula 1 must be known already, but not, apparently, very widely. NOTATION. xi, * ,X form a linear basis of P3R. G denotes the determinant of their Gramian (n X n matrix with i, j entry (xi, xj)). If the kth row of the Gramian is replaced by (z, xi), . . . , (z, x.), all other rows being left unchanged, the determinant of the resulting matrix will be denoted G(xk; z). Evident properties: G(xk; z) =0 if z= (1 -P)z or z=xi (i%k), while G(xk; Xk) =G; also G(xk; z) is linear in z. These may be summed up by saying that G(xk; z) = (z, x)G, where {x1, *, xI} is the basis of P3R biorthonormal with { xi, n, x FORMULA 1. Pz=G-1 Gk G(xk; Z)Xk. (This notation here and below means summation over all available values of the index.) PROOF. Uniquely z= Es aix+(1 -P)z. Substitute this on both sides, and use the evident properties of G(xk; z). FORMULA 2. tr(PAP) =G ZEk G(xk; Axk). PROOF. Let 1, * * * , in be orthonormal eigenvectors of PAP, and Xi, * * * 9 Xn their respective eigenvalues; then xi= E, Ti,ip, where T is some nonsingular matrix. Recall that

The Bethe-Salpeter equation is investigated under some general assumptions, without using specific approximations. We assume 1) the validity of integral representations for the interaction kernel and eigenfunctions, 2) the permissibility of the Wick transform, 3) the absolute boundedness of the transformed kernel and 4) the kernel to be of Hilbert-Schmidt type. In the rest frame the kernel is found to be a real analytic function of P4 in a strip region of the complex P4 plane, where P4 is the total energy multiplied by i. The eigenvalue problem for the hermitian analytic kernel is discussed. As a consequence, the sign of the norm of bound states represented by eigenfunctions with real eigenvalues for total energies in a real interval, is shown to be determined by three factors. This is a generalization of a result known in the equal-mass ladder models.

It is shown that in the Born-von Karman theory of lattice dynamics the dynamical matrix may, under particular conditions, be non-hermitian for low symmetry crystals. Normal modes can then be introduced by using both right and left eigenvectors of the matrix. The hermitian character of the matrix can be maintained by imposing constraints on the interatomic force constants.

The random phase approximation yields a description of collective nuclear excitation in terms of an eigenvalue problem involving a non-hermitian matrix. It is shown that there exists a related hermitian matrix H, satisfying the standard eigenvalue problem Hs = hs, whose solution directly yields the solution of the original problem.

By generalizing the relation TrHp = ΣλEλp between a Hermitian matrix H and its eigenvalues Eλ, moment relations are obtained for the energy spectrum of a non-relativistic particle undergoing elastic s-wave scattering by a spherically symmetric potential V(r). The energy moments of the phase shift δ and dδ/dE are obtained in terms of the potential V(r) and its radial derivatives. ¦V(r)¦ must be integrable, and V(r) non-singular and differentiable any number of times; its odd derivatives must vanish at the origin. The high-energy divergence of the energy moments is removed by means of an asymptotic expansion. The moment relations for dδ/dE are provided. General expressions for ap and bp are provided. The results of Dikii (1961) on the Sturm-Liouville problem are thus extended to systems with a continuous or mixed spectrum. Note added in proof. These moment relations have been obtained by analytic continuation by Buslaev and Faddeev in 1960, and extended to three dimensions by Buslaev in 1962. The connection with the Kirkwood-Wigner expansion is apparently new.

As for general kernels p(x), little has been proved but much has been suspected. Bellman and Latter [1] have obtained upper and lower bounds for the largest eigenvalue of (1.1) for quite general p, these bounds, however, being rather weak for most interesting kernels. More to the point is a conjecture of Kac, Murdock, and Szego [5]. Assume p(x) > 0 end f0(1 +x2)p(x)dx< c. Denote the positive eigenvalues of (1.1) by

The model in which a charged, elastically bound point electron interacts with the dipole part of the radiation field is studied within the framework of the new Tamm-Dancoff method. It is shown that assumption of the existence of eigenstates forces the conclusion that the Hamiltonian is not hermitian. Since this behaviour is shared by the Lee model when the coupling constant exceeds the critical value, the difficulties of the two models may well have a common origin. It is shown that, if imaginary values for energy differences are disregarded, the N.T.D. method gives the same energy shifts and phase shifts as the method of transformation to normal coordinates.

The purpose of this paper is to extend a well known theorem of A.M. Liapunov concerning Hurwitzian matrices (n x n matrices with eigenvalues in the half plane Re z < 0) to strongly continuous semi-groups of operators on a complex Hilbert space. Liapunov’s result may be phrased as follows: Let A be a complex n x n matrix and A* its adjoint. Then A has all its characteristic roots lying in the half plane Re z < 0 if and only if the solution in B of the matrix equation A*B + BA = -I (the identity matrix) is a unique positive definite Hermitian matrix (see e.g. ([I] p. 245)). A partial extension of Liapunov’s theorem has been given in [2]. This paper is a sequel to that paper in the sense that much sharper results than those in [2] are obtained if we restrict ourselves to Hilbert spaces instead of general Banach spaces.

This paper considers the sensitivity of the eigenvalues and eigenvectors of the generalized matrix eigenvalue problem $Ax = \lambda Bx$ to perturbations of A and B. The notion of a deflating subspace for the problem is introduced, and error bounds for approximate deflating subspaces obtained. The bounds also provide information about the eigenvalues of the problem. The resulting perturbation bounds estimate realistically the sensitivity of the eigenvalues, even when B is singular or nearly singular. The results are applied to the important special case where A is Hermitian and B is positive definite.

Das Iterationsverfahren von Jacobi dient bekanntlich dazu, gleichzeitig samtliche Eigenwerte und Eigenvektoren einer hermiteschen Matrix zu approximieren. Hier wird fur den Fall lauter einfacher Eigenwerte die Differenz der exakten und genaherten Eigenvektoren abgeschatzt, und zwar sowohl fur das klassische als auch fur das spezielle zyklische Jacobi-Verfahren. Fur diese Abschatzung wird in Satz 1 der kleinste Abstand zweier Eigenwerte der vorgelegten Matrix benotigt. In Satz 2 dagegen kommt nur der kleinste Abstand zweier Diagonalelemente derjenigen Matrix vor, die im Laufe des Jacobi-Verfahrens diagonalisiert wird. As is well known, the iterative Jacobi method is used to approximate simultaneously all eigenvalues and eigenvectors of a hermitian matrix. The purpose of this paper is to find bounds for the difference between the exact and the approximated eigenvectors in the case of distinct eigenvalues. The classical Jacobi method is treated as well as the special cyclic one. In theorem 1 the smallest distance between two eigenvalues is used. In theorem 2 only the smallest distance between two diagonal elements of the matrix is used, which is diagonalized by the Jacobi method.

We prove the existence of a transformation operator with a condition at infinity that sends a solution of the matrix equation−y″ + My=λ2y (M is a constant Hermitian matrix) into a solution of the matrix equation−y″+Q(x)y+My=λ2y (the matrix function Q(x) is continuously differentiable for 0 ≤ x