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On ground state of some non local Schrödinger 6 1 operators. ∗ 0 2 n Yuri Kondratiev Stanislav Molchanov Sergey Pirogov a † ‡ § J Elena Zhizhina ¶ 8 2 ] h p Abstract - h We study a ground state of some non local Schro¨dinger operator t a associated with an evolution equation for the density of population m in the stochastic contact model in continuum with inhomogeneous [ mortality rates. Wefoundaneweffectinthismodel,wheneven inthe 2 high dimensional case the existence of a small positive perturbation v 6 of a special form (so-called, small paradise) implies the appearance 3 of the ground state. We consider the problem in the Banach space of 1 boundedcontinuousfunctionsC (Rd)andintheHilbertspaceL2(Rd). 1 b 0 Keywords: dispersal kernel, discrete spectrum, spectral radius, . compact operator, contact model 1 0 6 1 1 Introduction : v i X The asymptotic behavior of stochastic infinite-particle systems in continuum r can be studied in terms of evolution equations for correlation functions. For a ∗The work is partially supported by SFB 701 (Universitat Bielefeld). The research of Sergey Pirogov and Elena Zhizhina (sections 2-3) was supported by the Russian Science Foundation, Project 14-50-00150, Stanislav Molchanov was supported by NSF grant DMS 1008132(USA). †Fakultat fur Mathematik, Universitat Bielefeld, 33615 Bielefeld, Germany ([email protected]). ‡Department of Mathematics and Statistics, UNC Charlotte(USA) and University Higher School of Economics, Russian Federation ([email protected]) §InstituteforInformationTransmissionProblems,Moscow,Russia([email protected]). ¶Institute for Information Transmission Problems, Moscow, Russia ([email protected]). 1 the stochastic contact model in the continuum [5] the evolution equation for the first correlation function (i.e., the density of the system) is closed and it can be considered separately from equations for higher-order correlation functions [4]. In this case we have the following evolution problem for u C([0, ); ) associated with a nonlocal diffusion generator L: ∈ ∞ E ∂u = Lu, u = u(t,x), x Rd,t 0, u(0,x) = u (x) 0. (1) ∂t ∈ ≥ 0 ≥ in a proper functional space . As , we consider in our paper two spaces: E E C (Rd),theBanachspaceofboundedcontinuousfunctionsonRd,andL2(Rd). b These spaces arecorresponding totwo different regimes inthe contact model: systems with bounded density and ones essentially localized in the space. The operator L has the following form: Lu(x) = m(x)u(x) + a(x y)u(y)dy, (2) − − Rd Z where a(x) 0, a L1(Rd) is an even continuous function such that: ≥ ∈ a(x)dx = 1. (3) Rd Z Since a(x) is bounded, then a(x) L2(Rd) and ∈ a˜(p) = e−i(p,x)a(x)dx L2(Rd) with a˜(p) 1. (4) ∈ | | ≤ Rd Z The function a(x y) is the dispersal kernel associated with birth rates in − the contact model. The function m(x) is related with mortality rates. We assume here that m(x) C (Rd), 0 m(x) 1, m(x) 1, x . (5) b ∈ ≤ ≤ → | | → ∞ The contact model with homogeneous mortality rates m(x) const has ≡ been studied in [4]. It was proved there that only in the case m(x) 1 ≡ there exists a family of stationary measures of the model (for d 3); if ≥ m(x) m = 1, then the density of population either exponentially growing ≡ 6 (supercritical regime: m < 1) or exponentially decaying (subcritical regime: m > 1). 