On the eigenvalues of spectral gaps of matrix-valued Schrödinger operators

This paper is a sequel to two articles of the second author, written over a period of more than 25 years. The first, [12], presents numerical methods for self-adjoint Sturm-Liouville-type equations with matrix coefficients, while the second, [13], analyses the application of a dissipative barrier scheme to a Schrödinger equation on a half-line. In the years since [12] appeared, there has been a lot of activity on Schrödinger-type equations with matrix-valued coefficients—see, e.g., Clark and Gesztesy [4] and Clark, Gesztesy, Holden and Levitan [5], together with the substantial bibliographies therein. Some results from the scalar case carry across to the matrix case in a straightforward way; some require new proofs; and some are simply no longer true. As a simple example, the usual spectral data only determines the Titchmarsh-Weyl coefficient, and hence the matrix-valued potential, up to unitary equivalence. Our concern in this article is to examine which of the results in [13] are still true in the case of a matrix-valued potential, and which not. At the end of the article, we indicate some directions for future work on quantum waveguide problems.

To fix notation, we consider the dissipative matrix Schrödinger equation on the half-line \([0,\infty )\),

with a regular self-adjoint-type boundary condition at the origin. The precise form of this condition is not important for our results, so we shall use

where α ∈ [0,π), even though this is not the most general form. Here \(\underline {u}\) is a vector-valued function in a subspace of \(L^{2}([0,\infty ))^{n}\), the parameter γ is a non-zero real, and the coefficients Q, S satisfy the following hypotheses:

• (A1): Q(x) is a Hermitian-valued, integrable over compact subsets of \([0,\infty )\), and is eventually periodic with period a > 0 i.e, there exists R₀ ≥ 0 such that

  $$ Q(x+a)=Q(x), \quad \forall x \geq R_{0}. $$

  (3)

• (A2): S is a cutoff function with support in [0,R] for some R ≥ R₀ : there exists 0 < c < 1 such that

  $$ S(x)= \left\{\begin{array}{ll} I, & x< cR;\\ 0, & x\geq R. \end{array}\right. $$

  (4)

When x ∈ (cR,R), we assume that elements of S are measurable and their values lie in [0,1].

The hypothesis (A2) in particular is stronger than is really needed for most of our results, but sufficient to analyse most dissipative barrier schemes.

We define an operator L₀ by:

with domain:

Our aim is to present a substantial analysis of the interaction between the dissipative barrier approach to the problem of numerical approximation of the spectrum of L₀, and interval truncation methods. Our methods will be based on the Floquet theory and Weyl-Titchmarsh functions.

We investigate the following results for an eigenvalue λ_γ of our problem with the dissipative term iγS(⋅) which develops from the eigenvalue λ when γ = 0 :

1. 1.

   For our non-truncated problem, if λ is an isolated eigenvalue of multiplicity ν, where 1 ≤ ν ≤ n, Q is a compactly supported perturbation of a Hermitian periodic function, and for a sufficiently small γ > 0, the approximation λ_γ,j for each j = 1,…,ν satisfies the bound:

   $$ |\lambda+i\gamma-\lambda_{\gamma,j}|\leq C_{1}\gamma \exp(-c C_{2}R), $$

   (7)

   where C₁ and C₂ are positive constants.

2. 2.

   If our problem is truncated to some interval [0,X], X > R, then by imposing a boundary condition at x = X, any eigenvalue λ_γ,X,good of the truncated problem which converges to λ_γ as \(X\to \infty \) satisfies:

   $$ |\lambda_{\gamma}-\lambda_{\gamma,X,\text{good}}|\leq C_{3} \exp\left( -C_{4}(X-R)\right), $$

   (8)

   where C₃ and C₄ are positive constants and depend on λ_γ. Moreover, the total algebraic multiplicities of all λ_γ,X,good converging to λ_γ is equal to the algebraic multiplicity of λ_γ.

3. 3.

   If our problem is truncated to some interval [0,X], X > R, then by imposing a boundary condition at x = X, then any eigenvalue λ_γ,X,bad of the truncated problem which converges as \(X\to \infty \) to a point which is not in the spectrum of L₀ + iγS satisfies

   $$ |\Im(\lambda_{\gamma,X,\text{bad}})| \leq C_{5} \exp(-C_{6}(X-R)), $$

   (9)

   where C₅ and C₆ are positive constants.

4. 4.

   One crucial difference between the operators considered here and the scalar-coefficient operators in [13] concerns the behaviour of eigenvalues of L₀ + iγS as γ ↘ 0. In [13], it is shown that if an eigenvalue λ_γ of L₀ + iγS evolves continuously to become an interior point of a spectral band of L₀ + iγS when γ = 0, then a threshold effect occurs: there exists γ_crit > 0 such that as γ ↘ γ_crit,λ_γ converges to the interior point of the spectral band. We give an example to show that this is not generally true for the operators considered in the present article.

