On the ground state for quantum graphs
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Abstract
Groundstate eigenfunctions of Schrödinger operators can often be chosen positive. We analyse to which extent this is true for quantum graphs—differential operators on metric graphs. It is shown that the theorem holds in the case of generalised delta couplings at the vertices—a new class of vertex conditions introduced in the paper. It is shown that this class of vertex conditions is optimal. Relations to positivity preserving and positivity improving semigroups are clarified.
Keywords
Quantum graphs Positivity preserving semigroups Ground stateMathematics Subject Classification
34L15 35R301 Introduction
Perron–Frobenius theorem [8, 9, 10, 17, 18, 22] states that for a symmetric matrix with positive entries the largest eigenvalue is nondegenerate and the corresponding eigenvector can be chosen having positive entries. This theorem has been generalised for differential operators, where instead of the largest eigenvalue one looks at the ground state, and is closely related to Courant nodal domain theorem stating that the nth eigenfunction has at most n nodal domains [5, 6]. This theorem has been extended for quantum graphs in [2, 11], see also [7]. Our aim is to examine to which extent a counterpart of Perron–Frobenius theorem holds for operators on metric graphs often called quantum graphs. This theorem is trivial for Laplacians with standard vertex conditions, since the ground state is a constant function. For Schrödinger operators with nonconstant potential, the ground state cannot in general be calculated explicitly even for standard vertex conditions and the corresponding statement seems to become a mathematical folklore, but no rigorous proof could be traced, despite extensive literature on the subject [3, 20]. One possible explanation is that metric graphs are locally onedimensional objects, but in one dimension the statement can be obtained as an easy corollary of Courant nodal domain theorem: if an eigenfunction has a zero, then there are two nodal domains. In fact, it is enough to use Sturm oscillation theory in that case. But for metric graphs single zeroes do not necessarily split the graphs into nodal domains and Courant theorem cannot be applied directly.
Our second source of inspiration is the intrinsic connection between positivity and nondegeneracy of the ground state and positivity preserving property of the operator or the corresponding semigroups [12, 21]. Using this connection, we shall be able to characterise a large class of differential operators on metric graphs for which it is guaranteed that the groundstate eigenfunction does not change sign. We just pick up the class of operators for which the quadratic form does not increase under taking the absolute value [see (6)]. The corresponding admissible vertex conditions have never been used before and allow to interpolate between (positive weighted) vertex delta interactions and most general vertex conditions. We call such vertex conditions generalised delta couplings. This class is a generalisation of vertex conditions already considered in [4].
The structure of the article is as follows. In Sect. 2, we prove positivity of the groundstate eigenfunction for standard vertex conditions. We believe that this form of our result will be widely used even outside the mathematical community, since standard vertex conditions are the most common conditions for quantum graphs. The case of general vertex conditions is considered in Sect. 3; an analogue of Perron–Frobenius theorem is established for the generalised delta couplings. Section 4 is devoted to positivity preserving semigroups generated by quantum graphs.
2 Positivity of the groundstate eigenfunction for standard vertex conditions
In this section, we consider Schrödinger operators on metric graphs with standard vertex conditions. To our opinion, such elementary proof is awaited by mathematical community and it will enable us to indicate important ingredients that will be used to motivate our main result (Theorem 3).
Theorem 1
The original proof of Pleijel [19] of Courant nodal domain theorem can be transferred to standard quantum graphs without many modifications. That proof would imply that the groundstate eigenfunction has a single nodal domain. But this property cannot exclude the possibility that the function has zeroes—not every zero leads to nodal domains as in the case of one interval. Consider for example a nonnegative function on a ring: if the function has just one zero, then there is just one nodal domain. Then, it is necessary to show that the groundstate eigenfunction cannot have zeroes without changing sign both inside the edges and at the vertices (see the second part of the proof below). Other standard methods developed for operators in infinitedimensional spaces cannot be applied directly either [12, 14]. Moreover, the proof presented below is based on the quadratic form analysis which will play essential role in the rest of the paper.
Proof
Any ground state is an admissible function minimising the Rayleigh quotient. If a function u is realvalued and admissible, then \( \vert u \vert \) is also admissible with the same Rayleigh quotient. Hence, the minimiser of (4) can be chosen not only real, but even nonnegative.
