CC BY
11
NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2016, 7 (5), P. 789-802
On resonances and bound states of Smilansky Hamiltonian
P Exner, V. Lotoreichik, M. Tater
Nuclear Physics Institute, Czech Academy of Sciences, 25068 Rez, Czech Republic [email protected], [email protected], [email protected]
PACS 02.30.Tb, 03.65.Db
DOI 10.17586/2220-8054-2016-7-5-789-802
We consider the self-adjoint Smilansky Hamiltonian H£ in L2(R2) associated with the formal differential expression — dX — — (d^ + y2) — V2eyS(x) in the sub-critical regime, e € (0, 1). We demonstrate the existence of resonances for H£ on a countable subfamily of sheets of the underlying Riemann surface whose distance from the physical sheet is finite. On such sheets, we find resonance free regions and characterize resonances for small e > 0. In addition, we refine the previously known results on the bound states of H£ in the weak coupling regime (e ^ 0+). In the proofs we use Birman-Schwinger principle for H£, elements of spectral theory for Jacobi matrices, and the analytic implicit function theorem.
Keywords: Smilansky Hamiltonian, resonances, resonance free region, weak coupling asymptotics, Riemann surface, bound states.
Received: 1 July 2016. Revised: 28 July 2016.
In memory ofB.S. Pavlov (1936-2016)
1. Introduction
In this paper we investigate resonances and bound states of the self-adjoint Hamiltonian He acting in the Hilbert space L2 (R2) and corresponding to the formal differential expression
~dl - 2 (дУ + y2) -V2eyS(x) on R2, (1.1)
in the sub-critical regime, e g (0,1). The operator He will be rigorously introduced in Section 1.1 below. Operators of this type were suggested by U. Smilansky in [1] as a model of irreversible quantum system. His aim was to demonstrate that the ‘heat bath’ need not have an infinite number of degrees of freedom. On a physical level of rigor he showed that the spectrum undergoes an abrupt transition at the critical value e = 1. A mathematically precise spectral analysis of these operators and their generalizations has been performed by M. Solomyak and his collaborators in [2-8]. Time-dependent Schrddinger equation generated by Smilansky-type Hamiltonian is considered in [9].
By now many of the spectral properties of He are understood. On the other hand, little attention has been paid so far to the fact that such a system can also exhibit resonances. The main aim of this paper is to initiate investigation of these resonances starting from demonstration of their existence. One of the key difficulties is that this model belongs to a class wherein the resolvent extends to a Riemann surface having uncountably many sheets. The same complication appears e.g. in studying resonances for quantum waveguides [10-13], [14, §3.4.2] and for general manifolds with cylindrical ends [15,16].
In this paper, we prove the existence and obtain a characterization of resonances of He on a countable subfamily of sheets whose distance from the physical sheet is finite in the sense explained below. On any such sheet we characterize a region which is free of resonances. As e ^ 0+, the resonances on such sheets are localized in the vicinities of the thresholds vn = n +1/2, n g N. We obtain a description of the subset of the thresholds in the vicinities of which a resonance exists for all sufficiently small e > 0 and derive asymptotic expansions of these resonances in the limit e ^ 0+. No attempt has been made here to define and study resonances on the sheets whose distance from the physical sheet is infinite.
As a byproduct, we obtain refined properties of the bound states of He using similar methods as for resonances. More precisely, we obtain a lower bound on the first eigenvalue of He and an asymptotic expansion of the weakly coupled bound state of He in the limit e ^ 0+.
Methods developed in this paper can also be useful to tackle resonances for the analog of Smilansky model with regular potential which is suggested in [17] and further investigated in [18,19].
Notations
We use notations N := {1, 2,... } and N0 := N U {0} for the sets of positive and natural integers, respectively. We denote the complex plane by C and define its commonly used sub-domains: Cx := C \ {0}, C± := {A g C: ± Im A > 0} and Dr(A0) := {A g C: |A - A0| < r}, DX(A0) := {A g C: 0 < |A - A0| < r}, Dr := Dr(0),
DX := DX(0) with r > 0. The principal value of the argument for A g Cx is denoted by arg A g (—п, п]. The
branches of the square root are defined by:
Cx э A ^ (Aj/2 := |A|1/2ei((1/2)argx+jn), j = 0,1.
If the branch of the square root is not explicitly specified, we understand the branch (^У1 2 by default. We also set
0 = (0,0) g c2 .
The L2-space over Rd, d = 1, 2, with the usual inner product is denoted by (L2(Rd), (•, •)Rd) and the L2-based first order Sobolev space by H 1(Rd), respectively. The space of square-summable sequences of vectors in a Hilbert space G is denoted by f2(N0; G). In the case that G = C we simply write f2(N0) and denote by (•, •) the usual inner product on it.
For £ = {£n} g f2(N0), we adopt the convention that £_1 = 0. Kronecker symbol is denoted by Jnm,
n, m g N0, we set en := {£nm}meNo G f2(N0), n g N0, and adopt the convention that e_1 := {0}. We understand
by diag({qn}) the diagonal matrix in f2(N0) with entries {qn}neNo and by J({an}, {bn}) the Jacobi matrix in f2(N0) with diagonal entries {an}neNo and off-diagonal entries {bn}neN'. We also set J0 := J({0}, {1/2}).
