Ulam stability for nonautonomous quantum equations

*Correspondence: onitsuka@xmath.ous.ac.jp 2Department of Applied Mathematics, Okayama University of Science, Okayama, 700-0005, Japan Full list of author information is available at the end of the article Abstract We establish the Ulam stability of a first-order linear nonautonomous quantum equation with Cayley parameter in terms of the behavior of the nonautonomous coefficient function. We also provide details for some cases of Ulam instability.

for q > 1. In this paper, we consider the nonautonomous Cayley quantum equation D q z(t) = α(t) z(t) β , t ∈ q N 0 , (1.1) where α(t) is a complex-valued time-varying coefficient, the q-difference operator is D q z(t) := z(qt)z(t) (q -1)t , q > 1, and the Cayley component is If β = 0, then the Cayley quantum equation reduces to the mere quantum equation It is well known that D q z(t) → z (t) as q 1, so we can say that the quantum equation is an approximate equation of the differential equation z (t) = α(t)z(t). Notice that equation (1.1) can be rewritten as This formula shows that the condition 1β(q -1)tα(t) = 0 = 1 + (1β)(q -1)tα(t) for t ∈ q N 0 (1.2) is necessary to keep the recurrence viable. For this reason, we assume this condition throughout this paper. For any ε > 0 and for any function ζ satisfying there is a solution z of (1.1) such that ζ (t)z(t) ≤ Cε for t ∈ q N 0 .
We call such C a Ulam constant for (1.1) on q N 0 .
The paper will proceed as follows. In the next section, we highlight the q-difference (quantum) exponential function and its properties and provide details on the solution to the related nonhomogeneous equation. In Sect. 3, we establish our main result, the Ulam stability of (1.1). In Sect. 4, we show some conditions under which (1.1) is Ulam unstable.

Exponential function and its properties
In this section, we introduce the exponential function of equation (1.1). Define e α (t) := log q t-1 for t ∈ q N 0 ; note that for a function f , we define which is the standard definition. We immediately have the following lemma.
Proof It is clear that e α (1) = 1. Now we will show that e α (t) solves (1.1). Substituting it into the left side of (1.1) gives On the other hand, substituting e α (t) into the right side, we get Hence e α (t) solves equation (1.1). By e α (1) = 1 we have z(1) = z 0 . From the linearity of the solutions of linear equations we can conclude that z(t) = z 0 e α (t) is also a solution of (1.1). This completes the proof.
Needless to say, the function e α (t) as defined above will play the role of the exponential function in q-difference equations. Define . (2. 2) The following lemma holds according to the method of variation of parameters.
is the general solution of (2.3).
Proof Let ζ (t) := η(t)e α (t) for t ∈ q N 0 , where η(t) is an unclear function here. We assume that ζ (t) is a solution of (2.3). Noting that we have

This implies
for t ∈ q N 0 . Hence the solution of this equation is inductively obtained in the following form: Conversely, it satisfies the above equation. Indeed, we can check that for t ∈ q N 0 . If we go back to the above calculation, we can see that ζ (t) = η(t)e α (t) is the solution of (2.3) with ζ (1) = z 0 . Hence we have η(t) ≡ z 0 + γ (t), and this completes the proof.

Ulam stability
The main Ulam stability result of this paper is as follows.

Ulam instability
What happens if the coefficient function α fails to satisfy (2.4)? In the following example, we show an example where (1.1) is unstable in the Ulam sense.

Conclusions
Using the properties of the exponential function for nonautonomous Cayley quantum equations, we established sufficient conditions for the Ulam stability of quantum equations with a variable coefficient under the assumptions that the Cayley parameter satisfies β = 1 2 and the absolute value of the variable coefficient does not approach zero. After that, these assumptions are elaborated. The situation is clarified by presenting an example where Ulam stability breaks down if the absolute value of the variable coefficient approaches zero. If the coefficient is a constant, it has already been shown in [8] that β = 1 2 means the Ulam instability, but with the variable coefficient, something interesting happens, that is, if the absolute value of the variable coefficient increases, the Ulam stability is derived. Therefore it turns out that both Ulam stable and unstable cases may occur for β = 1 2 . In this way, we found in this study that by considering variable coefficients there is a problem of balance between stability and instability, which does not appear in the case of constant coefficients.