On the orbital instability of excited states for the NLS equation with the $\delta$-interaction on a star graph

We study the nonlinear Schr\"odinger equation (NLS) on a star graph $\mathcal{G}$. At the vertex an interaction occurs described by a boundary condition of delta type with strength $\alpha\in \mathbb{R}$. We investigate an orbital instability of the standing waves $e^{i\omega t}\mathbf{\Phi}(x)$ of NLS-$\delta$ equation with attractive power nonlinearity on $\mathcal{G}$ when the profile $\Phi(x)$ has mixed structure (i.e. has bumps and tails). In our approach we essentially use the extension theory of symmetric operators by Krein - von Neumann, and the analytic perturbations theory, avoiding the variational techniques standard in the stability study. We also prove orbital stability of the unique standing wave solution of NLS-$\delta$ equation with repulsive nonlinearity.

Practically, equation (1.1) means that on each edge of the graph, i.e. on each half-line, we have i∂ t u j (t, x) + ∂ 2 x u j (t, x) + µ|u j (t, x)| p−1 u j (t, x) = 0, x > 0, j ∈ {1, ..., N}. A complete description of this model requires smoothness conditions along the edges and some junction conditions at the vertex ν = 0. The family of self-adjoint conditions naturally arising at the vertex ν = 0 of the star graph G has the following description (U − I)U(t, 0) + i(U + I)U ′ (t, 0) = 0, (1.2) where U(t, 0) = (u j (t, 0)) N j=1 , U ′ (t, 0) = (u ′ j (t, 0)) N j=1 , U is an arbitrary unitary N ×N matrix, and I is the N × N identity matrix . The conditions (1.2) at ν = 0 define the N 2 -parametric family of self-adjoint extensions of the closable symmetric operator ( [9,Chapter 17]) In this work we consider the matrix U which corresponds to so-called δ-interaction at vertex ν = 0. More precisely, the matrix where I is the N × N matrix whose all entries equal 1, induces the following nonlinear Schrödinger equation with δ-interaction (NLS-δ) on the star graph G where H α δ is the self-adjoint operator on L 2 (G) defined for V = (v j ) N j=1 by (1.4) Condition at ν = 0 can be considered as an analog of δ-interaction condition for the Schrödinger operator on the line (see [4]), which justifies the name of the equation. When α = 0, one arrives at the known Kirchhoff condition. Equation (1.1) models propagation through junctions in networks (see [8,17,19]). The analysis of the behavior of NLS equation on networks is not yet fully developed, but it is currently growing (see [1,2] for the references). In particular, models of BEC on graphs/networks is a topic of active research (see [10,19]).
It is worth noting that the quantum graphs (star graphs equipped with a linear Hamiltonian H) have been a very developed subject in the last couple of decades. They give simplified models in mathematics, physics, chemistry, and engineering, when one considers propagation of waves of various type through a quasi one-dimensional (e.g. meso-or nanoscale) system that looks like a thing neighborhood of a graph (see [8-10, 17, 19] for details and references).
The nonlinear PDEs on graphs have been studied in the last ten years in the context of existence, stability, and propagation of solitary waves. The main purpose of this work is the investigation of the stability properties of the standing wave solutions to NLS-δ equation (1.3). In a series of papers R. Adami, C. Cacciapuoti, D. Finco, D. Noja (see [1] and references therein) investigated variational and stability properties of standing wave solutions to equation (1.3) for µ = 1 (attractive nonlinearity). In [2] it was shown that all possible profiles Φ(x) belong to the specific family of N −1 2 + 1 vector functions (see Theorem 2.3 below) consisting of bumps and tails. It was proved that there exists a global minimizer of the constrained NLS action for −N √ ω < α < α * < 0. This minimizer coincides with the N-tails stationary state symmetric under permutation of edges, which consists of decaying tails (notice also that this profile minimizes NLS energy under fixed mass constraint for sufficiently small mass [3]). Using minimization property, the authors proved the orbital stability of this N-tails stationary state in the case −N √ ω < α < α * < 0.