By <r(K), we denote the spectrum of a closed (not necessarily self-adjoint) operator K in a Hilbert space. An isolated eigenvalue A g C of K having finite algebraic multiplicity is a point of the discrete spectrum for K; see [23, §XII.2] for details. The set of all the points of the discrete spectrum for K is denoted by <rd(K) and the essential spectrum of K is defined by aess(K) := <r(K) \<rd(K). For a self-adjoint operator T in a Hilbert space, we set Aess(T) := inf aess(T) and, for k G N, Ak(T) denotes the k-th eigenvalue of T in the interval (-TO,Aess(T)). These eigenvalues are ordered non-decreasingly with multiplicities taken into account. The number of eigenvalues with multiplicities of the operator T lying in a closed, open, or half-open interval Д c R satisfying aess(T)nA = 0 is denoted by N (Д; T). For A < Aess (T) the counting function of T is defined by Ny(T) := N ((-to, A); T).
1.1. Smilansky Hamiltonian
Define the Hermite functions:
Xn(y) := e_y2/2Hn(y), n G N0. (1.2)
Here, Hn(y) is the Hermite polynomial of degree n G N0 normalized by the condition ||хп||к = 12. For more details on Hermite polynomials see [20, Chap. 22] and also [21, Chap. 5]. As it is well-known, the family {xn}neN0 constitutes an orthonormal basis of L2(R). Note also that the functions xn satisfy the three-term recurrence relation:
Vn+Ixn+1(y) -V2yXn(y) + VnXn_1(y)=0, n G N0, (1.3)
where we adopt the convention x_1 = 0. The relation (1.3) can be easily deduced from the recurrence relation [20, eq. 22.7.13] for Hermite polynomials. By a standard argument any function U G L2(R2) admits unique expansion:
U(x,y)=^ Un(x)xn(y), Un(x) := / U(x,y)xn(y)dy, (1.4)
neNo R
where {un} G f2(N0; L2(R)). Following the presentation in [7], we identify the function U G L2(R2) and the sequence {un} and write U ~ {un}. This identification defines a natural unitary transform between the Hilbert spaces L2(R2) and H := f2(N0; L2(R)). For the sake of brevity, we denote the inner product on H by (-, •}. Note that the Hilbert space H can also be viewed as the tensor product f2(N0) (g> L2(R).
For any £ g R, we define the subspace De of H as follows: an element U ~ {un} g H belongs to De if, and only if
(i) un G H 1(R) for all n G N0;
(ii) {-(u^^i+ ® мП,_) + VnUn} G H with u„,± := Un|R± and Vn = n + 1/2 for n G N0;
1We do not distinguish between Jacobi matrices and operators in the Hilbert space ^2(No) induced by them, since in our considerations all the Jacobi matrices are bounded, closed, and everywhere defined in ^2(No).
2This normalization means that Hn (y) is, in fact, a product of what is usually called the Hermite polynomial of degree n G No with a normalization constant which depends on n.
(iii) the boundary conditions
«4(°+) - «4(°-) = e(Vn + 1un+i(0) + \fnun-1(0)) are satisfied for all n g No. For n = 0 only the first term is present on the right-hand side.
By [7, Thm. 2.1], the operator:
dom He .= De, He{un} .= { (un,+ ф un, — ) + vnun} , (1.5)
is self-adjoint in H. It corresponds to the formal differential expression (1.1). Further, we provide another way of defining He which makes the correspondence between the operator He and the formal differential expression (1.1) more transparent. To this aim, we define the straight line Я .= {(0, y) g R2. y G R}. Then, the Hamiltonain He, £ G (-1,1), can be alternatively introduced as the unique self-adjoint operator in L2(R2) associated via the first representation theorem [22, Thm. VI.2.1] with a closed, densely defined, symmetric, and semi-bounded quadratic form:
hs[u] .= IldxullR + 1 \\dvu|Ir2 + 1(yu,yu)R2 + £V2 (sign (y)|y|1/2u|s, |y|1/2u|s) ,
2 2 V 2r (1.6) dom he := {u G H 1(R2). yu G L2(R2), |y|1/2(u|s) G L2(R) j .
For more details and for the proof of equivalence between the two definitions of He, see [7, §9]. Since He commutes with the parity operator in y-variable, it is unitarily equivalent to H—e. We remark that the case £ = 0 admits separation of variables. Thus, it suffices to study He with £ > 0.
In the following proposition, we collect fundamental spectral properties of He, £ G (0,1), which are of importance in the present paper.
Proposition 1.1. Let the self-adjoint operator He, £ G (0,1), be as in (1.5). Then the following claims hold:
(i) CTess(He) = [1/2, +ж);
1£
(ii) inf a(He) > —;
(iii) 1 < N1/2(He) < ж;
(iv) N1/2(He) = 1 for all sufficiently small £ > 0.
Items (i)—(iii) follow from [6, Lem 2.1] and [7, Thm. 3.1 (1),(2)]. Item (iv) is a consequence of [6, Thm. 3.2] and [7, §10.1]. Although we only deal with the sub-critical case, £ G (0,1), we remark that in the critical case, £ = 1, the spectrum of H1 equals to [0, +ж) and that in the sup-critical case, £ > 1, the spectrum of He covers the whole real axis. Finally, we mention that in most of the existing literature on the subject not £ > 0 itself but a = V2£ is chosen as the coupling parameter. We choose another normalization of the coupling parameter in order to simplify formulae in the proofs of the main results.
