Spectral asymptotics of self-adjoint fourth order boundary value problems with eigenvalue parameter dependent boundary conditions

Sturm-Liouville problems have attracted extensive attention due to their intrinsic mathematical challenges and their applications in physics and engineering. However, apart from classical Sturm-Liouville problems, also higher order ordinary linear differential equations occur in applications, with or without the eigenvalue parameter in the boundary conditions. Such problems are realized as operator polynomials, also called operator pencils. Some recent developments of higher order differential operators whose boundary conditions depend on the eigenvalue parameter, including spectral asymptotics and basis properties, have been investigated in [1–4]. General characterizations of self-adjoint boundary conditions have been presented in [5, 6] for singular and (quasi-)regular problems. In all these cases, the minimal operator associated with an n th order differential equation must be symmetric, see [7, 8] for necessary and sufficient conditions. A more general discussion on the spectra of fourth order differential operators can be found in [9, 10].

depends quadratically on the eigenvalue parameter. This problem is represented by a quadratic operator pencil, in a suitably chosen Hilbert space, whose coefficient operators are self-adjoint. In [13] we have investigated a class of boundary conditions for which necessary and sufficient conditions were obtained such that the associated operator pencil consists of self-adjoint operators, while in [14] we have continued the work of [13] in the direction of [12] to derive eigenvalue asymptotics associated with boundary conditions which lead to self-adjoint operator representations. We have considered the particular case of boundary conditions which do not depend on the eigenvalue parameter at the left endpoint and depend on the eigenvalue parameter at the right endpoint.

In this paper, we extend the work of [14] to a class of boundary conditions where exactly one of the left endpoint boundary conditions does not depend on the eigenvalue parameter, while the remaining boundary conditions depend on the eigenvalue parameter.

We define the operator pencil in Section 2 and we discuss which boundary conditions are considered. In Section 3, the eigenvalue asymptotics for the case $g=0$ are derived. In Section 4, it is shown that the boundary value problems under consideration are Birkhoff regular, which implies that the eigenvalues for general g are small perturbations of the eigenvalues for $g=0$. Hence, in Section 5, the first four terms of the eigenvalue asymptotics are found and are compared to those obtained in [14].

On the interval $[0,a]$, we consider the boundary value problem

(2.1)

(2.2)

see [[8], p.26].

Define

Assumption 2.1 The numbers $p_1$, $p_2$, $q_j$ for $j\in {\mathrm{\Theta }}_1^0$ are distinct as well as the numbers $p_3$, $p_4$, $q_j$ for $j\in {\mathrm{\Theta }}_1^a$.

in the space $L_2(0,a)\oplus {\mathbb{C}}^k$ with the problem (2.1), (2.2).

The conditions under which the differential operator $A(U)$ is self-adjoint are given in

Theorem 2.2 ([13], Theorem 1.2)

Denote by $P_0$ the set of p in $y^{[p]}(0)=0$ for the λ-independent boundary conditions and by $P_a$ the corresponding set for $y^{[p]}(a)=0$. Then the differential operator $A(U)$ associated with this boundary value problem is self-adjoint if and only if $p+q=3$ for all boundary conditions of the form $y^{[p]}(a_j)+i\alpha {\epsilon }_j\lambda y^{[q]}(a_j)=0$ and ${\epsilon }_j=1$ if q is even in case $a_j=0$ or odd in case $a_j=a$, ${\epsilon }_j=-1$ otherwise, $\{0,3\}\not\subset P_0$, $\{1,2\}\not\subset P_0$, $\{0,3\}\not\subset P_a$ and $\{1,2\}\not\subset P_a$.

Proposition 2.3 The operator pencil $L(\cdot ,\alpha )$ is a Fredholm valued operator function with index 0. The spectrum of the Fredholm operator $L(\cdot ,\alpha )$ consists of discrete eigenvalues of finite multiplicities, and all eigenvalues of $L(\cdot ,\alpha )$, $\alpha \ge 0$, lie in the closed upper half-plane and on the imaginary axis and are symmetric with respect to the imaginary axis.