In [1] it was shown that although the constrained minimization problem does not admit global minimizers for large mass, the N-tails stationary state is still a local minimizer of the constrained energy which induces orbital stability for any −N √ ω < α < 0. The orbital stability of N-tails (bumps) profile was studied in [5] in the framework of the extension theory. Other example of application of the extension theory to stability study can be found in [6].
In the case α < 0 it was shown that the NLS action grows when the number of tails in the stationary state increases, i.e. one can call the rest of the profiles (except N-tails stationary state) excited stationary states (see Subsection 2.1). Nothing was known up to now about stability properties of the excited states. This is a subject of special interest because there are only few cases where excited states of NLS equations are explicitly known.
In the present paper we provide sufficient condition for orbital instability of the excited states of (1.3). Moreover, we obtain the novel result on the orbital stability/instability of the standing waves in the case α > 0. , and ω > α 2 (N −2k) 2 . Let also the profile Φ α k be defined by (2.3), and the spaces E and E k be defined in notation section.Then the following assertions hold.
(ii) Let α > 0, then , and e iωt Φ α k is orbitally stable in E k as ω ∈ (ω k , ∞); 3) for p ≥ 5 the standing wave e iωt Φ α k is orbitally unstable in E. In the case of p > 5, α < 0, and ω > ω * k our approach does not provide any information about the stability of the excited states Φ α k . Our approach contains new original technique. It does not use variational analysis, and it is based on the extension theory of symmetric operators, the analytic perturbations theory, and Weinstein-Grillakis-Shatah-Strauss approach (see [11,12]).
In Section 4 we proved instability result for the standing wave solution of classical halfsoliton type to the NLS equation with Kirchhoff condition at ν = 0 announced in [15] without proof (see Theorem 4.2).
The paper is organized as follows. In Section 2 we provide preliminary information on the existence of the standing waves and their orbital stability. Section 3 is devoted to the proof of Theorem 1.1. In Section 4 we consider NLS on G with Kirchhoff condition at ν = 0. Section 5 is concerned with the proof of Theorem 1.2. In Section 6 we discuss key facts of the extension theory of symmetric operators and also estimate Morse index of the Schrödinger operator associated with the linearization of the NLS-δ equation (1.3).
Notation. Let A be a densely defined symmetric operator in a Hilbert space H. The domain of A is denoted by dom(A). The deficiency subspaces and deficiency numbers of A are denoted by N ± (A) := ker(A * ∓ iI) and n ± (A) := dim N ± (A) respectively. The number of negative eigenvalues counting multiplicities is denoted by n(A) (the Morse index ). The spectrum of A is denoted by σ(A).
We denote by G the star graph constituted by N half-lines attached to a common vertex ν = 0. On the graph we define For instance, the norm in L p (G) is defined by By || · || p we denote the norm in L p (G), and (·, ·) 2 denotes the scalar product in L 2 (G). We also denote by E and L 2 k (G) the spaces and E k = E ∩ L 2 k (G). We also use the following notation , and E eq = E ∩ L 2 eq (G).
3) has a maximal solution U max (t) defined on an interval [0, T ⋆ ), and the following "blow-up alternative" holds: ei- Furthermore, the conservation of energy and mass holds, that is, for any t ∈ [0, T ⋆ ) we have where the energy E is defined by Using Gagliardo-Nirenberg estimate and conservation properties, the authors in [2] showed global well-posedness of the Cauchy problem for (1.3) in the case 1 < p < 5.
Let us discuss briefly the existence of the standing wave solutions U(t, x) = e iωt Φ(x) for (1.3). It is easily seen that the amplitude Φ ∈ D α satisfies the following stationary equation In [2] the authors obtained the following description of all solutions to equation (2.2) in the case µ = 1.