1.2. Main results
While we are primarily interested in the resonances, as indicated in the introduction, we have also a claim to make about the discrete spectrum which we present here as our first main result and which complements the results listed in Proposition 1.1.
Theorem 1.2. Let the self-adjoint operator He, £ G (0,1), be as in (1.5). Then the following claims hold.
(i) A^He) > 1 - \j - + £4 for all £ G (0,1).
(ii) A1 (He) = vo ~~ + O(£5) as £ ^ 0+.
16
Theorem 1.2 (i) is proven by means of Birman-Schwinger principle. The bound in Theorem 1.2 (i) is non-trivial for £4 < 3/4. This bound is better than the one in Proposition 1.1 (ii) for small £ > 0.
For the proof of Theorem 1.2 (ii) we combine Birman-Schwinger principle and the analytic implicit function theorem. We expect that the error term O(£5) in Theorem 1.2(ii) can be replaced by O(£6) because the operator He has the same spectral properties as H—e for any £ g (0,1). Therefore, the expansion of A1(He) must be invariant with respect to interchange between £ and —£. In Lemma 4.1 given in Section 4 we derive an implicit scalar equation on A1(He). This equation gives analyticity of £ ^ A1(He) for small £. It can also be used to compute higher order terms in the expansion of A1(He). However, these computations might be quite tedious.
Our second main result concerns the resonances of He. Before formulating it, we need to define the resonances rigorously. Let us consider the sequence of functions:
r„(A):=(v„ - A)1/2, n e N0. (1.7)
Each of them has two branches rn(A, l) := (vn — A);1/2, l = 0,1. The vector-valued function R(A) = (r0(A),r1 (A), r2(A),...) naturally defines the Riemann surface Z with uncountably many sheets. With each sheet of Z we associate the set E c N0 and the characteristic vector lE defined as:
lE :
lE, lE, lE, . . . , {l0 , l1 ,l2 , . . . J,
lE :
0,
1,
n e e,
n e E.
(1.8)
We adopt the convention that lE1 = 0. The respective sheet of Z is convenient to denote by ZE. Each sheet ZE of Z can be identified with the set C \ [v0, +to) and we denote by Z± the parts of ZE corresponding to C±. With the notation settled, we define the realization of R(-) on ZE as:
Re (A) :=(r0 (A ,l0E ),n (A, lE ),r2 (A ,l2E),...). (1.9)
The sheets ZE and ZF are adjacent through the interval (vn,vn+1) c R, n e N0, (ZE ^n ZF), if their
characteristic vectors lE and lF satisfy:
ljF = 1 — lE, for k = 0,1, 2,..., n
lF = lE, for k > n.
We set v_1 = —to and note that any sheet ZE is adjacent to itself through (v_1, v0). In particular, the function A ^ Re (A) turns out to be componentwise analytic on the Riemann surface Z.
The sequence E = {E1, E2,..., EN} of subsets of N0 is called a path if for any k = 1, 2,..., N — 1 the sheets ZEk and ZEfc+1 are adjacent. The following discrete metric:
p(E, F) := inf{N e N0: E = {E1, E2,..., En}, E1 = E, En = F}, (1.10)
turns out to be convenient. The value p(E, F) equals the number of sheets in the shortest path connecting ZE and ZF. Note that for some sheets ZE and ZF a path between them does not exist and in this case we have p(E, F) = to. We identify the physical sheet with the sheet Z0 (for E = 0). A sheet ZE of Z is adjacent to the physical sheet Z0 if p(E, 0) = 1 and it can be characterised by existence of N e N0 such that lE = 1 if, and only if n < N. Also, we define the component:
Z := UEesZe C Z E := {E c N0 : p(E, 0) < to}, (1.11)
of Z which plays a distinguished role in our considerations. Any sheet in Z is located on a finite distance from the physical sheet with respect to the metric p(-, •). The component Z of Z in (1.11) can alternatively be characterized as:
Z = UfefZf, F := {F C N0 : sup{n e N0 : lF = 1} < to}. (1.12)
The number of the sheets in Z is easily seen to be countable. In order to define the resonances of He on Z, we show that the resolvent of He admits an extension to Z in a certain weak sense.
Proposition 1.3. For any u e L2(R) and n e N0 the function:
A ^ r^e(A; u) := ((He — A)-1u <g> en,u ® ert) (1.13)
admits unique meromorphic continuation tE,e(-; u) from the physical sheet Z0 to any sheet ZE c Z.
The proof of Proposition 1.3 is postponed until Appendix. Now we have all the tools to define resonances of He on •Z.
Definition 1.4. Each resonance of He on ZE c Z is identified with a pole of rE,e(-; u) for some u e L2(R) and n e N0. The set of all the resonances for He on the sheet ZE is denoted by RE (e).
Our definition of resonances for He is consistent with [23, §XII.6], see also [14, Chap. 2] and [24] for multithreshold case. It should be emphasized that by the spectral theorem for self-adjoint operators the eigenvalues of He are also regarded as resonances in the sense of Definition 1.4 lying on the physical sheet Z0. This allows us to treat the eigenvalues and ‘true’ resonances on the same footing. Needless to say, bound states and true resonances correspond to different physical phenomena and their equivalence in this paper is merely a useful mathematical abstraction.
According to Remark 2.5 below, the set of the resonances for He on ZE is symmetric with respect to the real axis. Thus, it suffices to analyze resonances on Z—. Now, we are prepared to formulate the main result on resonances.