We shall prove now that if \( \psi _1 \) is a nonnegative minimiser for (4), then it is never equal to zero. This would imply that \( \psi _1 \) may be chosen strictly positive. We need to exclude that \( \psi _1 \) may have zeroes on the edges or at the vertices.
If \( \psi _1 \) is a minimiser of the Rayleigh quotient, then it is an eigenfunction of the corresponding Schrödinger equation, i.e. it satisfies the differential equation on the edges and as well as vertex conditions (see Appendix A).
Assume now that \( \psi _1 \) is equal to zero at a certain vertex \( V_m.\) Since \( \psi _1 \) is a minimiser for (4), it satisfies the standard vertex conditions at this vertex. The function \( \psi _1 \) is nonnegative and is equal to zero at the vertex. It follows that all normal derivatives are nonnegative, but their sum is equal to zero; hence, all normal derivatives are actually equal to zero. We see that as before \( \psi _1 \) satisfies a secondorder differential equation with zero Cauchy data on every edge incident to \( V_m\). It follows that \( \psi _1 \) is zero not only at this particular vertex \( V_m \) but at all neighbouring vertices as well. Repeating the argument, we conclude that \( \psi _1 \) is identically equal to zero on the whole \( \varGamma \) (which is assumed to be connected) and therefore is not an eigenfunction.
It remains to prove that the lowest eigenvalue is simple. Assume that the lowest eigenvalue is not simple and there exists two orthogonal eigenfunctions \( \psi _1 \) and \( \psi _2. \) One of these eigenfunctions can be chosen positive, say \( \psi _1 \), then the other one necessarily has zeroes, since it is continuous and attains both positive and negative values being orthogonal to \( \psi _1.\) Every such function is identically equal to zero as we have already proven. Hence, the lowest eigenvalue is in fact simple. \(\square \)
In a similar way, the following corollary can be proven:
Corollary 1
To prove the corollary, one needs to take into account that the domain of the quadratic form is again invariant under taking the complex conjugate and the absolute value. Moreover, if \( \psi _1 \) is equal to zero at a vertex, then it satisfies standard vertex conditions there.

for complexvalued functions the quadratic form is invariant under complex conjugation;

for realvalued functions the quadratic form is invariant under taking the absolute value.
3 Positivity of the groundstate eigenfunction for general vertex conditions
We start by providing a counterexample showing that in order to guarantee positiveness of the ground state the set of allowed vertex conditions has to be restricted.
3.1 Counterexample
It is not hard to provide examples of quantum graphs with positive nondegenerate ground states. Our aim here is to characterise a wider class of vertex conditions that guarantee positivity of the ground state. Considered counterexample should not give an impression that vertex conditions leading to violation of the positivity of the groundstate eigenfunction are pathological.
3.2 A few definitions
We shall need few definitions: two are coming from [21] (positive and strongly positive functions), and one (strictly positive) is new—we need it to formulate a stronger version of the theorem.
Definition 1
A function f is called positive if it is nonnegative \( f(x) \ge 0. \) A function \( f \in L_2 (\varGamma ) \) is called strongly positive if \( f(x) > 0 \) holds almost everywhere on \( \varGamma . \) Finally, a function f is called strictly positive if \( f(x) > 0 \) holds everywhere on \( \varGamma \) except at those vertices, where Dirichlet conditions are assumed.
3.3 Quadratic form
3.4 Generalised delta couplings
Consider a new subclass of Hermitian vertex conditions to be called generalised delta couplings. This class is a generalisation of weighted delta couplings (see for example [1] where approximations of diffusion in thin tubes of different sizes were considered) and vectorvalued Robin conditions (see for example [4], where such conditions were considered in connection with positivity preserving semigroups). It appears that precisely the new class of vertex conditions guarantees that the groundstate eigenfunction is positive. As a first step, we describe generalised delta conditions for just one vertex.