The paper is organised as follows. Section 3 is devoted to the representation of the Floquet theory and Glazman decomposition for the main equation. Section 4 and Section 5 contain the analysis of the method for the truncated and non-truncated problem respectively. Finally, Section 6 represents some numerical experiments to illustrate our results.

The essential spectrum of L₀ can be described using the Floquet theory, studying the solutions of the differential (1) over just one period. We shall review the elements of the Floquet theory below, primarily to introduce the notation which we require for an analysis of the domain truncation technique.

For the point spectrum of L₀ + iγS, we shall apply the Glazman decomposition technique [1, Appendix 2].

Recall the parameter R > 0 from hypothesis (A2). For a fixed \(\lambda \in \mathbb {C}\) and any non-zero constant vector \(\underline {h}\), consider the following two boundary value problems:

If these problems can be solved uniquely for every \(\underline {h}\) ^{Footnote 1}, then the maps \(\underline {v}(R) \longmapsto \underline {v}^{\prime }(R) \in \mathbb {C}^{n}\) and \(\underline {w}(R) \longmapsto -\underline {w}^{\prime }(R) \in \mathbb {C}^{n}\) are linear operators (Weyl-Titchmarsh operators or Dirichlet to Neumann maps) which admit representations by n × n matrices:

\(M_{\text {left}}(\lambda )\underline {v}(R)=\underline {v}^{\prime }(R);\) \(M_{\text {right}}(\lambda )\underline {w}(R)=-\underline {w}^{\prime }(R).\)

The matrices M_left(λ) and M_right(λ) are analytic functions of λ on suitable subsets of \({\mathbb C}\). In particular M_left is meromorphic with poles in \({\mathbb {C}}^{+}\) when γ > 0, and on the real axis when γ = 0; the function M_right is analytic outside Spec_ess(L₀) except at a set of poles (see [4, 9]).

If λ is an eigenvalue of L₀ + iγS, then there exists an eigenfunction \(\underline {u};\) we can take \(\underline {v}(x)=\underline {u}(x), x\in [0,R],\) and \(\underline {w}(x)=\underline {u}(x), x\in [R,\infty )\) so that:

\([ M_{\text {left}}(\lambda )+M_{\text {right}}(\lambda )]\underline {u}(R)=\underline {u}^{\prime }(R)+(-\underline {u}^{\prime }(R))=\underline {0}\).

Assuming that \(\underline {u}(R)\) is not zero, this leads to the condition that \({\ker }(M_{\text {left}}(\lambda )+M_{\text {right}}(\lambda ))\) is not trivial. In fact, if \(\underline {u}(R)\) were zero, then both M_left(⋅) and M_right(⋅) would be undefined at λ, so the condition that \(\underline {u}(R)\) be non-zero is satisfied automatically if M_left(λ) and M_right(λ) are well defined.

Conversely, suppose there exists \(\mu \in \mathbb {C}\) such that

Take \(\underline {h}\in \ker (M_{\text {left}}(\mu )+M_{\text {right}}(\mu ))\) and define a non-trivial vector \(\underline {u}\) by:

where \(\underline {v}\) and \(\underline {w}\) are the solutions of P_left and P_right respectively for the case λ = μ. Then \(\underline {u}\) is a solution for the differential equation \(-\underline {u}^{\prime \prime }+(Q+i\gamma S)\underline {u}=\mu \underline {u}\) on both the intervals (0,R) and \((R,\infty )\), which is continuous. Moreover, it has a continuous first derivative at x = R which follows from (12) since \(\underline {h}\in \ker (M_{\text {left}}(\mu )+M_{\text {right}}(\mu ))\) and using the definitions of M_left(μ) and M_right(μ). This means that \(\underline {u}\) is an eigenfunction of L₀ + iγS with eigenvalue μ. We have proved the following result.

Lemma 1

Suppose that M_left(λ) and M_right(λ) are well defined at λ = μ. Then μ is an eigenvalue of L₀ + iγS if and only if the kernel of M_left(μ) + M_right(μ) is non-trivial.

We now describe how to calculate M_left and M_right. For M_left, the procedure is (at least in principle) straightforward:

where U is the n × n matrix-valued solution of the initial value problem

In order to find M_right, we bear in mind that the Q(x) is periodic for x ≥ R₀ and hence for x ≥ R ≥ R₀; also the dissipative perturbation S(x) = 0 for x ≥ R. Therefore, we can apply the Floquet theory [6] for the system of differential equations. We rewrite (1) as a first-order differential system:

Let Φ(x,λ) be the fundamental matrix of this equation, i.e.

where I_2n×2n is the 2n × 2n identity. Define a non-singular matrix A(λ) by

Then we can find the eigenvalues of A(λ), say ϱ₁(λ),ϱ₂(λ),…,ϱ_2n(λ). These eigenvalues are called Floquet multipliers.

Suppose that A(λ) has a canonical Jordan form, i.e.

where J(λ) is a Jordan matrix and F(λ) is a non-singular matrix. It may be shown then that

so Φ(x + a,λ)F(λ) = Φ(x,λ)F(λ)J(λ). In fact, the columns of Φ(⋅,λ)F(λ) are called Floquet solutions of (15).