Theorem 1.5. Let the self-adjoint operator He, e G (0,1), be as in (1.5). Let the sheet ZE C Z of the Riemann surface Z be fixed. Define the associated set by:
S(E) := {n G N: (lE-i,lE,Ci) G {(1,0,0), (0,1,1)}}.
Let RE (e) be as in Definition 1.4 and set R—(e) := RE (e) П C_. Then, the following claims hold:
(i) R_(e) C U(e) := {A G C_ : |v„_i - A||v„ - A| < e4n2, Vn G N}.
(ii) For any n G S(E) and sufficiently small e > 0 there is exactly one resonance AE(He) G C_ of He on Z— lying in a neighbourhood of vn, with the expansion
e4
Ae (He) = vn — 16 [(2n + 1) + 2n(n + l^i] + 0(f), e ^ 0+ . (1.14)
(iii) For any n G N \ S(E) and all sufficiently small e,r > 0
R_(e) П Dr(vn) = 0.
Fig. 1.1. The region U(0.12) (for e = 0.12) from Theorem 1.5 (i) (in grey) consists of 6 connected components. The components located in the neighbourhoods of the points v0,v1, v2, v3, are not visible because of being too small. The plot is performed with the aid of Sagemath.
In view of Theorem 1.5 (i) for sufficiently small e > 0, the resonances of He on any sheet of Z are located in some vicinity of the thresholds vn (see Figure 1.1). Such behavior is typical for problems with many thresholds; see e.g. [11,13] and [14, §2.4, 3.4.2]. Note also that the estimate in Theorem 1.5 (i) reflects the correct order in e in the weak coupling limit e ^ 0+ given in Theorem 1.5 (ii). However, the coefficient of e4 in the definition of U(e) can be probably improved. Observe also that R— (e) C U(1) for any e G (0,1).
According to Theorem 1.5 (ii)—(iii), the existence of a resonance near the threshold vn, n g N, on a sheet ZE for small e > 0 depends only on the branches chosen for rn—1(A), rn(A), rn+1(A) on ZE. Although, one cannot exclude that higher order terms in the asymptotic expansion (1.14) depend on the branches chosen for other square roots. By exactly the same reason as in Theorem 1.2 (ii), we expect that the error term O(e5) in Theorem 1.5 (ii) can be replaced by O(e6). Theorem 1.5 (ii)-(iii) are proven by means of the Birman-Schwinger principle and the analytic implicit function theorem. The implicit scalar equation on resonances derived in Lemma 4.1 gives analyticity of e ^ AE(He) for small e > 0 and, as in the bound state case, it can be used to compute further terms in the expansion of AE (He).
We point out that according to numerical tests that we performed, some resonances emerge from the inner points of the intervals (v„,vn+1), n g No, as e ^ 1—. The mechanism for the creation of these resonances is unclear at the moment.
Example 1.6. Let E = {1,2,4,5}. In this case Iе = {0,1,1,0,1,1,0,0,0,0,0,... } and we get that S(E) = {1,4,6}. By Theorem 1.5 (ii)-(iii) for all sufficiently small e > 0 there will be exactly one resonance on Z— near v1; v4, v6 and no resonances near the thresholds vn with n G N \ {1,4,6}. We confirm this result by numerical tests whose outcomes are shown in Figures 1.2 and 1.3.
Fig. 1.2. Resonances of H£ with £ = 0.2 lying on Z— with E = {1,2,4, 5} are computed numerically with the help of Mathematica. Unique weakly coupled resonances near the thresholds vi = 1.5, V4 = 4.5, v6 = 6.5 are located at the intersections of the curves.
To plot Figure 1.2, we used the condition on resonances in Theorem 2.4 below. The infinite Jacobi matrix in this condition was truncated up to a reasonable finite size. Along the curves, respectively, the real and the imaginary part of the determinant of the truncated matrix vanishes. At the points of intersection of the curves the determinant itself vanishes. These points are expected to be close to true resonances3. We have also numerically verified that resonances do not exist near other low-lying thresholds vn with n G N \{1, 4, 6}, which corresponds well to Theorem 1.5. In Figure 1.3 we summarize the results of all the numerical tests.
Fig. 1.3. Resonances of H£ with £ = 0.2 lying on ZE with E = {1, 2,4,5}.
Finally, we mention that no attempt has been made here to analyze the multiplicities of the resonances and to investigate resonances lying on Z \ Z.
Structure of the paper
Birman-Schwinger-type principles for the characterization of eigenvalues and resonances of H£ are provided in Section 2. Theorem 1.2 (i) on a lower bound for the first eigenvalue and Theorem 1.5 (i) on resonance free region are proven in Section 3. The aim of Section 4 is to prove Theorem 1.2 (ii) and Theorem 1.5 (ii)—(iii) on weakly coupled bound states and resonances. The proofs of technical statements formulated in Proposition 1.3 and Theorem 2.4 are postponed until Appendix.
2. Birman-Schwinger-type conditions
The Birman-Schwinger principle is a powerful tool for analyzing the discrete spectrum of a perturbed operator in the spectral gaps of the unperturbed one. This principle also has other various applications. Frequently, it can be generalized to detect resonances, defined as the poles of a meromorphic continuation of the (sandwiched)
3
The analysis of convergence of the numerical method is beyond our scope.
resolvent from the physical sheet to non-physical sheet(s) of the underlying Riemann surface. In the model under consideration, we encounter yet another manifestation of this principle.