all coordinates of \( \mathbf {a}^j \) are nonnegative numbers^{2}\( \mathbf {a}^j \in \mathbb R_+^{d} ;\)
 the vectors have disjoint supports so that for all \( \ell : x_\ell \in V \)holds.$$\begin{aligned} \mathbf {a}^j (x_\ell ) \cdot \mathbf {a}^i (x_\ell ) = 0, \; \text{ provided } \; j \ne i \end{aligned}$$
Changing the order \( n, \; 0 \le n \le d \), of the delta couplings allows one to interpolate between the classical delta couplings and the most general vertex conditions, so that \( n=1 \) corresponds to the usual weighted delta couplings and \( n=d \)—to the most general vertex conditions with \( \mathcal B = \mathbb C^d.\)
Note that in Eq. (13) we introduced a new reduced vector \( \mathbf{u} = ( \mathbf u^1, \mathbf u^2, \ldots , \mathbf u^n)\)—it contains the common weighted values of the vector \( \mathbf {u} \). The dimension of the vector coincides with the dimension n of the linear subspace \( \mathcal B\).
Of course, only properly connecting vertex conditions should be considered. Vertex conditions are called properly connecting if and only if the vertex cannot be divided into several vertices so that equivalent vertex conditions connecting boundary values from each of the new vertices can be found.
 1.The union of supports of vectors \( \mathbf {a}^j \) coincides with all endpoints in V:$$\begin{aligned} \cup _{j=1}^n \mathrm{supp}\; (\mathbf {a}^j) = \{ x_l \}_{x_l \in V}. \end{aligned}$$(14)
 2.
The matrix \( \mathbf A = \{ A_{ji} \}_{j,i =1}^n \) is irreducible, i.e. it cannot be put into a blockdiagonal form by permutations.
If the second condition is not satisfied, then the vertex V can be chopped into two (or more) vertices preserving the vertex conditions. Such conditions correspond to the metric graph, where the vertex V is divided.
3.5 Invariance of the quadratic form domain
It is important for the future to realise that for generalised delta interactions the domain of the quadratic form is invariant under taking the absolute value, provided that the weights are all positive. We have this property for any matrix \( \mathbf A.\) One has to require that the matrix \( \mathbf A \) is negative Minkowski Mmatrix in order to ensure that the quadratic form does not increase under taking the absolute value. The following theorem states in particular that the quadratic form possesses these properties if and only if the vertex conditions are generalised delta couplings.
Theorem 2
Proof
We divide the proof into two steps.
Step 1The domain of the quadratic form is invariant under taking the absolute value if only if the subspace\( \mathcal B \)appearing in (10) is trivial or is generated by several vectors\( \mathbf {e}^j \)with nonnegative coordinates and disjoint supports.
In other words, we claim that the boundary values of functions from the domain of the quadratic form belong to the subspace in \( \mathbb C^{2N}\) (N is the number of edges in \( \varGamma \)) generated by the vectors \( \mathbf {e}^j \) having very special properties: all coordinates are nonnegative and their supports are disjoint ( i.e. no two vectors have positive coordinates with the same index).
The same analysis applies to any two vectors from the basis; hence, we conclude that all \( \mathbf {e}^j \) not only have disjoint supports but also their entries are nonnegative. It might happen that several vectors \( \mathbf {e}^j \) correspond to the same vertex. Thus, we have proven that vertex conditions should be the generalised delta couplings.
Without loss of generality, we normalise vectors \( \mathbf {e}^j \) and arrange them so the vectors \( \mathbf {a}^j_m \) are nonzero at the endpoints belonging to the vertex \( V_m \) only.
Step 2 The value of the quadratic form does not increase while taking the absolute value if and only if the Hermitian matrix \(  \mathbf A =  (\mathbf A_1 \oplus \mathbf A_2 \oplus \cdots \oplus \mathbf A_M) \) is from Minkowski class (all nondiagonal entries are nonnegative).
Summing up, the domain of the quadratic form is given by the requirement that at each vertex \( V_m \) the vector of boundary values belongs to a subspace spanned by \( n_m \le d_m \) vectors with disjoint supports and positive coordinates \( \mathbf {u}_m \in \mathcal B_m = \mathfrak L \{ \mathbf {a}_m^{i} \}_{i=1}^{n_m} \) and the matrix \( \mathbf A_m\) in \( \mathcal B_m \) is Hermitian with nonpositive nondiagonal entries. Remember that we consider only properly connecting vertex conditions implying that the supports of vectors \( \mathbf {a}^j_m \) span \( \mathbb C^{d_m}\) (equation (14)) and the matrix \( \mathbf A_m \) is irreducible.