The following Lemma summarises some standard facts (for more information about the Floquet theory for Hamiltonian systems, see [5, 16]):

Lemma 2

1. 1.

   The Floquet multipliers ϱ₁(λ),…,ϱ_2n(λ) satisfy ϱ₁(λ)…ϱ_2n(λ) = 1.

2. 2.

   For λ∉Spec_ess(L₀), there exist precisely n of the ϱ_j(λ); say ϱ₁(λ),…,ϱ_n(λ), such that |ϱ_j(λ)| < 1.

Proof 1

1. 1.

   This statement holds because \(\det (A(\lambda ))=1\), which follows from the fact that \(\det ({\Phi }(x,\lambda ))\) is a non-zero constant (the coefficient matrix on the right-hand side of (16) having zero trace).

2. 2.

   The (1) is in the limit-point case at infinity, and hence for λ outside the essential spectrum it has precisely an n-dimensional space of solutions in \(L^{2}(0,\infty )\). None of the Floquet multipliers ϱ₁(λ),…,ϱ_2n(λ) has absolute value 1, for otherwise it is possible to construct a Weyl singular sequence of oscillatory solutions from the corresponding Floquet solution; this is impossible as λ lies outside the essential spectrum. Thus, precisely n of the Floquet multipliers must have absolute value strictly less than 1, precisely n have absolute value strictly greater then 1, and we can order them so that \(|{\varrho _{1}(\lambda )}|,\dots ,|{\varrho _{n}(\lambda )}|<1\) and \(|\varrho _{n+1}(\lambda )|,\dots ,|\varrho _{2n}(\lambda )|>1.\)

□

It follows from Lemma 2 that if λ lies outside the essential spectrum, then the Jordan matrix J(λ) in (18) decomposes into n × n blocks as

where J₁(λ) corresponds to ϱ₁(λ),…,ϱ_n(λ) and J₂(λ) corresponds to ϱ_n+ 1(λ), …, ϱ_2n(λ). In this case, the matrix

has columns which span the n-dimensional space of square-integrable solutions; moreover, for \(N\in \mathbb {N}:\)

We can partition Γ(x,λ) as

where Ψ(x,λ) is an n × n solution of

whose columns (as we mentioned above) span the space of all square integrable solutions of (1). Hence, by direct verification, if Ψ(R,λ) is invertible, then the function

is the solution of the problem P_right. A simple calculation now shows that \(-\underline {w}^{\prime }(R) = M_{\text {right}}(\lambda )\underline {w}(R)\), where

We immediately have the following corollary to Lemma 1.

Corollary 1

Suppose that M_left(λ) is well defined and that Ψ(R,λ)^− 1 exists. Then λ is an eigenvalue of L₀ + iγS if and only if

In this section, we truncate the problem over \([0,\infty )\) to a problem on [0,X] for some X > R. At x = X we impose, for some \(\upbeta \in \mathbb {R}\), a self-adjoint artificial boundary condition

The operator L₀ is replaced by L_0,X defined by:

with domain:

In this case, the spectra of L_0,X and L_0,X + iγS are purely discrete. To characterise the eigenvalues of L_0,X + iγS, we replace P_right in (11) by:

Let Λ(x,λ) be a 2n × n matrix of non-\(L^{2}(0,\infty )\) solutions corresponding to the eigenvalues ϱ_n+ 1,…,ϱ_2n of (17), thus,

where J₂(λ) is the corresponding Jordan matrix introduced in (19). Moreover, for \(N\in \mathbb {N}:\)

We may partition Λ(λ,x) as

construct the matrix:

choosing C_X(λ) so that cos\((\upbeta ){\Psi }_{X}(X)-\textit {sin}(\upbeta ){\Psi }^{\prime }_{X}(X)=0\):

Hence, the solution \(\underline {w}\) of the boundary value problem (28) exists if the corresponding Ψ_X(R,λ)^− 1 exists and:

\(\underline {w}(x)= {\Psi }_{X}(x,\lambda ){\Psi }_{X}(R,\lambda )^{-1}\underline {h},\)

Thus, the eigenvalues of L_0,X + iγS may be characterised by an analogue of Lemma 1 if we replace M_right by

Corollary 1 also has the following analogue.

Lemma 3

Suppose that M_left(λ) and Ψ_X(R,λ)^− 1 exist. Then λ is an eigenvalue of L_0,X + iγS if and only if

We analyse the effect of interval truncation through a sequence of intermediate results and technical lemmas.