Proof As in [[12], Section 3], we can argue that for all $\lambda \in \mathbb{C}$, $L(\lambda ,\alpha )$ is a relatively compact perturbation of $L(0,0)$, where $L(0,0)$ is well known to be a Fredholm operator. The statement on the location of the spectrum now follows as in [[12], Lemma 3.1]. □

We now consider the particular cases that exactly one of the boundary conditions at 0 depends on λ, whereas both boundary conditions at a depend on λ. Therefore, taking Assumption 2.1 and Theorem 2.2 into account, we have the four boundary conditions

where $0\le p_1\le 3$, $0\le p_2\le 3$, $0\le q_2\le 3$, $p_2+q_2=3$, and $p_1\notin \{q_2,p_2\}$, while $\{p_3,q_3\}=\{1,2\}$ and $\{p_4,q_4\}=\{0,3\}$. Thus, we have 8 and 4 possible sets of boundary conditions at the endpoint 0 and a, respectively. Whence there are 32 different sets of boundary conditions. Recall that the parameter λ emanates from derivatives with respect to the time variable in the original partial differential equation, and it is reasonable that the highest space derivative occurs in the term without time derivative. Thus, the most relevant boundary conditions would have $q_2<p_2$, $q_3<p_3$ and $q_4<p_4$. This leaves us with four different cases for the boundary conditions $B_j(\lambda )y=0$.

These four cases are uniquely determined by the value of $p_1$, so that we will consider

Case 1: $p_1=3$; Case 2: $p_1=0$; Case 3: $p_1=1$; Case 4: $p_1=2$.

The corresponding boundary operators are then

(2.5)

(2.6)

(2.7)

(2.8)

the eigenvalues of the boundary value problem (2.1), (2.2) are the eigenvalues of the analytic matrix function M, where the corresponding geometric and algebraic multiplicities coincide, see [[15], Theorem 6.3.2].

The second row of $M(\lambda )$ has exactly two non-zero entries (for $\lambda \ne 0$), and these non-zero entries are:

In Cases 1 and 2, $B_2(\lambda )y_2(\cdot ,\lambda )=-i\alpha {\mu }^2$ and $B_2(\lambda )y_3(\cdot ,\lambda )=1$;

In Cases 3 and 4, $B_2(\lambda )y_1(\cdot ,\lambda )=i\alpha {\mu }^2$ and $B_2(\lambda )y_4(\cdot ,\lambda )=1$.

For the above four cases, we obtain:

We next give the asymptotic distributions of the zeros of ${\varphi }_0(\mu )$ with proper counting.

simple zeros at $-{\tilde{\mu }}_k$, ${\tilde{\mu }}_{-k}=i{\tilde{\mu }}_k$ and $-i{\tilde{\mu }}_k$ for $k=3,4,\dots$, and no other zeros.

simple zeros at $-{\tilde{\mu }}_k$, ${\tilde{\mu }}_{-k}=i{\tilde{\mu }}_k$ and $-i{\tilde{\mu }}_k$ for $k=2,3,\dots$ , and no other zeros.

simple zeros at $-{\tilde{\mu }}_k$, ${\tilde{\mu }}_{-k}=i{\tilde{\mu }}_k$ and $-i{\tilde{\mu }}_k$ for $k=2,3,\dots$, and no other zeros.

simple zeros at $-{\tilde{\mu }}_k$, ${\tilde{\mu }}_{-k}=i{\tilde{\mu }}_k$ and $-i{\tilde{\mu }}_k$ for $k=2,3,\dots$, and no other zeros.

Proof The result is obvious in Cases 1 and 3. Cases 2 and 4 only differ in the factor with the power of μ, and the multiplicity of the corresponding zero of ${\varphi }_0$ at 0 is easy to verify. The choice of the indexing for the non-zero zeros of ${\varphi }_0$ in each case will become apparent later.

