Proof of the completeness of the system of eigenfunctions for one boundary-value problem for the fractional differential equation

1.   Introduction

The problems of completeness of eigenfunctions and associated functions of the operators, generated by the ordinary differential expressions of fractional order and boundary conditions of Sturm-Liouville type are in the focus of many researchers. This mainly deals with the fact that such problems arise in solving boundary value problems for fractional differential equations for advection-diffusion using the method of separation of variables. In present paper, we resolve this very important problem by well-known Livshits theorem on spectral decomposition of linear nonself-adjoint operators.

2.   Main Results

In paper ^[1] was studied operator in the space $L_{2}(0, 1)$

which was first considered in ^[2,3], where $0 < \rho < 2$ and

is the Green function of the following problem $S$ (with $\lambda = 0$):

($n-1\leq\rho^{-1} < n, \,$ $n = [\rho^{-1}]+1$, where $[\rho^{-1}]$ is the integer part of $\rho^{-1}$)

In this case ^[1,4], if $\gamma_{0} = \gamma_{1} = \cdots = \gamma_{n} = 1$, then problem $S$ takes the form

Its Green function $G(x, t)$ (for $\lambda = 0$) reads

In particular, in this paper we provide very important proof of the completeness of the system of eigenfunctions and associated functions in $L_{2}$ of the operator $A_{\rho}$ for $0 < \rho < 2$ (this fact plays main role in solving boundary value problems for advection-diffusion equation of fractional order by the method of separation of variables ^[5], based on well-known Livshits theorem ^[6].

Theorem (Livshits):

If $K(x, y)$ $(a\leq x, y\leq b)$ – is a limited kernel, and "real part" $\frac{1}{2}(K+K^*)$ of it is non-negative kernel, then the inequality is hold

where $\lambda_{j}$ – is the characteristic numbers of kernel $K$. The system of main functions of the kernel $K$ is complete in domain of values of the integral operator $Kf$ if and only if, when there is an equal sign in inequality above.

In his paper ^[7] M. M. Dzhrbashian wrote, that "the question about the completeness of the systems of eigenfunctions of the operator $A_{\rho}$ or a finer question about whether these systems compose a basis in $L_{2}$, has a certain interest but its solving is apparently associated with significant analytic difficulties". The questions of the completeness of the systems of eigenfunctions and associated functions for similar problems were studied by A. V. Agibalova in ^[8,9]. Undoubtedly, we shall note the fundamental results of M. M. Malamud and L. L. Oridoroga ^{[10,11,12,13]}, obtained in this direction. In ^[14,15] (see also ^[2]), using the theorem of Matsaev and Palant, it was established that the system of eigenfunctions of the operator $A_{\rho}$ is complete in $L_{2}$.

As noted above, in this paper, a similar result was obtained using the well-known Livshchits theorem ^[6].

The following proof of the completeness of the system of eigenfunctions is simpler than the previously presented proofs, which makes the results of this paper very significant.

Now we give the main result of paper.

Theorem 1. The system of eigenfunctions and associated functions of the operator $A_{\rho}$, where $0 < \rho < 2$, is complete in $L_{2}$.

Proof. Let us designate the kernel of $A_{\rho}$ as $K(x, y)$. In ^[14] the authors have proved that this kernel is non-negative by the following way: Let us rewrite $A_{\rho}$ as

Clearly, for $\rho > 1$, the kernel of $A_{\rho}$ is non-negative.

By the same way, we may show that the kernel $K^*(x, y)$ for adjoint operator

is non-negative too. Thus $\frac{1}{2}(K+K^*)$ is non-negative. Let us show that the following expression holds

From ^[14], we know that the value $\lambda_{j}$ is an eigenvalue of the operator $A_{\rho}$ if and only if $\lambda_{j}$ is a zero of the function $E_{\rho}(\lambda_{j}; \frac{1}{\rho})$, where ^[7]

The asymptotics of zeros for function $E_{\rho}(\lambda_{j}; \frac{1}{\rho}) = 0$ is well known. In particular, we have the following well-known Dzhrbaschian-Nersisian lemma [16, p.142].

Lemma (Dzhrbaschian-Nersisian):1. All zeros of functions $E_{\rho}(z; \mu)$ (where $\rho > \frac{1}{2}, \rho\neq 1; Im \mu = 0$) with largest absolute values, are prime.

2. The following asymptotic formulas are valid

So, if $\lambda_{j} = \alpha_{j}+i\beta_{j}$ is an eigenvalue of the operator $A_{\rho}$, the adjoint number $\overline{\lambda}_{j} = \alpha_{j}-i\beta_{j}$ will be an eigenvalue of the operator $A_{\rho}$. Therefore

To find the trace $sp A_{\rho}$ of the operator $A_{\rho}$, let's rewrite $A_{\rho}$ as $A_{\rho}u = A_{1}u-A_{0}u$ where

Clearly, for $0 < \rho < 2$, the operators $A_{0}$ and $A_{1}$ are trace class. Hence

Moreover, it's clear that $sp(A_{0}) = 0.$ Thus

Since operator $A_{1}$ is one-dimensional, it's easy to find a trace. Consider the equation

The Fredhold determinant

where

From above follow that

which proves the Theorem 1.

Remark. Since the operator $A_{\rho}$ doesn't generate associated functions ^[17], we proved that the system of functions

is complete in $L_{2}$ (but the system of these functions, unfortunately, is not orthogonal).

Similarly, it is possible to provide a spectral analysis of the operator

considered in ([14], see the references therein).

Theorem 2. Let $0 < \rho < 2$, $\alpha < \frac{1}{\rho}$. Then, the system of eigenfunctions and associated functions of the operator $A^{[\alpha^{-1}, \rho]}_{\rho}$ is complete in $L_{2}$.

Proof. We carry out the proof of Theorem 2 in the same way as the proof of Theorem 1. It can be easily shown that the kernel $M(x, t)$ of the operator $A^{[\alpha^{-1}, \rho]}_{\rho}$ is non-negative. Elementary calculations show that the kernel $M^{*}(x, t)$ of the operator adjoint to the operator $A^{[\alpha^{-1}, \rho]}_{\rho}$ will be non-negative too. Thus $\frac{1}{2}(M+M^*)$ will be non-negative too. The fact that

where $\mu_{j}$ are eigenvalues of the operator $A^{[\alpha^{-1}, \rho]}_{\rho}$, shown in the same way as in Theorem 1.

3.   Conclusion

In present paper, we provide the proof of the completeness of eigenfunctions and associated functions of the operators, generated by the ordinary differential expressions of fractional order and boundary conditions of Sturm-Liouville type.

Conflict of interest

The authors declare no conflict of interest.