Proposition 1

If X = R + Na where \(N\in \mathbb {N} \), and J₁(λ) and J₂(λ) are defined as in (19) then

Furthermore,

in particular,

where C is a positive constant, \(\varrho (\lambda )=\max \limits (|\varrho _{1}(\lambda )|,\ldots ,|\varrho _{n}(\lambda )|)\) and \(\tilde {\varrho }(\lambda )=\min \limits (|\varrho _{n+1}(\lambda )|,\ldots ,|\varrho _{2n}(\lambda )|).\)

Proof 2

Equation (35) follows directly from (20, 29) upon using the definitions (21, 30). Substituting (35) into (32) yields (36). In order to estimate the norm of C_X(λ), we need to find the norm of J₁(λ)^N. The expression for J₁(λ)^N may have terms, of worst case scenario, like:

where \(\varrho (\lambda )=\max \limits (|\varrho _{1}(\lambda )|,\ldots ,|\varrho _{n}(\lambda )|).\) Thus,

A similar analogue would be followed to estimate the norm of J₂(λ)^−N. Therefore,

where \(\tilde {\varrho }(\lambda )=\min \limits (|\varrho _{n+1}(\lambda )|,\ldots ,|\varrho _{2n}(\lambda )|).\) Finally, (37) follows from the expressions of the norms of J₁(λ)^N and J₂(λ)^−N. □

Lemma 4

Let Ψ(x,λ) be defined as in (22) and suppose I(λ) > 0. Then Ψ(R,λ) is invertible.

Proof 3

Assume that Ψ(R,⋅) is not invertible: then we can find a non-zero vector \(\underline {c} \in \mathbb {C}^{n}\) such that \({\Psi }(R,\lambda )\underline {c}=\underline {0}.\) Define a function \(\underline {u}(x,\lambda )={\Psi }(x,\lambda )\underline {c},\) which satisfies the differential equation for all x. Also, \( \underline {u}(R,\lambda )=\underline {0}.\) However, the fact that |ϱ_j(λ)| < 1 for j = 1,…,n, means that \(\underline {u}(\cdot ,\lambda )\in L^{2}(R,\infty ).\) Hence \(\underline {u}(x,\lambda )\) is an eigenfunction for the problem \(-\underline {u}^{\prime \prime }+Q\underline {u}=\lambda \underline {u},\) on \([R,\infty )\) with Dirichlet condition at R. This problem is self-adjoint, so I(λ) = 0, which is a contradiction. □

Lemma 5

Let Ψ_X(x,λ) be defined as in (31) and suppose I(λ) > 0. Then Ψ_X(R,λ) is invertible.

Proof 4

Assume Ψ_X(R,λ) is not invertible. Then ∃ a non-zero \(\underline {c}\in \mathbb {C}^{n}\) such that \({\Psi }_{X}(R,\lambda )\underline {c}=\underline {0}. \) Define a function \(\underline {u}_{X}(x,\lambda )={\Psi }_{X}(x,\lambda )\underline {c},\) so that \(\underline {u}_{X}(R,\lambda )=\underline {0}.\) Hence, \(\underline {u}_{X}(x,\lambda )\) is an eigenfunction of the problem on [R,X] with Dirichlet condition at R and the boundary condition (25) at X with \(\upbeta \in \mathbb {R}.\) This is also a self-adjoint problem, so again we have the contradiction I(λ) = 0. □

Finally, we have a quantitative lemma on continuity of determinants, which will be needed in the proof of Theorem 1. We shall use this lemma with X = R + Na and large N.

Lemma 6

Suppose that \(A,A_{X} \in \mathbb {M}_{n}(\mathbb {C})\) have the property:

where b is a positive constant and 0 < τ < 1. Then

where \(\tilde {b}\) is a positive constant.

Proof 5

The proof follows directly from [8], by letting \(\tilde {b} =nb[\left \lVert A\right \rVert +b\tau ]^{n-1}\). □

Theorem 1

Suppose that assumptions (A1) and (A2) hold. For γ > 0, let λ_γ be an eigenvalue of the non-self-adjoint Schrödinger operator L₀ + iγS defined in (5, 6). Then there exist approximations λ_γ,X,good to λ_γ, whose total algebraic multiplicity is equal to the algebraic multiplicity of λ_γ, obtained as eigenvalues of the operator L_0,X + iγS defined in (26, 27), which satisfies

Here C₃ and C₄ are positive constants which depend on λ_γ.

Proof 6

Without loss of generality, it is sufficient to check the cases X = R + Na where \(N\in \mathbb {N}\) is sufficiently large. The other cases follow by exploiting the freedom in the choice of the constants c and R in (26) and (27). For example, if X = R + Na + b, with 0 < b < a, then we can replace R by R + b and use a smaller constant c in (26).

First, we observe that for γ > 0, λ_γ has strictly positive imaginary part. If \(\underline {u}_{\gamma }\) is the corresponding normalised eigenfunction, then a standard integration by parts yields: \(\Im (\lambda _{\gamma })=\gamma {{\int \limits }_{0}^{R}} \underline {u}^{*}_{\gamma }(x)S(x)\underline {u}_{\gamma }(x) dx >0,\) where \( \underline {u}^{*}_{\gamma }(x)\) is the Hermitian conjugate of \(\underline {u}_{\gamma }(x).\) Next, we observe consequently from Lemma 4 and Lemma 5, Ψ(R,⋅) and Ψ_X(R,⋅) are invertible for all λ in a neighbourhood of λ_γ. It follows that M_right(⋅) and M_right,X(⋅) are well defined in a neighbourhood of λ_γ.

