Abstract
We consider the spectral problem for a Dirac operator with arbitrary two-point boundary conditions and an arbitrary complex-valued integrable potential. The existence of nontrivial boundary value problems of this type with an unbounded growth of the multiplicity of eigenvalues is established.
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INTRODUCTION
In the present paper, we study the Dirac system
where \(\mathbf {y}=\mathrm {col}\thinspace (y_1(x),y_2(x)) \), \(\lambda \in \mathbb {C} \) is the spectral parameter,
and the functions \(p, q\in L_1(0,\pi ) \) are complex-valued, with the two-point boundary conditions
where
the coefficients \(a_{ij} \) can be any complex numbers, and the rows of the matrix
are linearly independent.
We denote by \(\|f\|=(|f_1|^2+|f_2|^2)^{1/2} \) the norm of an arbitrary vector \(f=\mathrm {col}\thinspace (f_1,f_2)\in \mathbb {C}^2\) and set \(\langle f,g\rangle =f_1g_1+f_2g_2\). We denote the norm of an arbitrary \( 2\times 2\) matrix \(W \) by \(\|W\|=\sup \limits _{\|f\|=1}\|Wf\|\). Let \(L_{2,2}(a,b) \) be the space of two-dimensional vector functions \(f(t)=\mathrm {col}\thinspace (f_1(t),f_2(t))\) with the norm \( \|f\|_{L_{2,2}(a,b)}=(\int \nolimits _a^b\|f(t)\|\thinspace dt)^{1/2} \), and let \(L_{2,2}^{2,2}(a,b) \) be the space of \(2\times 2 \) matrix functions \(W(t) \) with the norm \( \|W\|_{L_{2,2}^{2,2}(a,b)}=(\int \nolimits _a^b\|W(t)\|\thinspace dt)^{1/2} \). We treat the operator \(\mathbb {L}\mathbf {y}=B\mathbf {y}^{\prime }+V\mathbf {y}\) as a linear operator in the space \( L_{2,2}(0,\pi )\) with domain \(D(\mathbb {L})=\{\mathbf {y}\in W_1^1[0,\pi ]:\mathbb {L}\mathbf {y}\in L_{2,2}(0,\pi ) \), \(U_j(\mathbf {y})=0\) \((j=1,2)\} \).
Let
be the fundamental matrix of Eq. (1) with the boundary condition \(E(0,\lambda )=I \), where \(I \) is the identity matrix, and let \(E_0(x,\lambda ) \) be the fundamental matrix of the unperturbed equation \(B\mathbf {y}^{\prime }=\lambda \mathbf {y}\) with the boundary condition \(E_0(0,\lambda )=I\). It is obvious that
It is well known that the entries of the matrix \(E(x,\lambda )\) are related by
for any \(x \) and \(\lambda \). Let \(J_{ij} \) be the determinant formed by the \(i \)th and \(j \)th columns of \(A \). Set \(J_0=J_{12}+J_{34} \), \(J_1=J_{14}-J_{23}\), and \( J_2=J_{13}+J_{24}\).
It was shown in [1] by the transformation operator method that the characteristic determinant \(\Delta (\lambda ) \) of problem (1), (2), which is equal to
can be reduced to the form
where the function
\( C_1=(J_1+iJ_2)/2\), \(C_2=(J_1-iJ_2)/2 \), is the characteristic determinant of the unperturbed problem
and the functions \(r_j \) belong to the space \(L_1(0,\pi ) \), \(j=1,2\). If \(p, q\in L_2(0,\pi )\) (for short, we write \(V\in L_2(0,\pi ) \)), then \(r_j\in L_2(0,\pi ) \). It follows that the function \(\Delta (\lambda ) \) is an entire function of exponential type; therefore, we only have the following possibilities for the operator \(\mathbb {L} \) of problem (1), (2):
-
1.
The spectrum is empty.
-
2.
The spectrum is a finite nonempty set.
-
3.
The spectrum is a countable set without finite limit points.
-
4.
The spectrum fills the entire complex plane.
Relations (5) and (6) imply that case 1 is realized for problem (7), for example, with the boundary conditions defined by the matrix
and case 4, with the boundary conditions defined by the matrix
Let us prove that case 2 is impossible. Let the equation
have finitely many roots \(\lambda _k \), \(k={1,\ldots ,n}\). If \(C_1C_2\ne 0 \), then conditions (2) are regular and problem (1), (2) has a countable set of eigenvalues; therefore, \(C_1C_2=0\). Set \(P(\lambda )=\prod _{k=1}^n(\lambda -\lambda _k)\). By [2],
where \(a \) and \(b \) are some constants. Assume, for example, that \(C_2=0 \). Setting \(\lambda =-iy \) in relation (5), where \(y>0\), we obtain
which implies that
According to [3, p. 36], the expression on the left-hand side in relation (8) tends to \(C_1 \) as \(y\to \infty \). If \(\mathrm {Im}\thinspace a-\pi \ge 0\), then the expression on the right-hand side in relation (8) tends to infinity in absolute value, and if \(\mathrm {Im}\thinspace a-\pi <0\), then it tends to zero. It follows that \( C_1=0\). If \(C_1=C_2=0 \), then
Obviously, the left-hand side of relation (9) is bounded on the real axis, while the right-hand side is not; that is, we arrive at a contradiction.