In order to formulate a Birman-Schwinger-type condition on the bound states for He, we introduce the sequence of functions:
b„(A) :
2(v„
n
1/2
A)1/4(v„_i
A)1/4’
n € N,
(2.1)
and the off-diagonal Jacobi matrix:
J(A) = J ({0}, {b„(A)}), A € (0,vo) . (2.2)
Recall that we use the same symbol J(A) for the operator in f2(N0) generated by this matrix. It is straightforward to check that the operator J(A) is bounded and self-adjoint. It can be easily verified that the difference J(A) - J0 is a compact operator. Therefore, one has aess(J(A)) = aess(J0) = [-1,1]. Moreover, the operator J(A) has simple eigenvalues ±^„, p„ > 1, with the only possible accumulation points at p = ±1.
Theorem 2.1. [6, Thm. 3.1] Let the self-adjoint operator He, e € (0,1), be as in (1.5) and let the Jacobi matrix J(A) be as in (2.2). Then, the relation:
N((0, A); He) = N((1/e, +rc); J(A)), (2.3)
holds for all A € (0, v0).
Remark 2.2. A careful inspection of the proof of [6, Thm 3.1] yields that Theorem 2.1 can also be modified, replacing (2.3) by:
N((0, A]; He) = N([1/e, +rc); J(A)). (2.4)
In other words, the right endpoint of the interval (0, A) and the left endpoint of the interval (1/e, +to) can be simultaneously included.
The following consequence of Theorem 2.1 and of the above remark will be useful further.
Corollary 2.3. Let the assumptions be as in Theorem 2.1. Then the following claims hold:
(i) e ^ Ak (He) are continuous non-increasing functions;
(ii) dimker (He — A) = dimker (i + eJ(A)) for all A € (0, v0). In particular, since the eigenvalues of J(A) are simple, the eigenvalues of He are simple as well.
Proof. (i) Let e1 € (0,1). For A = Ak(Hei), k € N, we have by Theorem 2.1 and Remark 2.2
N([1/ei, +rc); J(A)) = N((0, A]; Hei) > k.
Hence, for any e2 € (e1,1), we obtain:
N((0, A]; He2 ) = N ([1/e2 , +to); J(A)) > N([1/e1, +to); J(A)) > k.
Therefore, we get Ak(He2) < A = Ak(Hei). Recall that He represents the quadratic form he defined in (1.6). Continuity of the eigenvalues follows from [22, Thms. VI.3.6, VIII.1.14] and from the fact that the quadratic form:
dom he Э u ^ eV2 (signy|y|1/2u|s, |y|1/2u|s) , e € (0,1),
V /R
is relatively bounded with respect to
dom he Э u ^ ||джиУк2 + 1 ||dyu||r2 + 1(yu,yu)R2
with a bound less than one; cf. [6, Lem. 2.1].
(ii) By Theorem 2.1, Remark 2.2, and using symmetry of <r(J(A)) with respect to the origin we get: dimker (He — A) = N((0, A]; He) — N((0, A); He)
= N([1/e, +to); J(A)) — N((1/e, +to); J(A)) = dimker (I + eJ(A)). □
For resonances of He, one can derive a Birman-Schwinger-type condition analogous to the one in Corollary 2.3 (ii). For the sheet ZE C Z of the Riemann surface Z, we define the Jacobi matrix:
Je(A) := J({0}, {bE(A)}), A € C \ К +~), (2.5)
where
bE (A) :
1 ( n
П r„(A,lE )r„_1(A,/E_1)
1/2
n € N.
(2.6)
The Jacobi matrix JE(A) in (2.5) is closed, bounded, and everywhere defined in f2(N0), but in general nonselfadjoint. For E = 0 and A G (0, v0) the Jacobi matrix J0(A) coincides with J(A) in (2.2). In what follows it is also convenient to set bf (A) = 0. In the next theorem, we characterize resonances of He lying on the sheet ZE.
Theorem 2.4. Let the self-adjoint operator He, e G (0,1), be as in (1.5). Let the sheet ZE C Z be fixed, let RE (e) be as in Definition 1.4 and the associated operator-valued function JE (A) be as in (2.5). Then, the following equivalence holds:
A gRe(e) ^ ker (I + eJe(A)) = {0}. (2.7)
For E = 0, the claim of Theorem 2.4 follows from Corollary 2.3 (ii). The proof of the remaining part of Theorem 2.4 is postponed until Appendix. The argument essentially relies on Krein-type resolvent formula [7] for He and on the analytic Fredholm theorem [25, Thm. 3.4.2].
Remark 2.5. Thanks to compactness of the difference JE(A) — J0 we get by [23, Lem. XIII.4.3] that aess(eJE(A)) = o-ess(eJo)) = [—e, e]. Therefore, the equivalence (2.7) can be rewritten as:
A g Re(e) —1 G ct^eJe(A)).
Identity JE(A)* = JE(A) combined with [22, Rem. III.6.23] and with Theorem 2.4 yields that the set RE(e) is symmetric with respect to the real axis.
3. Localization of bound states and resonances
In this section we prove Theorem 1.2 (i) and Theorem 1.5 (i). The idea of the proof is to estimate the norm of JE(A) and to apply Corollary 2.3 (ii) and Theorem 2.4.