If M_left(λ_γ) is well defined, then from Corollary 1,

from Lemma 3, we seek points λ_γ,X,good which satisfy (38) together with the truncated problem eigenvalue condition:

Using (31) and the definitions of M_right and M_right,X for a fixed λ, we obtain

Now we exploit the fact that X = R + Na, which allows us to use Proposition 1 (Eq. (37)). By Lemma 2 part 2, since λ is outside the essential spectrum, and since the Floquet multipliers may be chosen to be continuous functions of λ, there exist constants c₋ < 1 and c₊ > 1 such that \(\max \limits (|\varrho _{1}(\lambda )|,\ldots ,|\varrho _{n}(\lambda )|)<c_{-}<1\) and \(\min \limits (|\varrho _{n+1}(\lambda )|,\ldots ,|\varrho _{2n}(\lambda )|)=(\max \limits (|\varrho _{1}(\lambda )|,\ldots ,|\varrho _{n}(\lambda )|))^{-1} >c_{+}>1\) uniformly with respect to λ in a neighbourhood of λ_γ which does not lie in a spectral band. Thus, in addition to (39), we have from (37) and (41),

uniformly with respect to λ in a neighbourhood of λ_γ, where b is a positive constant. Using Lemma 6 and since \(0< (\frac {c_{-}}{c_{+}} )<1,\)

uniformly with respect to λ in a neighbourhood of λ_γ, where \(\tilde {b}\) is a positive constant. It follows by a standard zero-counting argument for analytic functions (Lemma 3 in [13]) that there exist points λ_γ,X,good which satisfy (40) and are such that

where C > 0 and ν is the order of the zero of \(\det (M_{\text {left}}(\cdot )+M_{\text {right}}(\cdot ))\) at λ_γ. Moreover, the total order of the λ_γ,X,good as zeros of \( \det (M_{\text {left}}(\cdot )+M_{\text {right,X}}(\cdot ))=0\) in a neighbourhood of λ_γ, is ν. In view of (43), given 𝜖 > 0 such that \((1+\epsilon )\frac {c_{+}}{c_{-}}<1,\) we have

for all sufficiently large N. Hence (43) becomes:

furthermore, any solutions of (40) which converges to λ_γ must satisfy (44). This completes the proof. □

Remark 1

When M_left(λ) has a pole at λ, it is still possible for λ to be an eigenvalue of L₀ + iγS. However, M_right(λ) and M_right,X(λ) cannot have poles off the real axis, as Lemma 4 and Lemma 5 show that Ψ(R,λ) and Ψ_X(R,λ) are invertible for I(λ)≠ 0.

Theorem 2

Suppose that assumptions (A1) and (A2) hold. For γ > 0, let λ_γ,X,bad be an eigenvalue of the non-self-adjoint Schrödinger operator L_0,X + iγS defined in (5, 6) which converges, as \(X\rightarrow +\infty \), to a point which is not in the spectrum of L₀ + iγS. Then for some positive constants C₅ and C₆:

Proof 7

Similarly to the proof of Theorem 1, it is sufficient to consider the case X = R + Na where \(N\in \mathbb {N}\). Since λ_γ,X,bad has the property:

in particular M_left(λ_γ,X,bad) + M_right,X(λ_γ,X,bad) has an eigenvalue 0. It is known (see, e.g., [3]) that spectral pollution must lie on the real axis, since L₀ + iγS is a relatively compact perturbation of the semi-bounded self-adjoint operator L₀: thus, \(\lambda _{\gamma ,X,\text {bad}}\rightarrow \mu \in \mathbb {R}\cap \{\text {Spectral Gap}\}\) as \(X\rightarrow \infty \). Additionally, since μ is not in the spectrum of L₀ + iγS, the matrix M_left(μ) + M_right(μ) is invertible if it is well defined, i.e. if Ψ(R,μ) and U(R,μ) are invertible (see (14, 23)). We assume without loss of generality that this is indeed so, for if not then one may simply use a greater value of R for the Glazman decomposition.

Hence, there is a compact neighbourhood of μ, say \(\overline {B(\mu ,r)}; r>0\) such that M_left(λ) + M_right(λ) is invertible \(\forall \lambda \in \overline {B(\mu ,r)}\). Following the reasoning which led to (42), we deduce that provided C_R(μ) is well defined, then M_right,X will converge locally uniformly to M_right in a neighbourhood of μ. Since such a uniform convergence excludes spectral pollution, it follows that C_R(μ) must not be well defined, i.e.