The location of the zeros on the other three half-axes follows by repeated application of ${\varphi }_0(i\mu )=-{\varphi }_0(\mu )$.

Putting $(1+i)\mu a=x+iy$, $x,y\in \mathbb{R}$, it follows for $\mu \ne 0$ that

(3.2)

is either real or pure imaginary. □

Proposition 3.2 For $g=0$, there exists a positive integer $k_0$ such that the eigenvalues ${\hat{\lambda }}_k$, counted with multiplicity, of the problem (2.1), (2.5)-(2.8), where $k\in \mathbb{Z}\setminus \{0\}$ in Cases 1 and  2 and $k\in \mathbb{Z}$ in Cases 3 and 4, can be enumerated in such a way that the eigenvalues ${\hat{\lambda }}_k$ are pure imaginary for $|k|<k_0$, and ${\hat{\lambda }}_{-k}=-\overline{{\hat{\lambda }}_k}$ for $k\ge k_0$. For $k>0$, we can write ${\hat{\lambda }}_k={\hat{\mu }}_k^2$, where the ${\hat{\mu }}_k$ have the following asymptotic representation as $k\to \mathrm{\infty }$:

In particular, the number of pure imaginary eigenvalues is even in Cases 1 and 2 and odd in Cases 3 and 4.

By periodicity, there are numbers $C_2(\epsilon )>0$ and $C_3(\epsilon )>0$ such that $|tan(\mu a)|<C_2(\epsilon )$ and $|cos(\mu a)|>C_3(\epsilon )$ for all $\mu \in R_{k,\epsilon }$ and all k. Observing $|cosh(\mu a)|\ge |sinh((Re\mu )a)|$, it follows that there is $k_2(\epsilon )\ge k_1(\epsilon )$ such that for all μ on the squares $R_{k,\epsilon }$ with $k>k_2(\epsilon )$ the estimate $|{\varphi }_1(\mu )|<1$ holds. Further, we may assume by Lemma 3.1 that ${\tilde{\mu }}_k$ is inside $R_{k,\epsilon }$ for $k>k_2(\epsilon )$ and that no other zero of ${\varphi }_0$ has this property. Hence, it follows by Rouché’s theorem that there is exactly one (simple) zero ${\hat{\mu }}_k$ of ϕ in each $R_k$ for $k\ge k_2(\epsilon )$. Replacing μ with iμ only changes the sign of the second term in (3.3) and thus the sign of ${\varphi }_1$. Hence, the same estimates apply to corresponding squares along the other three half-axes, and we therefore have that ϕ has zeros $\pm {\hat{\mu }}_k$, $\pm {\hat{\mu }}_{-k}$ for $k>k_2(\epsilon )$ with the same asymptotic behavior as the zeros $\pm {\tilde{\mu }}_k$, $\pm i{\tilde{\mu }}_k$ of ${\varphi }_0$ as discussed in Lemma 3.1.

Furthermore, we are going to make use of the estimates

(3.9)

(3.10)

which hold for all $k\in \mathbb{Z}$ with $|k|\ge {\hat{k}}_0$ and all $\gamma \in \mathbb{R}$. Therefore, it follows from (3.6), (3.8)-(3.10) and the corresponding estimates with μ replaced by iμ that there is ${\hat{k}}_1$ such that $|{\varphi }_1(\mu )|<1$ for all $\mu \in S_k$ with $k>{\hat{k}}_1$. Again from the definition of ${\varphi }_1$ in (3.4) and Rouché’s theorem, we conclude that the functions ${\varphi }_0$ and ϕ have the same number of zeros in the square $S_k$, for $k\in \mathbb{N}$ with $k\ge {\hat{k}}_1$.