Thus, the vertex conditions coincide with the generalised delta couplings determined by (12). \(\square \)
It is straightforward to see that generalised delta couplings guarantee that (16) holds. Using Beurling–Deny criterion one may characterise possible vertex conditions with the help of Theorem 6.85 in [16], but our characterisation is much more explicit. We return to this question in Sect. 4.
3.6 Positivity of the ground state
We are ready to generalise the theorem to Schrödinger operators with generalised delta couplings.
Theorem 3
Assume that all assumptions of Theorem 2 are satisfied and the graph \( \varGamma \) is connected. Then, the ground state is unique and may be chosen real, in which case it is strictly positive.
Proof
For the generalised delta couplings and Dirichlet conditions, the domain of the quadratic form is invariant under taking the absolute value and under complex conjugation. Moreover, the value of the quadratic form does not increase under these operations. Hence, as in Sect. 2 the ground state may be chosen real and nonnegative. It remains to prove that such a groundstate eigenfunction is strictly positive, i.e. it is equal to zero only at the vertices, where Dirichlet conditions are assumed (Dirichlet points).

\( x_0 \) is an inner point on an edge,

\( x_0 \) belongs to a vertex.
It follows that on each edge incident to \( V_m\) the function \( \psi _1 \) is a solution of the secondorder differential equation satisfying trivial Cauchy data. Hence, the function is identically equal to zero on all edges incident to \( V_m. \)
Repeating the argument for the vertices connected to \( V_m \) by an edge, we conclude that \( \psi _1 \) is zero on all edges incident to those vertices. Continuing this procedure, we shall prove that \( \psi _1 \equiv 0 \) on the whole \( \varGamma \), since the graph is connected. \(\square \)
In fact we have proven that if the quadratic form of the Schrödinger operator on a connected finite compact metric graph does not increase under taking the absolute value, then the corresponding ground state is strictly positive, i.e. the eigenfunction is equal to zero only at the points where the Dirichlet conditions are assumed. We not only proved that \( \psi _1 \) is strongly positive, but characterised explicitly all points where \( \psi _1 \) is equal to zero. If there are no Dirichlet points, then \( \psi _1 \) is separated from zero \( \psi _1(x) \ge \delta > 0. \)
3.7 Generalised delta couplings are optimal
The goal of this subsection is to show that the assumptions of Theorem 2 are in some sense necessary to guarantee positivity of the groundstate eigenfunction: given a vertex with conditions violating these assumptions, one may construct a quantum graph with the groundstate eigenfunction either complexvalued or not sign definite.
Theorem 4
The class of generalised delta couplings and Dirichlet conditions is optimal to guarantee positivity of the groundstate eigenfunction for quantum graphs in the following sense:
Assume that Hermitian vertex conditions not from the selected class are given. Then, there exists a metric graph \( \varGamma \) and a Laplace operator on it satisfying given vertex conditions at one of the vertices and generalised delta couplings and Dirichlet conditions at the other vertices such that its groundstate eigenfunction cannot be chosen nonnegative.
Proof
Let \( \varGamma \) be dstar graph with the edge lengths \( \ell _j, j= 1,2, \ldots , d\) (to be specified later). Consider the Laplace operator on \( \varGamma \) assuming Dirichlet vertex conditions at the degree one vertices and given vertex conditions at the central vertex V (not generalised delta couplings, not Dirichlet conditions).
Both determinants are given by analytic functions, convergence of analytic functions imply convergence of their zeroes in any bounded domain (Cauchy formula). Therefore, for sufficiently small \( \ell _j, j= 1,2, \ldots , \ell _K \) the zeroes of the secular equation for dstar graph are close to the spectrum of the Dirichlet–Dirichlet Laplacian on \( [0, \ell _0]\) with multiplicity \( dK \). Moreover, the corresponding eigenfunctions can also be obtained in the limit.