Now \(\cos \limits (\upbeta ){\Theta }(R,\lambda ) - \sin \limits (\upbeta ){\Theta }^{\prime }(R,\lambda )\) has at worst branch-cut singularities as a function of λ, so its inverse must admit a bound

for some C, ν > 0, in a neighbourhood of λ = μ. Combining this with (37) and using the notation of (42), we see that, for X = R + Na,

The fact that the eigenvalues λ_γ,X,bad form a polluting sequence means that M_right,X(λ_γ,X,bad) cannot converge to M_right(μ) as \(X\rightarrow \infty \), so in view of (41) the norms ∥C_X(λ_γ,X,bad)∥ cannot converge to zero. This implies a bound

and the required result follows since the exponential decay of \(\biggl (\frac {c_{-}}{c_{+}}\biggr )^{N/\nu }\) overcomes the power N^{2(n− 1)/ν} which completes the proof. □

The purpose of this section is to study the evolution of the point spectrum of L₀ + iγS with respect to the coupling constant γ. Throughout this section, we drop the assumption of eventual periodicity (A1).

Theorem 3

Suppose that assumption (A2) holds (see (4)). Let λ be an isolated eigenvalue of L₀ with multiplicity ν, where 1 ≤ ν ≤ n, and normalised eigenvectors u_j, j = 1,…,ν. For each sufficiently small γ > 0, let λ_γ,j;j = 1,…,ν, be eigenvalues of the non-self-adjoint operator L₀ + iγS defined in (5, 6) with eigenvectors u_γ,j, j = 1,…,ν, and suppose \(\lambda _{\gamma ,j}\rightarrow \lambda \) as \(\gamma \rightarrow 0.\) Then for each 1 ≤ j ≤ ν, the projection of u_γ,j onto Span{u₁,…,u_ν} remains bounded away from zero, uniformly with respect to R and γ for sufficiently small γ.

If, additionally, the assumption

• (A1’):

  $$ \left\lVert u_{j}(x)\right\rVert \leq C\exp(-C_{2}x),{\kern2.2pt}{\kern2.2pt} x\in[0,\infty),{\kern2.2pt}{\kern2.2pt}j=1,\ldots,\nu, $$

  (46)

  holds for some positive constants C and C₂, then there exists C₁ > 0, independent of R, such that for all R > 0,

  $$ |\lambda+i\gamma-\lambda_{\gamma,j}|\leq C_{1}\gamma \exp(-c C_{2}R), $$

  (47)

  where c ∈ (0,1) is the constant appearing in assumption (A2).

Proof 8

In fact, we prove more than stated: only the estimate (48) depends on (A2); the rest of the theorem requires only that S to be bounded independently of R as a multiplication operator. The existence of λ_γ,j with \(|\lambda _{\gamma ,j}-\lambda |\rightarrow 0\) as \(\gamma \rightarrow 0\) is a consequence of results in [11] on analytic families. Since γ is sufficiently small, let Γ be a contour which encloses the spectral point λ of L₀ and suppose that λ_γ,j, j = 1,…,ν are the only spectral points of L₀ + iγS inside Γ. Clearly, \(\left \lVert S\right \rVert =1\) independently of R and since L₀ is a self-adjoint operator then |λ − λ_γ,j|≤ γ independently of R; thus, the contour Γ can be chosen independently of R. Suppose u_γ,j, j = 1,…,ν are eigenvectors of L₀ + iγS, linearly independent with \(\left \lVert u_{\gamma ,j}\right \rVert =1.\) Following Kato [11, VII,§3], let P(γ) be the projection onto the eigenspace of L₀ + iγS spanned by the u_γ,j, j = 1,…,ν, and P(0) be the projection onto the eigenspace of L₀ associated with λ; the projection P(γ) is analytic as a function of γ, so that

Since

taking the norm of (49) and using (48), we conclude that P(0)u_γ,j is bounded away from zero uniformly with respect to R and γ for sufficiently small γ.

Now, since (L₀ + iγS)u_γ,j = λ_γ,ju_γ,j and (L₀ − iγI)u_k = (λ − iγ)u_k, using the inner product for the first equation with u_k, we obtain:

Similarly, using the inner product for the second equation with u_γ,j, we obtain:

Because L₀ and S are self-adjoint and u_k and u_γ,j are in the domain of L₀ and it is contained in the domain of S then from (51), we have:

From (50) and (52), we obtain:

Since P(0)u_γ,j is bounded away from zero, we may choose k (possibly depending on γ) such that 〈u_γ,j,u_k〉 is bounded away from zero for small γ, uniformly with respect to R; furthermore, from the assumption (46) and (A2), we deduce

for some positive constants C and C₂. The result is proved. □

Remark 2

We may also ask what happens when an eigenvalue λ_γ merges into the essential spectrum with decreasing γ. For the scalar problem, Theorem 5 in [13] states that in this situation, there is a strictly positive value γ_crit > 0 of the the coupling constant at which λ_γ merges into the essential spectrum.