2 Here we are interesting in local perturbations of the stationary regime, when m(x) is an inhomogeneous in space non-negative function. We prove that local fluctuations of the mortality with respect to the critical value m(x) 1 can push the system away from the stationary regime. As a ≡ result of such local perturbations, we will observe exponentially increasing density of population everywhere in the space. The goal of this paper is to obtainconditions on the mortality rateswhich give theexistence of a positive discrete spectrum of the operator (2) in the spaces C (Rd) and L2(Rd), and b to prove the existence and uniqueness of a positive eigenfunction ψ(x) > 0 corresponding to the maximal eigenvalue λ > 0 of the operator L. We call 0 this function the ground state of the operator L. The existence of not only the principal eigenvalue λ but also the cor- 0 responding positive eigenfunction ψ is very important in the analysis of the long-time behavior of non-local evolution problems. The so-called principal eigenpair problem (λ ,ψ) for some nonlocal dispersal operators has been 0 studied recently in many papers, see e.g. [2, 1] and references therein. How- ever the results of these papers were obtained under quite restrictive conditions, namely the assumption that the dispersal kernel a(x y) is a function − with a compact support, or under the condition that the eigenvalue problem is studied in a bounded domain Ω Rd. We consider in our paper the case ⊂ of unbounded domain Ω = Rd, and the only assumption on the dispersal kernel is the integrability condition. We prove in our paper that the ground state appears in two cases: either there exists such a region of any (small) positive volume, where the fluctuation V(x) is equal to 1 (Theorem 6), or V(x) is positive and less than 1 in a large enough region (Theorem 8). We stress that the function V(x) should be bounded from above by 1, since the mortality m(x) 0 is a non- ≥ negative function. Thus we observe for the nonlocal operator (2) new effects different from those of the Shro¨dinger operators. We found that in the high dimensional case some small (in the integral sense) perturbations V(x) of the mortality m(x) from the critical value m(x) 1 imply existence of the ≡ positive eigenvalue and the corresponding positive eigenfunction. In small dimensions d = 1,2, the ground state of the operator L exists for any small local positive fluctuation V(x) = 1 m(x) of the mortality m(x) from the − critical value under condition that the dispersal kernel has the second mo- ment (Theorem 9). We also proved the analogous results for the model in the subcritical regime (Theorem 11). 3 2 Spectral properties of L In this section we describe a general approach to study a discrete spectrum of the operator L in both spaces C (Rd) and L2(Rd). This approach is based on b the analytic Fredholm theorem and the study of a spectral radius of compact operators. The operator L can be rewritten as Lu(x) = L u(x) + V(x)u(x), u(x) C (Rd), (6) 0 b ∈ L u(x) = a(x y)(u(y) u(x))dy, V(x) = 1 m(x). 