The limit spectrum is the union of the spectrum of the Dirichlet–Dirichlet Laplacian on \( [0, \ell _0 ] \) (with multiplicity \( dK \)) and the spectrum of the Laplacian on the interval \( [0, \ell _0 ] \) with Robin condition \( u'(0) = a_{11} u(0) \) at the left endpoint and Dirichlet condition at the right. The latter spectrum is described by the equation \( \cos k \ell _0 + \frac{1}{k} a_{11} \sin k \ell _0 = 0. \) The groundstate eigenfunction is given by the groundstate eigenfunction of the Robin–Dirichlet Laplacian multiplied by \( \mathbf {e}^1. \) Therefore, all coordinates in \( \mathbf {e}^1 \) should be nonnegative, otherwise the groundstate eigenfunction of the Laplacian on the dstar graph is not nonnegative.
Similarly, we prove that the coordinates of all vectors \( \mathbf {e}^2, \ldots , \mathbf {e}^K \) forming basis in \( \mathcal B \) are nonnegative.
Considering different pairs of edges, we conclude that to guarantee positivity of the groundstate eigenfunction for all graphs one should require that \( \mathbf A \) is negative Minkowski Mmatrix. \(\square \)
Our result does not imply that the groundstate eigenfunction cannot be chosen nonnegative if vertex conditions at one of the vertices are not generalised delta couplings or Dirichlet. To prove necessity, we considered different graphs. Taking the dstar graph with \( \ell _{K+1} = \cdots = \ell _d = \ell _0 \) and \( \ell _1, \ell _2, \ell _3, \ldots , \ell _K \ll \ell _0.\) As we have seen, the ground state is determined by the ground state of the Dirichlet–Dirichlet Laplacian on \( [0, \ell _0] \) and therefore is independent of the matrix \( \mathbf A \); hence, it is always nonnegative.
4 On positivity preserving semigroups
The nondegeneracy and positivity of the ground state are often connected to the fact that the corresponding semigroup is positivity preserving. In what follows, we shall explore this direction, but first we recall a few wellknown facts (see [21], some of the formulations are slightly shortened in order to fit our needs).
Proposition 1
 1.
E is a simple eigenvalue with a strongly positive eigenvector.
 2.
\( e^{tH}\) is positivity improving for all \( t > 0. \)
Proposition 2
 1.
\( e^{tH} \) is positivity preserving for all \( t > 0. \)
 2.
\( \langle \vert u \vert , H \vert u \vert \rangle \le \langle u, H u \rangle \) for all \( u \in L_2. \)
With these propositions in mind, we can prove the following statements.
Theorem 5
Assume that all conditions of Theorem 2 are fulfilled. Then, the corresponding Schrödinger operator with \( q \in L_1 (\varGamma ) \) is a generator of positivity improving semigroup.
Proof
The operator is semibounded with discrete spectrum. The quadratic form does not increase when taking the absolute value; hence, Proposition 2 implies that the corresponding semigroup is positivity preserving.
Moreover, we have proven that the ground state is strictly positive and hence is also strongly positive. Therefore, Proposition 1 implies that the semigroup generated by the operator is positivity improving. \(\square \)
It is also possible to turn our reasoning to prove the opposite result:
Theorem 6
Assume that the finite compact metric graph \( \varGamma \) is connected and the operator \( L_q (\varGamma ) \) with \( q \in L_1 (\varGamma )\) is a generator of a positivity preserving semigroup in \( L_2 (\varGamma ). \) Then, the ground state is strictly positive, the vertex conditions at each vertex are either Dirichlet or of generalised deltatype with all weights positive and the matrix \( \mathbf A \) is a Minkwoski Mmatrix. Moreover, the semigroup is positivity improving.
Proof
The semigroup is positivity preserving if and only if the domain of the quadratic form is invariant and does not increase when taking the absolute value (see Proposition 2). We have already characterised all corresponding vertex conditions proving Theorem 2—all such vertex conditions are of generalised delta type. Then, Theorem 5 implies that the semigroup is also positivity improving. \(\square \)
This statement can be found in [16] (Theorem 6.85) (see also [4, 15]), but without an explicit description of the vertex conditions as we have in terms of generalised delta couplings.
Footnotes
Notes
Acknowledgements
Open access funding provided by Stockholm University. The author would like to thank Rami Band and Delio Mugnolo for numerous discussions leading to the birth of this work. Special thanks go to anonymous referees who came with suggestions leading to improvements of the manuscript, especially the referee who forced him to include Sect. 3.7.
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