In our current setup, this is false. Consider the following system:

with \(Q(x)=\left (\begin {array}{cc} 0 & 0 \\ 0 & \frac {-40}{1+x^{2}}\chi _{[0,40]}(x)+\sin \limits (x) \end {array} \right )\) and \(S(x)=\left \{\begin {array}{ll} I_{2} & in \quad [0,50], \\ 0 & in \quad (50,\infty ). \end {array}\right .\) The system can be seen as two scalar problems with q₁(x) = 0 and \(q_{2}(x)=\frac {-40}{1+x^{2}}\chi _{[0,40]}(x)+\sin \limits (x)\). The problem with potential q₂ and γ = 0 has a spectral gap which is approximately (− 0.340363,0.595942), with infinitely many eigenvalues accumulating from below at the top end of the gap [15]. However, these eigenvalues all lie in the essential spectrum \([0,\infty )\) for the problem with potential q₁ and hence also lie in the essential spectrum of the matrix Schrödinger system. Nevertheless, they emerge from the real axis with positive speed, into the upper half-plane, as soon as γ is increased from zero, following the estimate in Theorem 3, (47).

In this section, we present some numerical examples to demonstrate our theoretical results. We generally apply the 3-point finite difference scheme to our system; the number of steps-per-period is fixed when the number of periods increases. The main advantage of using ‘low tech’ finite differences rather than a spectral method or Galerkin method is that there exists a Floquet theory for periodic Jacobi matrices, so the theorems of the previous section may be expected to have analogues for the (infinite) discretised system.

Example 1

Consider the following matrix Schrödinger systems:

1. (P):
   $$ -\underline{u}^{\prime\prime}(x)+\biggl(\frac{-40}{1+x^{2}}\chi_{[0,40]}(x)I_{3}+Q(x)\biggr)\underline{u}(x)=\lambda\underline{u}(x), \quad \underline{u}(0)=\underline{0}; $$
2. (P’):
   $$ -\underline{u}^{\prime\prime}(x)+\biggl(\frac{-40}{1+x^{2}}I_{3}+Q(x)\biggr)\underline{u}(x)=\lambda\underline{u}(x), \quad \underline{u}(0)=\underline{0}; $$

   problem P satisfies hypothesis (A1) while problem P’ satisfies (A1’). The fact that P’ satisfies (A1’) can be proved by an ODE version of the Agmon-type results in Janas, J. et al [10]. In both these equations, we use:

Both problems have exactly the same essential spectrum, though their point spectra are slightly different due to the compactly supported potential well in P. The first two spectral bands are approximately [13]:

For problem P’, we expect an eigenvalue close to λ ≈− 0.1076 in a gap, and an eigenvalue embedded in a spectral band near λ ≈ 0.6336. For both problems, we use the perturbation:

and perceive the resulting eigenvalues with γ = 1/4. Figure 1 shows eigenvalue computations for both P and P’. They show that our estimate (47) holds with \(C_{1}^{\prime } \approx 1293.6\) and \(C_{2}^{\prime } \approx 0.5386\) for (P’), and C₁ ≈ 1291.9 and C₂ ≈ 0.5384 for (P). These figures were calculated using X = 100 and a step-size h = 0.1, both of which were chosen to ensure that the effects of discretisation error would not be seen in the estimated constants.

a, b Logarithmic plot of |λ + iγ − λ_γ| against R

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For the embedded eigenvalues, Fig. 2 shows the effects of interval truncation. In particular we observe that for P the prediction of Theorem 1 holds with C₃ ≈ 0.0039 and C₄ ≈ 0.3842. Theorem 1 has not been proved for P’, which does not have an eventually periodic Q. However, the experiments indicate that the result still holds, with \(C_{3}^{\prime }\approx 0.0047\) and \(C_{4}^{\prime }\approx 0.3842\). For these experiments, the step-size was h = 0.1.

a, b Logarithmic plot of |λ_γ − λ_γ,X,good| against X − R

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Finally, we shall show that the prediction of Theorem 2 for a spurious eigenvalue also holds; in fact, Fig. 3 indicates that it holds not only for P, but also for P’, for which it is not proved. In these figures, we consider the value λ_bad ≈− 0.1847, which lies in a spectral gap but is not an eigenvalue of either problem. We fixed X = 55 and varied R from 19 to 43 in steps of 4. The value of h was again chosen small enough to suppress effects of discretisation error. The constants C₅ and C₆ predicted by Theorem 2 are C₅ ≈ 0.0017 and C₆ ≈ 0.5345; it seems also that \(C_{5}^{\prime }\approx 0.0017\) and \(C_{6}^{\prime }\approx 0.5345\) for P’.

a, b Logarithmic plot of |I(λ_γ,X,bad)| against X − R

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Example 2

Consider the system:

with the following perturbed periodic potential:

in which k(x) = sech²(x)χ_[0,5](x).