Definition.
We say that problem (1), (2) has the classical spectral asymptotics if its spectrum is a countable set and the multiplicities of the eigenvalues are uniformly bounded.
The present paper is aimed at constructing problems (1), (2) for which case 3 is realized and the multiplicities of the eigenvalues grow unboundedly, i.e., problems with nonclassical spectral asymptotics.
MAIN RESULTS
Set \( c_j(\lambda )=c_j(\pi ,\lambda )\) and \(s_j(\lambda )=s_j(\pi ,\lambda )\), \(j=1,2 \). In addition, let \(PW_\sigma \) be the class of entire functions \(f(z) \) of the exponential type \(\le \sigma \) such that \(\|f\|_{L_2(R)}<\infty \). It is well known [4] that the functions \(c_j(\lambda ) \) and \(s_j(\lambda ) \) admit the representation
where \(g_j,h_j\in PW_\pi \), \(j=1,2 \).
Lemma 1 [5].
The functions \(u(\lambda ) \) and \( v(\lambda )\) admit the representations
where \( h,g\in PW_\pi \) , if and only if
where \( \lambda _n=n+\varepsilon _n\) and \(\{\varepsilon _n\}\in l_2 \) , and
where \( \lambda _n=n-1/2+\kappa _n\) and \(\{\kappa _n\}\in l_2 \) .
Consider the Dirac system with the boundary conditions defined by the matrix
We will assume that \(V\in L_2(0,\pi ) \). It follows from the representation (4) that the characteristic determinant \(\Delta (\lambda ) \) of problem (1), (2) with matrix \(A \) defined in (10) can be reduced to the form
where \(r\in L_2(0,\pi )\), and \(f\in PW_\pi \). The converse statement holds true as well.
Theorem.
For each function \(f\in PW_\pi \), there exists a potential \(V\in L_2(0,\pi )\) such that the characteristic determinant \(\Delta (\lambda ) \) of problem (1), (2) with the matrix \(A\) defined by relation (10) and the potential \(V(x) \) is identically equal to \( f(\lambda )\).
Proof. Let \(f(\lambda ) \) be an arbitrary function in the class \(PW_\pi \). It follows from the Paley–Wiener theorem and [3, p. 36] that
consequently, there exists a positive integer \(N_0\) so large that \(|f(\lambda )|<1/100\) if \(\mathrm {Im}\thinspace \lambda =0\) and \(|\mathrm {Re}\thinspace \lambda |\ge N_0\).
Let \(\{\lambda _n\}\) , \(n\in \mathbb {Z}\) , be a strictly monotone increasing sequence of real numbers such that \( N_0<\lambda _n<N_0+1/100\) if \(1\le n\le N_0\) , \(\lambda _n=n-1/2\) if \(n>N_0\) , and \( \lambda _n=-\lambda _{-n+1}\) for any \(n\). Set
Lemma 1 implies the relation
where \(g\in PW_\pi \). It follows from the Paley–Wiener theorem and [3, p. 36] that
therefore,
(\(c_0=\mathrm {const}>0 \)) for \(|\mathrm {Im}\thinspace \lambda |\ge M\), where \(M \) is a sufficiently large number.
Differentiating relation (12), we obtain
Since the function \(\dot g \) belongs to the class \(PW_\pi \), we have, according to [6],
where
Based on this, by the definition of the numbers \(\lambda _n \), we obtain
where
Consequently, for all even \(n \) sufficiently large in modulus one has the inequality \(\dot c(\lambda _n)>0\). One can readily see that the inequality \(\dot c(\lambda _n)\dot c(\lambda _{n+1})<0\) holds for all \(n\in \mathbb {Z}\). It follows that
for all \(n\in \mathbb {Z}\). Note that (15) implies the relation
where
Consider the quadratic equation
It has the roots
By \(\Gamma (z,r)\) we denote the disk of radius \(r\) centered at point \(z \). One can readily see that all numbers \(s_n^{+} \) lie inside the disk \(\Gamma (1,1/10) \) and all numbers \(s_n^{-} \) lie inside the disk \(\Gamma (-1,1/10) \). Let \(s_n=s_n^{+} \) if \(n \) is odd and \(s_n=s_n^{-} \) if \(n \) is even. Since [6] \(\{f(\lambda _n)\}\in l_2 \), it follows from the definition of the numbers \(s_n \) that
where \(\{\vartheta _n\}\in l_2 \). It also follows from the definition of the numbers \(s_n \) and inequality (16) that all numbers \(z_n={s_n}/{\dot c(\lambda _n)} \) lie strictly to the left of the imaginary axis, while (17) and (19) imply the relation
where \(\{\rho _n\}\in l_2 \). Let \(\beta _n=s_n-\sin (\pi \lambda _n)\); then \(\{\beta _n\}\in l_2 \) in view of (19). Set
According to [7, p. 120], the function \(h \) belongs to the class \(PW_\pi \), and \(h(\lambda _n)=\beta _n \). Set \(s(\lambda )=\sin (\pi \lambda )+h(\lambda )\); then \(s(\lambda _n)=s_n\ne 0 \), and consequently, the functions \(s(\lambda ) \) and \(c(\lambda ) \) do not have common roots.