Proof of Theorem 1.2 (i) and Theorem 1.5 (i). The square of the norm of the operator JE (A) in (2.5) can be estimated from above by:
IIJe(A)||2 < sup ||Je(A)£||2 < sup
«e-«2(No),||eil=i ?er2(No),||£|| = 1
E |bE(A)£n-1 + bE+i(A)£n+i|2
neNo
< suP (2E (|bE (A)|2|£n-1|2 + |bE+1(A)|2|£n+1|2) ) (3Л)
«e^2(No),|«|=A neNo '
< 4 suP |bE(A)|2 suP ||£||2 = 4sup|bE(A)|2,
neNo £er2(No),||£|| = 1 neN
where (A), n g N0, are defined as in (2.6).
If ||eJe(A)| < 1 holds for a point A G C_, then the condition ker (I + eJE(A)) = {0} is not satisfied. Thus, A cannot by Theorem 2.4 be a resonance of He lying on Z_ in the sense of Definition 1.4. In view of estimate (3.1) and of (2.6) to fulfil ||eJE(A)|| < 1, it suffices to satisfy:
or, equivalently,
n1
|v„-1 - A|1/2|v„ - A|1/2 <E*,
V n G N,
|vn — A| • |vn-1 — A| > e4n2, V n G N.
Thus, the claim of Theorem 1.5 (i) is proven. If ||eJ0(A)|| < 1 holds for a point A G (0,1/2) then the condition ker (I + eJ0(A)) = {0} is not satisfied. Thus, by Corollary 2.3 (ii), A is not an eigenvalue of He. In view of (3.1) and (2.6) to fulfil ||eJ0(A)|| < 1, it suffices to satisfy:
(vn_1 — A (vn — A = A2 — 2nA + n2 — 1/4 > n2e4, V n G N.
(3.2)
The roots of the equation A2 — 2nA + n2 — 1/4 — n2e4 = 0 are given by A±(e) = n ± v21/4 + n2e4. Since A+ (e) > 1/2 for all n G N, the condition (3.2) yields A1(He) > min A_(e). For n G N we have:
n ne N n
A—+1(e) A— (e)
1
(2n + 1)e4
(4 + n2e4)1/2 + (1 + (n + 1)2e4)
1/2
1
(2n + 1)e4 (2n + 1)e2
1 — e2 > 0.
Hence, min A_ (e) = A_ (e) and the claim of Theorem 1.2 (i) follows.
ne N n 1
□
4. The weak coupling regime: e ^ 0+
In this section, we prove Theorem 1.2 (ii) and Theorem 1.5 (ii)-(iii). Intermediate results of this section given in Lemmata 4.1 and 4.3 are of an independent interest.
First, we introduce some auxiliary operators and functions. Let n g N0 and the sheet ZE C Z be fixed. We make use of notation Pki := en+fc_2(-, en+i-2) with k, l g {1, 2,3}. Note that for n = 0 we have Pk1 = P1k = 0 for k = 1,2, 3. It will also be convenient to decompose the Jacobi matrix JE (A) in (2.5) as:
Je (A) = S n,E (A) + T„,e (A), (4.1)
where the operator-valued functions A ^ Tn,E(A), Sn,E(A) are defined by:
T„,e (A) := &E+ i(A) [P23 + P32] + &E (A) [P21 + P12], Sn,E (A) := Je (A) - Tn,E (A). (4.2)
Clearly, the operator-valued function Sn,E(•) is uniformly bounded on D1/2(vn). Moreover, for sufficiently small r = r(n) g (0,1/2) the bounded operator I + eSn,E(A) is at the same time boundedly invertible for all (e, A) g Hr (n) := Dr x Dr (vn). Thus, the operator-valued function:
Rn,E (e,A):= (I + eSn,E(A)) , (4.3)
is well-defined and analytic on Qr(n) and, in particular, Rn,E(0, vn) = I. Furthermore, we introduce auxiliary scalar functions Qr (n) э (e, A) ^ fEl (e, A) by:
fkl (e, A) := (Rn,E(^ ^^Gk^ en+i-2), k, l g {1, 2, 3}. (4.4)
Thanks to Rn,E(0, vn) = I we have fjEl (0, vn) = Ski. Finally, we introduce 3 x 3 matrix-valued function:
Dr x D^ (vn) э (e, A) ^ An,E (e A) := (nEl (e, A)) kl = 1 (4.5)
with the entries given for k, l = 1, 2, 3 by:
aEi(e, A) := bE(A)(fEk(e, A)S2i + f2Efc(e, A)Su) + bE+^Af(e, A)S3i + fEk(e, A)S2i).
(4.6)
We remark that rank An,E(e, A) < 2 due to linear dependence between the first and the third columns in An,E(e, A).
In the first lemma, we derive an implicit scalar equation which characterizes those points A g C \ [v0, +to) near vn for which the condition ker (I + eJE(A)) = {0} is satisfied under additional assumption that e > 0 is small enough. This equation can be used to characterize the ‘true’ resonances for He as well as the weakly coupled bound state if n = 0 and E = 0.
Lemma 4.1. Let the self-adjoint operator He, e g (0,1), be as in (1.5). Let n g N0 and the sheet ZE C Z be fixed. Let r = r(n) > 0 be chosen as above. Then for all e g (0, r) a point A g Dr (vn) \ [v0, to) is a resonance of He on ZE if, and only if
det (I + eAn,E(e, A^ — 0.