This rather unwieldy formula was obtained by an expression

in which \(\tilde {Q}(x) = -6k(x) I_{5} + \text {diag}(n^{2}/4,n=1,\ldots ,5)\) and T is the matrix of orthonormal eigenvectors of a randomly chosen real, symmetric 5 × 5 matrix. According to [2], one eigenvalue for the scalar problem with q(x) = − 6k(x) is − 1; from this, we know that one of the eigenvalues of our Hamiltonian system is 1.25 which is an embedded eigenvalue in the essential spectrum \([1/4,\infty )\) of this multi-channel problem. We expect that if the dissipative barrier method is working well, then we should see an eigenvalue close to 1.25 + 0.25i when γ = 1/4. Figure 4 shows the plot of the corresponding eigenfunction, for which the calculated eigenvalue was approximately 1.24998 + 0.24993i. This illustrates the usefulness of the dissipative barrier method for lifting embedded eigenvalues out of the essential spectrum and making them easier to calculate, even bearing in mind Remark 2.

Plot of a finite difference approximation of a genuine eigenfunction

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Example 3

We consider an equation of the form

Here W is a continuous, periodic matrix, strictly positive definite, so that the corresponding weighted L² space is equivalent to \(L^{2}(0,\infty )\). The matrix S(x) was chosen to be compactly supported and bounded. The matrix D is a diagonal matrix with

qualitatively this is a second-order differentiation matrix.

The main differences compared to the cases considered in our theorems are, firstly, the weight W(x) is no longer the identity; and secondly, the compactly supported dissipative barrier S(x) is now also multiplied by the spectral parameter. The form of this problem is inspired by work in optics with complex index of refraction (see, e.g., [7]).

It is not difficult to show that the Floquet theory still holds for the problem on \([R,\infty )\), and so M_right,X(λ) still converges exponentially fast to M_right(λ) as \(X\to \infty \). For the problem on [0,R], there is now an additional λ-dependence of the barrier term, but M_left(λ) is still meromorphic. We therefore expect to see fast convergence of the eigenvalues lying well away from the real axis.

Figure 5 shows the results of computations in the purely diagonal case

with all matrices being of dimension 5 × 5. These results were computed using the Numerov discretisation [14], with a uniform mesh of 80 intervals per period (mesh size 2π/80). Because the values of λ in Fig. 5 are not large, this mesh size is sufficient to ensure that the points plotted in Fig. 5 will not move in the ‘eyeball norm’ if the mesh size is halved.

Eigenvalue approximations for coefficients in (53) computed using [0,20π] and [0,40π] as approximations to \([0,\infty )\). Eigenvalues are marked with asterisks for [0,20π] and circles for [0,40π]

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In Fig. 5, we see that the asterisks (shorter interval approximations) and circles (longer interval approximations) are essentially coincident for the eigenvalues well away from the real axis. We expected this, due to the exponentially small error which interval truncation causes. The more interesting parts are the ‘loops’ in the upper half plane, one of which starts at approximately λ = 1.75 and returns to the real axis around λ = 2.6. Here the asterisk loop (approximating interval [0,20π]) is approximately twice as far from the real axis as the circle loop (approximating interval [0,40π]). These loops are approximations to a spectral band. Though we have not proved this, they appear to converge at a rate 1/X, as the interval [0,X] goes to infinity. Note that there are also other approximations to (parts of) spectral bands, due to the fact that the spectral multiplicity of the higher bands can be greater than 1. It seems that, in this picture, only the bands near 0.5, and from 0.75 to just below 1, are simple.

In Fig. 6, we repeated the experiments using non-diagonal W(x) given by

The same phenomena are noted as in the diagonal coefficient case, though the different scale on the vertical axis makes the slow convergence to the essential spectrum more stark.

Eigenvalue approximations for coefficients in (54) computed using [0,20π] and [0,40π] as approximations to \([0,\infty )\). Eigenvalues are marked with asterisks for [0,20π] and circles for [0,40π]

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We have introduced a method to calculate eigenvalues in gaps of matrix-valued Schrödinger operators. Theoretically, we have shown that the relatively compact dissipative perturbation technique together with domain truncation obtain approximations of isolated eigenvalues close to the ones of the original problem. Moreover, spurious eigenvalues can be predicted using this method and are characterised by exponentially small imaginary parts. These approximations have been computed using finite difference schemes for some numerical examples and have shown excellent agreement with the theoretical part. An additional remark on this procedure is that the approximating results of the implementations of both fast decaying periodic potentials and compactly supported periodic potentials (see, e.g., Example 1) are mostly the same.

We have also observed the effectiveness of the presented method when the weighted matrix is different from the identity and the dissipative barrier is multiplied by the spectral parameter as in Example 3. The only caveat for these cases is that the approximations to the spectral bands do not converge fast, so if very high accuracy is required then one should use λ-dependent non-reflecting boundary conditions [7].

One of the main sources of Hamiltonian eigenvalue problems is the use of semi-discretisation for PDE problems on waveguides. In future work, we will consider the PDE:

on a semi-infinite waveguide. These give rise to a Hamiltonian system in which Q(x) in our problem is an operator in \(l^{2}(\mathbb {N})\); they can also be studied directly in PDE form.