(2.3)
Remark 2.4. (i) Note that in the case α < 0 vector Φ α k = (ϕ α k,j ) N j=1 has k bumps and N −k tails. It is easily seen that Φ α 0 is the N-tails profile. Moreover, the N-tails profile is the only symmetric (i.e. invariant under permutations of the edges) solution of equation (2.2). In the case N = 5 we have three types of profiles: 5-tails profile, 4-tails/1-bump profile and 3-tails/2-bumps profile. They are demonstrated on Figure 1 (from the left to the right).
(ii) In the case α > 0 vector Φ α k = (ϕ α k,j ) N j=1 has k tails and N − k bumps respectively. For N = 5 we have: 5-bumps profile, 4-bumps/1-tail profile, 3-bumps/ 2-tails profile. They are demonstrated on Figure 2 (from the left to the right).  In [2] it was shown that for any p > 1 there is α * < 0 such that for −N √ ω < α < α * the N-tails profile Φ α 0 minimizes the action functional on the Nehari manifold Namely, the N-tails profile Φ α 0 is the ground state for the action S on the manifold N . In [1] the authors showed that Φ α 0 is a local minimizer of the energy functional E defined by (2.1) among functions with the same mass.

Review of the previous results on the orbital stability
Crucial role in the stability analysis is played by the symmetries of NLS equation (1.3). The basic symmetry associated to the mentioned equation is phase-invariance (in particular, translation invariance does not hold due to the defect at ν = 0). Thus, it is reasonable to define orbital stability as follows.
Definition 2.5. The standing wave U(t, x) = e iωt Φ(x) is said to be orbitally stable in a Hilbert space H if for any ε > 0 there exists η > 0 with the following property: if U 0 ∈ H satisfies ||U 0 − Φ|| H < η, then the solution U(t) of (1.3) with U(0) = U 0 exists for any t ∈ R, and sup Otherwise, the standing wave U(t, x) = e iωt Φ(x) is said to be orbitally unstable in H.
In [2] the following orbital stability result has been shown for N-tails profile Φ α 0 defined by (2.3).
In [5] we proposed a short proof (based on the extension theory of symmetric operators) of the orbital stability of e iωt Φ α 0 for any α < 0. It's worth mentioning that our proof of stability is very short (comparatively with one in [1]). Moreover, using extension theory of symmetric operators, we showed the following instability theorem for the N-bumps profile as α > 0.
For the sake of completeness, we remark that in [5] NLS equation with δ ′ -interaction on G was considered. Namely, we considered the case when In [5] we studied the orbital stability of the standing wave In particular, we proved the following result.
3 Orbital stability of standing waves of NLS-δ equation with attractive nonlinearity

Stability framework
One of the main approaches in stability analysis of standing waves for nonlinear Schrödinger models is that developed by Weinstein [23] and Grillakis, Shatah and Strauss [11,12]. To formulate the stability theorem for NLS-δ equation (1.3) in the framework of this theory we will establish its basic objects. Let Φ α k be defined by (2.3). In what follows we will use the notation Φ k := Φ α k . We start verifying that the profile Φ k is a critical point of the action functional S defined by (2.4). Indeed, for U, V ∈ E, In the approach by [12] crucial role is played by spectral properties of the linear operator associated with the second derivative of S calculated at Φ k (linearization of (1.3)). Thus, splitting U, V ∈ E into real and imaginary parts U = U 1 + iU 2 and V = V 1 + iV 2 , with the vector functions U j , V j , j ∈ {1, 2}, being real valued, we get Then it is easily seen that S ′′ (Φ k )(U, V) can be formally rewritten as where ϕ k,j = ϕ α k,j . Next, we determine the self-adjoint operators associated with the forms B α j,k in order to establish a self-contained analysis. First note that the forms B α j,k , j ∈ {1, 2}, are bilinear bounded from below and closed. Thus, there appear self-adjoint operators L α 1,k and L α 2,k associated (uniquely) with B α 1,k and B α 2,k by the First Representation Theorem (see [18, Chapter VI, Section 2.1]), namely, In the following theorem we describe the operators L α 1,k and L α 2,k in more explicit form.
where δ i,j is the Kronecker symbol.