Proof. Using the decomposition (4.1) of JE(A) and the auxiliary operator in (4.3), we find:
dimker (I + eJk(A)) = dimker (I + eSn,E(A) + eTn,E(A)) = dimker (I + eRn,E(e, A)Tn,E(A)). (4.7)
Note that:
rank (Rn,E(e, A)Tn,E(A)) < rank(Tn,E(A)) < 3 and, hence, using [26, Thm. 3.5(b)], we get:
dimker (I + eRn,E(e,A)Tn,E(A)) > 1 det (I + eRn,E(e, A)Tn,E(A)) = 0. (4.8)
For the orthogonal projector P := P11 + P22 + P33 the identity Tn,E(A) = Tn,E(A)P is straightforward. Hence, employing [27, IV. 1.5] we find:
det (I + eRn,E(e, A)Tn,E(A)) = det (I + eRn,E(e, A)Tn,E(A)P) = det (I + ePRn,E(e, A)Tn,E(A)). (4.9)
For k, l g {1,2, 3} we can write the following identities:
PkkPRn,E(e, A)Tn,E(A)Pii
Pkk Rn,E(e, A)(bE(A) [P21 + P12] + bE+1(A) [P23 + P32Q Pii
Pkk Rn,E (e, A^bE (A) [P2iS1i + P 1iS2i] + bE+1(A) [P2iS3i + P3iS2i] j
PkibE(A) [f2Ek(e, A)S1i + fEk(e, A)S2i] + PkibE+1(A) [f2k|(e, A)S3i + f3Ek(e, A)S2i]
with /Ц as in (4.4), and as a result we get
3 3
PRn,E (e,A)T n,E (A) = aki(£,^)pkh
k=11=1
with a?Ae , A) as in (4.6). Hence, the determinant in (4.9) can be expressed as:
det (I + eRn,E(e, A)T„,e(A)) = det(I + eAn,E(e, A))
where on the right-hand side we have the determinant of the 3 x 3 matrix I + eAn,E(e, A); cf (4.5). The claim of lemma then follows from (4.7), (4.8), and Theorem 2.4. □
In the second lemma, we establish the existence and investigate properties of solutions of the scalar equation in Lemma 4.1. To this aim it is natural to try to apply the analytic implicit function theorem. The main obstacle that makes a direct application of the implicit function theorem difficult lies in the fact that A ^ det(I + eAn,E(e, A)) is not analytic near vn due to the cut on the real axis. We circumvent this obstacle by applying the analytic implicit function theorem to an auxiliary function which is analytic in the disc and has values in different sectors of this disc that are in direct correspondence with the values of A ^ det(I + eAn,E(e, A)) on the four different sheets in Z which are mutually adjacent in a proper way.
Assumption 4.2. Let n G N0 and the sheet ZE c Z be fixed. Let the sheets ZF, ZG and ZH be such that ZE ~n_i ZF, ZF ^n ZG and ZG ~n_i ZH. For r > 0 let the matrix-valued function Dr x В* э (e, к) ^ Bn,E (e, к) be defined by:
Bn,E (e, к)
An,E (e vn к ),
An,F(e vn к ),
An,G(e, vn к ),
An,H(e vn к ),
arg к G ФЕ arg к G Ф? arg к G Фа arg к G Фн
(-E
(_3n
( 4
(-n (-n
( 4,
-3П] и (о, n],
, - 2 ] и (4,2 ], - П ] и (2, 3п ], о] и (3f ,п].
Tracing the changes in the characteristic vector along the path ZE ^n-1 ZF ^n ZG ^n-1 ZH, one easily verifies that ZH ^n ZE. Thus, Bn,E is analytic on Br x В* for sufficiently small r > 0 which is essentially a consequence of componentwise analyticity in Br of vector-valued function:
к ^ R, (vn - к4), • G {E, F, G, H} for arg к G Ф.,
where R, is as in (1.9).
Lemma 4.3. Let n G N0 and the sheet ZE c Z be fixed. Set (p, q, r) := (1E_1,1E, ^Е+1). Let the matrix-valued function Bn,E be as in Assumption 4.2. Then the implicit scalar equation:
det (I + eBn,E(e, к)) = 0
has exactly two solutions кп,е,5-(•) analytic near e = 0 such that кп,е,5-(0) = 0, satisfying det(I + eBn,E(e, кп Е,^(e))) = 0 pointwise for sufficiently small e > 0, and having asymptotic expansions:
where zn,E
кп,Е j (e)
e
(Zn,E j/2 2
+ O(e2),
(-1)q+r(n + 1) + (-1)p+q+1ni.
e ^ 0+,
(4.10)
Proof. First, we introduce the shorthand notations:
и(к) := bn (vn - к4), «(к) := ЬП+^п - к4), • G {E, F, G, H} for arg к G Ф,.
Let bkl with k, l g {1, 2, 3} be the entries of the matrix-valued function Bn E. Furthermore, we define the scalar functions X = X(e, к), Y = Y(e, к), and Z = Z(e, к) by:
X := bn + b22 + Ьзз,
Y := ЬцЬ22 + Ь22Ьзз + ЬцЬзз - Ь1зЬз1 - b^b21 - Ь2зЬз2, (4.11)
Z := ЬцЬ22Ьзз + Ь1зЬз2Ь21 + Ь^2зЬз1 - Ь1зЬз1Ь22 - Ь12Ь21Ьзз - ЬцЬ2зЬз2.