0 − − − Rd Z The potential 0 V(x) 1 describes local (negative) fluctuations of the ≤ ≤ mortality, and we assume that V(x) C (Rd) and lim V(x) = 0. (7) b ∈ |x|→∞ The operator L is bounded and dissipative in C (Rd): 0 b (λ L )f λ f for λ 0. 0 k − k ≥ k k ≥ Lemma 1 The operator L has only discrete spectrum in the half-plane = λ C Reλ > 0 . D { ∈ | } Proof is based on the analytic Fredholm theorem for analytic operator- valued functions, see [3], [7]. Since for any λ the resolvent (L λ)−1 is 0 ∈ D − a bounded operator, we have (λ L V) = (λ L )(1 (λ L )−1V), (8) 0 0 0 − − − − − and (λ L V)−1 = (1 (λ L )−1V)−1 (λ L )−1. (9) 0 0 0 − − − − − Using the Neumann series for the operator (λ L )−1 we get: 0 − 1 1 (λ L )−1 = + A , (10) − 0 λ+1 λ+1 λ where ∞ a∗n A = (λ+1)(λ L )−1 1 = , (11) λ − 0 − (λ+1)n n=1 X 4 and A is a bounded convolution operator when λ . The decomposition λ ∈ D (10) for (λ L )−1 is the crucial point of our reasoning. The kernel of A is 0 λ − ∞ a∗n 1 a˜(p) dp G (x y) = (x y) = e−i(p,x−y) , λ − (λ+1)n − (2π)d λ+1 a˜(p) ! Rd n=1 Z − X (12) with a˜(p) G˜ (p) = L2(Rd) when λ , (13) λ λ+1 a˜(p) ∈ ∈ D − and, in particular, G (u) C (Rd) L2(Rd) and G (u) du < , λ . (14) λ b λ ∈ ∩ ∞ ∀ ∈ D Rd Z We denote by W an operator of multiplication by the function λ V W = 1 . λ − λ+1 ItisaboundedoperatorwiththeboundedinverseoperatorW−1 whenλ . λ ∈ D Then (10) implies 1 1 (λ L )−1V = W A V, 0 λ λ − − − λ+1 and we can rewrite (9) in the following way: (λ L V)−1 = (1 Q )−1 ((λ L )W )−1, (15) 0 λ 0 λ − − − − where 1 Q = W−1A V. (16) λ λ+1 λ λ The relations (12) - (14) for the kernel G (x y) of the operator A and λ λ − the conditions on the potential V(x) imply that Q is the compact operator. λ Really, let us define the truncated operator of multiplication on the potential V(r)(x) = χ (x)V(x) and the convolution operator A(r) with the truncated r λ kernel G(r)(x) = χ (x)G (x), where χ (x) is the indicator function of the λ r λ r ball B = x : x < r . Obviously, the operators A(r)V(r) are compact for r { | | } λ any r > 0, and A(r), V(r) converge in norm to A , V correspondingly. Thus λ λ we conclude that A V is the compact operator. λ 5 Consequently, Q is an analytic operator-valued function, such that Q λ λ is a compact operator for any λ . Then using the analytic Fredholm ∈ D theorem [7] (Theorem VI.14) we get that the function (1 Q )−1 is a mero- λ − morphic function in . Since ((λ L )W )−1 is a bounded operator, we can 0 λ conclude that the opDerator L has−only a discrete spectrum in . (cid:4) D Remark 2 The representations (11)-(12) imply that 1)A isa positivityimprovingoperatorforallλ > 0, sinceG (x y) > 0, x,y λ λ − ∀ ∈ Rd, 2) G (x y) is monotonically decreasing with respect to λ > 0, λ − 3) formulas (16), (11), (12) imply that Q , λ > 0, is a positivity improving λ compact integral operator in C (Rd) with the kernel b G (x y)V(y) (x,y) = λ − . (17) λ Q λ+1 V(x) − We study next the behavior of the spectral radius r(Q ) of the operator λ Q as a function of λ when λ > 0. λ Remark 3 From the known formula for the spectral radius it follows that if Q is a positive operator, and if there exists a function ϕ(x) , ϕ 0, ∈ E ≥ ϕ = 1, such that k k Qϕ(x) c ϕ(x), (18) 0 ≥ then r(Q) c , 0 ≥ (see e.g. [6], Theorem 6.2) Lemma 4 The spectral radius r(Q ) is continuous and monotonically de- λ creasing with respect to λ > 0. Moreover, r(Q ) 0 for λ + . λ → → ∞ Proof. The continuity of the spectral radius follows from the compactness and continuity in λ of the operator Q . λ The second statement of the lemma follows from the fact, that if A,B are positive operators, such that A B in the order sense defined by the cone of ≤ positive functions, then r(A) r(B). Really, it is easy to see that for both ≤ our spaces norms of positive operators A,B are attained on non negative E functions: Af A = sup k k, k k f f≥0 k k 6 and the same for B. Thus, A B . k k ≤ k k Analogously, for any n N we have An Bn , and consequently, ∈ k k ≤ k k r(A) r(B)duetotheformulaforthespectralradiusr(A) = lim n An . n→∞ ≤ k k The positive kernel (x,y) of the operator Q is the monotonically decreas- Qλ λ p ing function of λ > 0. Thus we have 0 Q Q , when λ > λ , ≤ λ1 ≤ λ2 1 2 and consequently, the spectral radius r(Q ) is monotonically decreasing with λ respect to λ > 0. The convergence to 0 follows from (16). (cid:4) As follows from (8), the equation on the eigenfunction ψ (L +V λ)ψ = 0, λ > 0, (19) 0 − can be written as Q ψ(x) = ψ(x). (20) λ where Q is a compact positive operator defined by (16). λ Using Lemma 4 we can conclude that if lim r(Q ) > 1, (21) λ λ→0+ then there exists such λ > 0 that r(Qλ) = 1, and r(Qλ′) < 1 for λ′ > λ. (22) If r(Q ) < 1 for all λ > 0, then the positive spectrum of L is absent. For λ example, the positive spectrum of L is absent in C (Rd) if d 3, V b ≥ ∈ L1(Rd), 0 V(x) 1 ε < 1, and L1-norm of V is small enough. ≤ ≤ − From(22)itfollowsbytheKrein-Rutmantheorem([6], Theorem6.2)that 1 is the eigenvalue of Q with a positive eigenfunction ψ (x) > 0. Obviously, λ λ λ is the maximal positive eigenvalue of the operator L, and ψ (x) is the λ groundstateoftheoperatorL. Theuniqueness ofthegroundstateψ (x) > 0 λ of the operator L in the space L2(Rd) follows from the positivity improving property of the semigroups etL0 and etL, see e.g. [8], Theorem XIII.44. The last semigroup is positivity improving due to the Feynman-Kac formula. Lemma 5 1. If the ground state ψ (x) C (Rd), then ψ (x) 0 as x λ b λ ∈ → | | → . ∞ 7 2. If the ground state ψ (x) L2(Rd), then ψ (x) C (Rd) L2(Rd) and λ λ b ∈ ∈ ∩ ψ (x) 0 as x . λ → | | → ∞ 3. If the ground state ψ (x) C (Rd) and V(x) L2(Rd), then ψ (x) λ b λ ∈ ∈ ∈ L2(Rd). In particular, if the potential V(x) C (Rd) then ψ (x) L2(Rd). 0 λ ∈ ∈ Proof. 1. The ground state ψ (x) satisfies the equation (20) with Q λ λ defined by (16). Since V(x) 0 as x and G (x) L1(Rd) by (14), λ → | | → ∞ ∈ then from the Lebesgue convergence theorem it follows that the convolution of the functions Vψ andG tends to 0 for x . Thus, all eigenfunctions λ λ | | → ∞ ψ (x) C (Rd) for positive λ tend to 0 when x . λ b ∈ | | → ∞ 2. We again use the equation (20) for the function ψ (x). Since V(x) λ is bounded, then (Vψ )(x) L2(Rd) and (Vψ )(p) L2(Rd). Using (13) λ λ ∈ ∈ G (p) L2(Rd) (λ > 0), we have G (p)Vψ (p) L1(Rd) as a product λ λ λ ∈ ∈ of two functions from L2(Rd). Consequentlgy, (A Vψ )(x) C (Rd), and λ λ b ∈ ψf(x) = Q ψ (x) C (Rd), since W−1f C g(Rd) for any λ > 0. λ λ λ ∈ b λ ∈ b (cid:4) 3. The statement directly follows from (20). 