Proof. Since the proof for L α 2,k is similar to the one for L α 1,k , we deal with L α We denote by L α (resp. L 1,k ) the self-adjoint operator on L 2 (G) associated (by the First Representation Theorem) with B α (resp. B 1,k ). Thus, The operator L α is the self-adjoint extension of the following symmetric operator , which yields the claim. Arguing as in the proof of Theorem 3.6(iii), we can show that the deficiency indices of L 0 are given by n ± (L 0 ) = 1. Therefore, there exists one-parametric family of self-adjoint extensions of L 0 . Similarly to [4, Theorem 3.1.1], we can prove that all self-adjoint extensions of L 0 are given by Indeed, due to [4, Theorem A.1], any self-adjoint extension L of L 0 is defined by

From the last equalities it follows that
which induces that dom( L) ⊆ dom(L β ). Using the fact that L β defined on dom(L β ) is self-adjoint, we arrive at dom( L) = dom(L β ) fore some β ∈ R.
Instability of excited states for NLS-δ equation on a star graph Finally, we need to prove that β = α. Take V ∈ dom(L α ) with V(0) = 0, then we obtain Note that L 1,k is the self-adjoint extension of the following multiplication operator Theorem is proved.
It is easily seen from (3.1) that formally S ′′ (Φ α k ) can be considered as a self-adjoint 2N × 2N matrix operator (see [11,12] for the details) Having established Assumptions 1, 2 (i.e. well-posedness of the associated Cauchy problem and the existence of C 1 in ω standing wave) in [12] (see [ , ω > α 2 (N −2k) 2 , and n(H α k ) be the number of negative eigenvalues of H α k . Suppose also that 1) ker(L α 2,k ) = span{Φ k }, 2) ker(L α 1,k ) = {0}, 3) the negative spectrum of L α 1,k and L α 2,k consists of a finite number of negative eigenvalues (counting multiplicities), 4) the rest of the spectrum of L α 1,k and L α 2,k is positive and bounded away from zero. Then the following assertions hold.
Remark 3.3. Below we will use the above theorem for the smaller space E k .
3.2 Spectral properties of L α 1,k and L α

2,k
Below we describe the spectra of the operators L α 1,k and L α 2,k which will help us to verify the conditions of Theorem 3.2. Our ideas are based on the extension theory of symmetric operators and the perturbation theory.
The main result of this subsection is the following.
Then the following assertions hold.
(iii) The positive part of the spectrum of the operators L α 1,k and L α 2,k is bounded away from zero.
Proof. (i) It is obvious that Φ k ∈ ker(L α 2,k ). To show the equality ker(L α 2,k ) = span{Φ k } let us note that any V = (v j ) N j=1 ∈ H 2 (G) satisfies the following identity Thus, for V ∈ D α we obtain .
Below using the perturbation theory we will study n(L α 1,k ) in the space L 2 k (G) for any k ∈ {1, ..., N −1 2 }. For this purpose let us define the following self-adjoint matrix Schrödinger operator on L 2 (G) with Kirchhoff condition at ν = 0 where ϕ 0 represents the half-soliton solution for the classical NLS model, .., ϕ 0 ). As we intend to study negative spectrum of L α 1,k , we first need to describe spectral properties of L 0 1 (which is "limit value" of L α 1,k as α → 0). (3.7) (iii) The operator L 0 1 has one simple negative eigenvalue in L 2 (G), i.e. n(L 0 1 ) = 1. Moreover, L 0 1 has one simple negative eigenvalue in L 2 k (G) for any k, i.e. n(L 0 1 | L 2 k (G) ) = 1. (iv) The positive part of the spectrum of L 0 1 is bounded away from zero. Proof. The proof repeats the one of Theorem 3.6 in [5]. We give it for self-contentedness.
(i) The only L 2 (R + )-solution to the equation up to a factor). Thus, any element of ker It is easily seen that continuity condition is satisfied since ϕ ′ 0 (0) = 0. Condition (ii) Arguing as in the previous item, we can see that ker(L 0 1 ) is one-dimensional in L 2 k (G), and it is spanned on Φ 0,k .