Set
In the subsequent exposition, we need the following elementary assertion.
Lemma 2.
If function systems \( \{\varphi _n\}\) and \( \{\psi _n\}\) are complete in \(L_2(a,b)\) \((n\in \mathbb {N}) \) , then the system of vectors
is complete in \( L_{2,2}(a,b)\) .
Proof. Assume that there exists a vector \(f(x)=\mathrm {col}\thinspace (f_1(x),f_2(x))\ne 0\) such that
for all \(n\in \mathbb {N}\). Then
consequently, \( f_1(x)\equiv f_2(x)\equiv 0\). The proof of the lemma is complete.
It follows from [8] that the function systems \( \{\cos (\lambda _n x)\}\) and \(\{\sin (\lambda _n x)\} \) \((n\in \mathbb {N})\) are complete in \( L_2(0,\pi )\). Based on this, it follows from the definition of the numbers \(\lambda _n\) and Lemma 2 that the system of vectors
(\(n\in \mathbb {Z}\)) is complete in \( L_{2,2}(0,\pi )\). Set
It follows from [4] that
where \(C \) is a constant independent of \(x \). Let us prove that for each \(x\in [0,\pi ] \) the homogeneous equation
where \(f(t)=\mathrm {col}\thinspace (f_1(t),f_2(t))\), \(f\in L_{2,2}(0,x)\), \(f(t)=0 \) for \(x<t\le \pi \), has only the trivial solution. Multiplying Eq. (21) by \(\overline {f^{\mathrm {T}}(t)} \) and integrating the resulting equation over the segment \([0,x] \), we obtain
Taking into account definition (20), by simple calculations we find that
which implies that
In view of Parseval’s identity, we obtain
therefore,
Since \(\mathrm {Re}\thinspace z_n<0 \) for each \(n \), Eq. (22) implies that \(\int \nolimits _0^x\langle f(t),Y_0(t,\lambda _n)\rangle \thinspace dt=0 \). The latter and the completeness of the system of vectors \( \{Y_0(t,\lambda _n)\}\) in \(L_{2,2}(0,\pi ) \) imply the identity \(f(t)\equiv 0 \). The unique solvability of Eq. (21) implies [4] that the functions \(c(\lambda ) \) and \(-s(\lambda ) \) are the entries of the first row of the monodromy matrix
of problem (1), (2) with the matrix \(A \) defined in (10) and some potential \(\tilde V\in L_2(0,\pi ) \); i.e.,
By virtue of (4), the characteristic determinant \(\tilde \Delta (\lambda )\) of this problem has the form
where \(\tilde f\in PW_\pi \). Relations (3), (18), and (23) imply the equality
It follows from the last equality that the function
is entire. Since
we conclude in view of inequality (13) that \(|\Phi (\lambda )|\le c_2=\mathrm {const} \) if \(|\mathrm {Im}\thinspace \lambda |\ge M\).
Let \(H\) stand for the union of vertical segments \( \{z:|\mathrm {Re}\thinspace z|=n,|\mathrm {Im}\thinspace z|\le M\} \), where \(|n|=N_0+1, N_0+2,\ldots \) Since the function \(c(\lambda ) \) is a sine-type function [9], we have \(|c(\lambda )|>\delta >0 \) for \(\lambda \in H \). The last inequality, the estimate (24), and the maximum principle imply the inequality \(|\Phi (\lambda )|<c_3=\mathrm {const}\) in the strip \(|\mathrm {Im}\thinspace \lambda |\le M\). Consequently, the function \(\Phi (\lambda )\) is bounded in the entire complex plane and is constant by Liouville’s theorem. Let \(|\mathrm {Im}\thinspace \lambda |=M \). Then, in view of relation (11), we have \(\lim \limits _{|\lambda |\to \infty }(f(\lambda )-\tilde f(\lambda ))=0\); therefore, \(\Phi (\lambda )\equiv 0 \), and hence \(f(\lambda )\equiv \tilde \Delta (\lambda )\). The proof of the theorem is complete.
Examples of functions in the class \(PW_\pi \) with roots of arbitrarily high multiplicity are known in the literature (see, e.g., [10, 11]). Note that the existence of one-dimensional boundary value problems with an unboundedly increasing multiplicity of eigenvalues was previously established for the Sturm–Liouville operator and an ordinary differential operator of any even order [10,11,12].
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Makin, A.S. On the Spectrum of Two-Point Boundary Value Problems for the Dirac Operator. Diff Equat 57, 993–1002 (2021). https://doi.org/10.1134/S0012266121080036
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DOI: https://doi.org/10.1134/S0012266121080036