3 Ground state in C (Rd) b Now we formulate conditions on V(x) which give the existence of the ground state ψ(x). We omit the subscript λ in ψ (x) in the subsequent text. λ Theorem 6 (small paradise) Assume that there exists δ > 0 such that V(x) = 1 when x B , where B is a ball of a radius δ. Then the ground δ δ ∈ state of L exists. Proof. It is enough to show (21). Let us take for a given 0 < δ < 1 a ”continuous approximation” of the indicator function χˆ (x): Bδ χˆ (x) = 0, x Rd B , χˆ (x) = 1, x B , Bδ ∈ \ δ Bδ ∈ 0.9δ whereB B , and0 χˆ (x) 1, x B B . Thenforanyλ (0,1) 0.9δ ⊂ δ ≤ Bδ ≤ ∈ δ\ 0.9δ ∈ and any x B we get: δ ∈ 1 vol(B ) Q χˆ (x) = (x,y)χˆ (y)dy G (x y)dy 0.9δ κ , λ Bδ Qλ Bδ ≥ λ λ − ≥ λ 1 ZBδ ZB0.9δ (23) 8 where vol(B ) is the volume of the ball B B , and κ is defined as 0.9δ 0.9δ δ 1 ⊂ κ = min G (x y) < min min G (x y) (24) 1 1 λ x,y∈B1 − λ∈(0,1)x,y∈Bδ − Thus lim r(Q ) = for any δ > 0. (cid:4) λ→0+ λ ∞ Remark 7 For any δ > 0 there is ε > 0 such that if V(x) 1 ε for ≥ − x B , then the ground state of L exists. δ ∈ The proof of the statement follows the similar reasoning as above in The- orem 6. Theorem 8 Assume that for some β (0,1) there exists R > 0 such that ∈ β V(x) 1, x B . (25) R ≤ ≤ ∈ Then the ground state of the operator L exists for R = R(β) sufficiently large. Proof. We will prove that there exists a ground state ψ of the operator L, such that ψ L2(Rd). Then from Lemma 5 it follows that ψ C (Rd). b ∈ ∈ To prove the existence of ψ L2(Rd) it is sufficient to verify that the ∈ quadratic form (Lf,f) is positive for some f L2(Rd). Let us take f = χ . ∈ BR Then (Lχ ,χ ) = (L χ ,χ )+(Vχ ,χ ), (26) BR BR 0 BR BR BR BR and (Vχ ,χ ) β vol(B ). (27) BR BR ≥ R For the operator L we have: 0 (L f,f) = a(y x)(f(x) f(y))f(x)dydx= 0 − − − Z Z 1 a(y x)(f(x) f(y))2dydx. 2 − − Z Z Consequently, the first term in (26) can be written as (L χ ,χ ) = a(y x)dxdy a(z)dxdz = − 0 BR BR − ≤ Z|x|<RZ|y|>R Z|x|<RZ|x|+|z|>R R R C rd−1 a(z)dzdr = C a(z) rd−1drdz = d d Z0 Z|z|>R−r ZRd Z(R−|z|)+ 9 C Rd z d d a(z) 1 1 | | dz. d − − R ZRd (cid:18) (cid:19)+! Here CdRd = vol(B ). Thus, d R 1 (L χ ,χ ) 0 as R . (28) vol(B ) 0 BR BR → → ∞ R Finally, (26)-(28)imply thattheoperatorLhasapositive discrete spectrum and a ground state if R is taken sufficiently large. (cid:4) Theorem 9 Let d = 1,2, and x 2a(x)dx < . (29) | | ∞ Rd Z Then for any V 0 the ground state of L exists. 6≡ Proof. Conditions on the function a(x) imply that a˜(p) = 1 (Cp,p)+ − o( p 2)as p 0. Letustakeafunctionϕ(x) C (Rd), ϕ 0, ϕ = 1such 0 | | | | → ∈ ≥ k k that(Vϕ)(0) = V(x)ϕ(x)dx > 0. Usingthepropertiesofthefunctionsa(x) and V(x), see (3)-(4), (7), we get that R f a˜(p), (Vϕ)(p) L2(Rd) C (Rd), b ∈ ∩ and consequently, a˜(p)(Vϕ)(fp) L1(Rd) Cb(Rd). ∈ ∩ Then for any x supp ϕ ∈ f 1 (Q ϕ)(x) = G (x y)V(y)ϕ(y) dy = λ λ λ+1 V(x) − Rd − Z 1 a˜(p) dp e−ipx+ipy V(y)ϕ(y) dy = λ+1 V(x) λ+1 a˜(p) Rd Rd − Z Z − 1 e−ipx a˜(p) (Vϕ)(p) dp as λ 0. λ+1 V(x) λ+1 a˜(p) → ∞ → Rd − Z − f The continuous functions U (x) = (Q ϕ)−1(x) tend to 0 monotonically λ λ as λ 0+ by Remark 2 and uniformly on supp ϕ by the Dini theorem. → Thus for any c there exists λ > 0 for which (18) is fulfilled. Consequently, 0 lim r(Q ) = and (21) holds. (cid:4) λ→0+ λ ∞ 10