Since ${\varphi }_0$ has $4k+2$ zeros inside $S_k$ and thus $4k+2+4$ zeros inside $S_{k+1}$, it follows that ϕ has no large zeros other than the zeros $\pm {\hat{\mu }}_k$ found above for $|k|$ sufficiently large, and that ${\hat{\lambda }}_k={\hat{\mu }}_k^2$ account for all eigenvalues of the problem (2.1)-(2.2) since each of these eigenvalues gives rise to two zeros of ϕ, counted with multiplicity. By Proposition 2.3, all eigenvalues with non-zero real part occur in pairs ${\hat{\lambda }}_k$, $-\overline{{\hat{\lambda }}_k}$, which shows that we can index all such eigenvalues as ${\hat{\lambda }}_{-k}=-\overline{{\hat{\lambda }}_k}$. Since there is an odd number of remaining indices, the number of pure imaginary eigenvalues must be odd.

Then all the estimates are as in Case 4, and the result in Case 2 immediately follows from that in Case 4 if we observe that each $S_k$ for k large enough contains two fewer zeros of ϕ than in Case 4.

The result follows with reasonings and estimates as in the proof of Case 4, replacing μ by $\mu \pm \frac{\pi }{2}$ and $\mu \pm i\frac{\pi }{2}$, respectively.

and a reasoning as in Case 1 completes the proof. □

We refer to [[15], Definition 7.3.1] for the definition of the Birkhoff regularity.

Proposition 4.1 The boundary value problem (2.1), (2.5)-(2.8) is Birkhoff regular for $\alpha >0$ with respect to the eigenvalue parameter μ given by $\lambda ={\mu }^2$.

where

in Cases 3 and 4. Thus, the problem (2.1), (2.5)-(2.8) is Birkhoff regular. □

Let D, as a function of μ with $\lambda ={\mu }^2$, be the characteristic function of the problem (2.1), (2.5)-(2.8) with respect to the fundamental system $y_j$, $j=1,2,3,4$, with $y_j^{[m]}(0)={\delta }_{j,m+1}$ for $m=0,1,2,3$, where δ is the Kronecker delta. Denote by $D_0$ the corresponding characteristic function for $g=0$. Note that the characteristic functions $D_0$ and ${\varphi }_0$ considered in Section 3 have the same zeros counted with multiplicity. Due to the Birkhoff regularity, g only influences lower order terms in D. Therefore, it can be inferred that outside the interior of the small squares $R_k$, $-R_k$, $i{R}_k$, $-i{R}_{-k}$ around the zeros of $D_0$, $|D(\mu )-D_0(\mu )|<|D_0(\mu )|$ if $|\mu |$ is sufficiently large. Since the fundamental system $y_j$, $j=1,2,3,4$, depends analytically on μ, also D and $D_0$ are analytic functions. Hence, applying Rouché’s theorem both to the large squares $S_k$ and to the small squares which are sufficiently far away from the origin, it follows that the eigenvalues of the boundary value problem for general g have the same asymptotic distribution as for $g=0$. Whence Proposition 3.2 leads to

Proposition 5.1 For $g\in C^1[0,a]$, there exists a positive integer $k_0$ such that the eigenvalues ${\hat{\lambda }}_k$, counted with multiplicity, of the problem (2.1), (2.5)-(2.8), where $k\in \mathbb{Z}\setminus \{0\}$ in Cases  1 and  2 and $k\in \mathbb{Z}$ in Cases 3 and 4, can be enumerated in such a way that the eigenvalues ${\lambda }_k$ are pure imaginary for $|k|<k_0$, and ${\lambda }_{-k}=-\overline{{\lambda }_k}$ for $k\ge k_0$. For $k>0$, we can write ${\lambda }_k={\mu }_k^2$, where the ${\mu }_k$ have the following asymptotic representation as $k\to \mathrm{\infty }$:

In particular, the number of pure imaginary eigenvalues is even in Cases 1 and 2 and odd in Cases 3 and 4.