Let us show that the operator L 0 0 is non-negative. First, note that every component of the vector V = (v j ) N j=1 ∈ H 2 (G) satisfies the following identity Using the above equality and integrating by parts, we get for V ∈ dom(L 0 0 ) where the non-integral term becomes zero by the boundary conditions for V and the fact that x = 0 is the first-order zero for ϕ ′ 0 (i.e. ϕ ′′ 0 (0) = 0). Indeed, Due to the von Neumann decomposition given in Theorem 6.1, we obtain Since n ± (L) = 1, by [20, Chapter IV, Theorem 6], it follows that n ± (L 0 0 ) = 1. Next, due to Proposition 6.3, n(L 0 1 ) ≤ 1. Taking into account that (L 0 ..., ϕ 0 ), we arrive at n(L 0 1 ) = 1. Finally, since Φ 0 ∈ L 2 k (G) for any k, we have n(L 0 1 | L 2 k (G) ) = 1. (iv) follows from Weyl's theorem.
Remark 3.7. Observe that, when we deal with deficiency indices, the operator L 0 0 is assumed to act on complex-valued functions which however does not affect the analysis of negative spectrum of L 0 1 acting on real-valued functions. The following lemma states the analyticity of the family of operators L α 1,k . Lemma 3.8. As a function of α, (L α 1,k ) is real-analytic family of self-adjoint operators of type (B) in the sense of Kato.
Proof. By Theorem 3.1 and [18, Theorem VII-4.2], it suffices to prove that the family of bilinear forms (B α 1,k ) defined in (3.2) is real-analytic of type (B). Indeed, it is immediate that it is bounded from below and closed. Moreover, the decomposition of B α 1,k into B α and B 1,k , implies that α → (B α 1,k V, V) is analytic. Combining Lemma 3.8 and Theorem 3.6, in the framework of the perturbation theory we obtain the following proposition. (ii) For all α ∈ (−α 0 , α 0 ), λ k (α) is the simple isolated second eigenvalue of L α 1,k in L 2 k (G), and F k (α) is the associated eigenvector for λ k (α).
(iii) α 0 can be chosen small enough to ensure that for α ∈ (−α 0 , α 0 ) the spectrum of L α 1,k in L 2 k (G) is positive, except at most the first two eigenvalues. Proof. Using the structure of the spectrum of the operator L 0 1 given in Theorem 3.6(ii)−(iv), we can separate the spectrum σ(L 0 1 ) in L 2 k (G) into two parts σ 0 = {λ 0 1 , 0}, λ 0 1 < 0, and σ 1 by a closed curve Γ (for example, a circle), such that σ 0 belongs to the inner domain of Γ and σ 1 to the outer domain of Γ (note that σ 1 ⊂ (ǫ, +∞) for ǫ > 0). Next, Lemma 3.8 and the analytic perturbations theory imply that Γ ⊂ ρ(L α 1,k ) for sufficiently small |α|, and σ(L α 1,k ) is likewise separated by Γ into two parts, such that the part of σ(L α 1,k ) inside Γ consists of a finite number of eigenvalues with total multiplicity (algebraic) two. Therefore, we obtain from the Kato-Rellich Theorem (see [22,Theorem XII.8]) the existence of two analytic functions λ k , F k defined in a neighborhood of zero such that the items (i), (ii) and (iii) hold. Now we investigate how the perturbed second eigenvalue moves depending on the sign of α. Proposition 3.10. There exists 0 < α 1 < α 0 such that λ k (α) < 0 for any α ∈ (−α 1 , 0), and λ k (α) > 0 for any α ∈ (0, α 1 ). Thus, in L 2 k (G) for α small, we have n(L α 1,k ) = 2 as α < 0, and n(L α 1,k ) = 1 as α > 0. Proof. From Taylor's theorem we have the following expansions (3.12) From (3.11) we obtain 14) The operations in the last equality are componentwise. Equations (3.14), (3.12), and Φ 0,k ∈ D α induce It follows that λ 0,k is positive for sufficiently small |α| (due to negativity of ϕ ′ 0 on R + ), which in view of (3.11) ends the proof. Now we can count the number of negative eigenvalues of L α 1,k in L 2 k (G) for any α, using a classical continuation argument based on the Riesz-projection.

