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Quadrupole mass filters are normally operated as narrow band pass filters by appropriate choices of rf and DC voltages corresponding to Mathieu a and q values near the apex of the first stability region. We add an auxiliary quadrupole excitation potential to the main drive voltage. As a result, stability islands appear on the ([Formula: see text]) plane. The method of the islands mapping on the ([Formula: see text]) plane is discussed in detail. The DC electric field's effect responsible for removing “shadowing” islands has been studied using a combination of analytic theory and computer modeling. We call a narrow stability region elongated along an iso-[Formula: see text] line an X-island. Two such stability islands were found where operation is possible without mass spectra interference when the DC potential is used. Those islands are formed by quadrupole potentials with relative excitation frequencies [Formula: see text] and [Formula: see text], where [Formula: see text] Many other stability X-islands are presented in detail and illustrated by their transmission contours. This data is necessary for experimental testing of the separation mode at those islands. The study demonstrates how to use X-islands to achieve relatively high resolution 4000-12,000 at the transmission of about 28-15%, respectively.

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The results of modeling AC and DC dipole excitation of ion oscillations in a quadrupole mass filter are presented. The simulation is done by numerical integration of the ion motion equations, ions’ initial coordinates and velocities are distributed normally. For AC dipole excitation the instability bands on the ( a, q) stability diagram follow along the isolines [Formula: see text] and [Formula: see text], creating regular dips on the transmission contour. We show that AC excitation at frequency Ω/2 makes it possible to control the resolution of the mass filter by either changing the AC amplitude or phase. Instability bands can be used for mass selective excitation in a linear ion trap. Options for the joint use of DC and AC dipole excitation to form the mass peak shape are considered.

The solvability of the Cauchy problem for second-order parabolic equations in the strip D = R n × (0, T ) in anisotropic Hölder spaces C l,α D is well known (e.g., see [1] for l ≥ 2 and [2] for l = 0, 1). In these papers, it was assumed that α ∈ (0, 1). The Cauchy problem for second-order parabolic equations in the Sobolev spaces W 2l,l p (D), 1 < p < ∞, was considered in [1]. In the anisotropic spaces C l D (α = 0) and the Lipschitz anisotropic spaces C l−1,1 D (either α = 1 or p = ∞), theorems similar to the assertions in the above-mentioned papers are not valid. For example, if f is a continuous function in the domain Ω, then the (generalized) solution of the heat equation Lu ≡ u t − ∆u = f does not necessarily belong to C 2 (Ω) [3, p. 385]. This is a general fact. It was shown in [4, p. 296 of the Russian translation] that for each differential operator P m of order m ≥ 1 with constant coefficients in R n , n ≥ 2, there exists a function f ∈ C (R n ) such that there exists a generalized solution u of the equation P m u = f in R n and u ∈ C m−1 (R n )\C m (R n ). Moreover, f and u can be chosen so as to ensure that for each multi-index k, |k| = m, the derivative ∂ k x u is not continuous unless P m is a multiple of ∂ k x . For the solution of the heat equation to be classical, it is sufficient that the right-hand side f satisfies the Dini condition [5]. It was shown in [6] that this condition is, in a sense, necessary: for any modulus of continuity not satisfying the Dini condition, there exists a continuous function f such that the modulus of continuity of f does not exceed the given modulus of continuity and the equation Lu = f has a generalized solution with unbounded second spatial derivatives. Consequently, the solution of the heat equation with a continuous right-hand side does not necessarily belong even to the anisotropic Lipschitz space C 1,1 (Ω) = W 2,1 ∞,loc (Ω). In the present paper, we consider the Cauchy problem in the anisotropic Zygmund spaces H l D . For solutions of the heat equation, these spaces are, in a sense, the natural "closure" of the scale of anisotropic Hölder spaces C l,α D in the case of integer smoothness exponents (α = 0) and of the Sobolev-Slobodetskii spaces W l,l/2 p