Slope analysis
In this subsection we evaluate p(ω) defined in (3.4).
(ii) Let α > 0. It is easily seen that J k (ω) < 0 for p ≥ 5, thus, 3) holds. Let 1 < p < 5. It can be easily verified that and lim ω→ α 2 Let 1 < p ≤ 3, using the fact that J ′ k (ω) < 0 we get from (3.18)-(3.19) the inequality J k (ω) > 0, and (ii) − 1) holds. Let 3 < p < 5, then J ′ k (ω) > 0, therefore, from (3.18)-(3.19) it follows that there existsω k > α 2 First, we note that equation (1.3) is well-posed in E k due to the uniqueness of the solution to the Cauchy problem. Below we sketch the proof. Applying Duhamel's formula to (1.3), one obtains where e −iH α δ t denotes the evolution group of Let us show that e −iH α δ t preserves the space E k , i.e. for U 0 ∈ E k we have e −iH α δ t U 0 ∈ E k . Without loss of generality we may assume that α > 0. Thus, by formula (24) in [7] we have, where R µ U 0 = (H α δ + µ 2 I) −1 U 0 . By formula (17) in [7] we get where constantsc j are defined in [7,Appendix]. To findc j we need to describe matrices A 1 and B 1 which induce δ-interaction on the star graph G, that is A 1 and B 1 such that It is easily seen that Let U 0 ∈ E k . Then, using the equality t j (0, ω) = ∞ 0 (U 0 ) j (y)e −|y|µ dy 2 , we get Therefore, from the first formula on page 924 of [7] it follows that  where g, h are constants depending on t j (0, ω). From (3.23) it follows thatc 1 = ... =c k and c k+1 = ... =c N , therefore, by (3.22) we obtain that U 0 ∈ E k implies R µ U 0 ∈ E k . Thus, formula (3.21) implies that e −iH α δ t preserves the space E k . Finally, from the uniqueness of the solution to the Cauchy problem for (1.3) in E and formula (3.20) we get that for U 0 ∈ E k the solution U(t) to the Cauchy problem lies in E k for any t.
In [14] the authors proved the following instability result.
Below we will show the following result for any N ≥ 2.
Proof. The action functional for α = 0 has the form Our idea is to apply the stability Theorem 3.2 (substituting L α 1,k and L α 2,k by L 0 1 and L 0 2 respectively, and Φ k by Φ 0 ). The spectrum of L 0 1 was studied in Theorem 3.6. Note that in L 2 eq (G) the kernel of L 0 1 is empty, moreover, n(L 0 1 | L 2 eq (G) ) = 1 since Φ 0 ∈ L 2 eq (G) and (L 0 It is easy to show that L 0 2 ≥ 0 and ker(L 0 2 ) = span{Φ 0 } (see the proof of Proposition 3.5(i)). To complete the proof we need to study the sign of ∂ ω ||Φ 0 || 2 2 . From (3.16) for k = 0 and α = 0 it follows that which is obviously positive for 1 < p < 5, and is negative for p > 5. Finally, using n(H 0 | L 2 eq (G) ) = 1, by Theorem 3.2, for 1 < p < 5 we get stability of e iωt Φ 0 (x) in E eq , and for p > 5 instability of e iωt Φ 0 (x) in E eq and consequently in E.

Orbital stability of standing waves of NLS-δ equation with repulsive nonlinearity
In this section we study the orbital stability of the standing waves of the NLS-δ equation with repulsive nonlinearity (µ = −1 in (1.3)). The profile Φ(x) of the standing wave e iωt Φ(x) satisfies the equation Equivalently Φ is a critical point of the action functional defined as In the following theorem we describe the solutions to equation (5.1).
Theorem 5.1. Let α < 0 and 0 < ω < α 2 N 2 . Then equation (5.1) has a unique solution (up to permutations of the edges of G) Φ α = (ϕ α ) N j=1 , where Instability of excited states for NLS-δ equation on a star graph Proof. Notice that H α δ acts componentwise as the Laplacian, thus if Φ = (ϕ j ) N j=1 is the solution to (5.1), then ϕ j is the L 2 (R + )-solution to the equation The most general where σ ∈ C, |σ| = 1 and y ∈ R (see [16]). Therefore, the components ϕ j of the solution Φ to (5.1) are given by In order to solve (5.1) we need to impose boundary conditions (1.4). The continuity condition in (1.4) and existence of the limits lim Proof of Theorem 1.2. The well-posedness of the Cauchy problem on E for equation (1.3) in the case µ = −1 can be established analogously to the case of focusing nonlinearity (see [2]). Analogously to the previous case, the second variation of S rep at Φ α can be written formally where δ i,j is the Kronecker symbol. Let us show that ker(L α 2,rep ) = span{Φ α }. It is obvious that Φ α ∈ ker(L α 2,rep ). Any V = (v j ) N j=1 ∈ D α satisfies the following identity Instability of excited states for NLS-δ equation on a star graph Then we get for any Thus, ker(L α 2,rep ) = span{Φ α }. The inequality implies immediately that L α 1,rep ≥ 0 and ker(L α 1,rep ) = {0}. By Weyl's theorem, the essential spectrum of L α 1,rep and L α 2,rep coincides with [ω, ∞). Moreover, there can be only finitely many isolated eigenvalues in (−∞, ω). Thus, except the zero eigenvalue, the spectrum of L α 1,rep and L α 2,rep is positive and bounded away from zero. Therefore, using the classical Lyapunov analysis and noting that H α rep is non-negative due to (5.4), we obtain that e iωt Φ α is orbitally stable.

Appendix
For convenience of the reader we formulate the following two results from the extension theory (see [20]) essentially used in our stability analysis. The first one reads as follows.
Remark 6.2. The direct sum in (6.1) is not necessarily orthogonal. Proposition 6.3. Let A be a densely defined lower semi-bounded symmetric operator (that is, A ≥ mI) with finite deficiency indices n ± (A) = n < ∞ in the Hilbert space H, and let A be a self-adjoint extension of A. Then the spectrum of A in (−∞, m) is discrete and consists of at most n eigenvalues counting multiplicities.
Below, using the above abstract results, we provide an estimate for the Morse index of the operator L α 1,k defined in Theorem 3.1 in the whole space L 2 (G). We expect that the result established below can be useful for a future study of the stability properties of the standing waves e iωt Φ α k in the cases left open (see Theorem 1.1).
Proof. (i) In what follows we will use the notation l α k = − d 2 dx 2 + ω − p(ϕ k,j ) p−1 δ i,j . First, note that L α 1,k is the self-adjoint extension of the following symmetric operator where a k is defined in Theorem 2.3. Below we show that the operator L 0,k is non-negative and n ± ( L 0,k ) = k + 1. First, let us show that the adjoint operator of L 0,k is given by Using standard arguments, one can prove that L * 0,k = l α k (see [20, Chapter V, §17]). We denote It is easily seen that the inclusion D * 0,k ⊆ dom( L * 0,k ) holds. Indeed, for any U = (u j ) N j=1 ∈ D * 0,k and V = (v j ) N j=1 ∈ dom( L 0,k ), denoting U * = l α k (U) ∈ L 2 (G), we get which, by definition of the adjoint operator, means that U ∈ dom( L * 0,k ) or D * 0,k ⊆ dom( L * 0,k ). Let us show the inverse inclusion D * 0,k ⊇ dom( L * 0,k ). Take U ∈ dom( L * 0,k ), then for any V ∈ dom( L 0,k ) we have Instability of excited states for NLS-δ equation on a star graph Thus, we arrive at the equality for any V ∈ dom( L 0,k ).