Employing an elementary formula for the determinant of 3 x 3 matrix, the equation det(I + eBn,E(e, к)) = 0 can be equivalently written as:
By a purely algebraic argument, one can derive from (4.6) that Z = 0. Hence, (4.12) simplifies to 1 + eX(e, к) + £2Y(e, к) = 0. Introducing new parameter t := e/к, we can further rewrite this equation as:
1+ 7кХ(е,к)+ t2^Y(e, к) = 0. (4.13)
Note also that the coefficients (е,к) ^ кХ(e, k),k2Y(e, к) of the quadratic equation (4.13) are analytic in Dj!. For each fixed pair (e, к) the equation (4.13) has (in general) two distinct roots tj (e, к), j = 0,1. The condition det(I + eBn,E(e, к)) = 0 with к = 0 holds if, and only ifat least one of the two conditions:
fj(e, к) := e - к^- (e, к) = 0, j is satisfied. Using analyticity of u(-) and v(-) near к = 0, we compute:
n1/2 rein/8
lim ки = lim rein/8u(rein/8) = lim
к—0 r—0+ r 'n 1
0,1,
(4.14)
,l/2ein/8
lim кг = lim rein/8v(rein/8) = lim
K —0 r—0+ r ' n 1
—0+ 2 ((-1+ir4)J/2(ir4)q/2)i/2 2((-1)p+qiein/4)1/2 ’
(n + 1)1/2 rein/8 (n + 1)1/2ein/8
—0+ 2 ((ir4)1/2(1 +ir4)1/2)1/2 2(( —1)q+rein/4)1/2 "
Hence, we get:
lim rein/86fc;(e,rein/8) = lim rein/8u(rein/8) f(0)^ + f3k(0)^)
£,r—— 0 +
r—0+
+ lim rein/8v(rein/8)(fffc(0)^ + f2l(0)^1i)
r—0+
n1/2ein/8( <^<*3! + <*2i) + (n + 1)1/2 ein/8( ^1fc ^ + <*2fc <*1i)
2((-1)p+qiein/4)1/2
Combining this with (4.11) we end up with:
lim кХ = lim rein/8X(e, rein/8)= lim rein/8 [b11 + 622 + 633] (e,
(£,«) —0
lim к2Y
(£,«) —0
>0+
lim r2ein/4Y (e
,r—0+
£,r —0+
2((-1)q+rein/4)1/2
/8) =
T/8)
lim r2ein/4 [611622 + 622633 + 611633 - 613631 - 612621 - 623632] (e, rein/8)
>0+
= g lim + r2ein/4 [ - 612621 - 623632] (e, rein/8)
,1/2ein/8
2((-1)p+qiein/4)1/2
(n + 1)1/2ein/8 2((-1)q+rein/4)1/2
= (-1)p+q+1ni (-1)q+r(n + 1) = z„,e
= 4 4 = ^.
Hence, the roots t-(e,к) of (4.13) converge in the limit ^,к) ^ 0 to the roots 2[(zn,E)1/2] 1, j = 0,1, of the
quadratic equation zn,Et2 - 4 = 0. Moreover, analyticity of the coefficients in equation (4.13), the above limits, and the formula for the roots of a quadratic equation imply analyticity of the functions (e, к) ^ t- (e, к) near 0. Step 2. The partial derivatives of f- in (4.14) with respect to e and к are given by d£f- = 1 - кд£^- and dKfj = -tj - кдк^. Analyticity of t- near 0 implies (d£f-)(0) = 1 and (dKf-)(0) = -1-. In particular, we have shown that (df)(0) = 0. Since the functions f-(•) are analytic near 0 and satisfy f-(0) = 0, we can apply the analytic implicit function theorem [25, Thm. 3.4.2] which yields existence of a unique function к-(•), analytic near e = 0 such that к- (0) = 0 and that f-(e, к- (e)) = 0 holds pointwise. Moreover, the derivative of к- at e = 0 can be expressed as:
1 (4.15)
к-(0) = - (d£f-)(0)
(dKf- )(0) t- (0)'
Hence, we obtain Taylor expansion for к- near e = 0:
(z )1/2
к-(e) = к-(0) + к -(0)e + O(e2) = —^- + O(e2) = e n,E ----------+ O(e2)
t- (0)
2
e ^ 0 + .
The functions к-, j = 0,1, satisfy all the requirements in the claim of the lemma.
□
0
2
2
Now we are prepared to prove Theorem 1.2 (ii) and Theorem 1.5 (ii)-(iii) from the introduction.
Proof of Theorem 1.2 (ii). By Proposition 1.1 (iv) we have Ni/2(He) = 1 for all sufficiently small e > 0. Recall that we denote by A1(He) the corresponding unique eigenvalue. Thus, we have by Lemma 4.1:
det(I + eAo,0(e, Ai(He))) = 0.
Using the construction of Assumption 4.2 for the physical sheet and n = 0, we obtain
det(| + eBo,0(e, (v0 — A1(He))1/4)) = det(| + eAO,0(e, A1(He))) = 0,
where we have chosen the principal branch for (-)1/4. Thus, by Lemma 4.3, we get:
(v0 — A1 (He)) 1/4 = 2 + O(e2) e ^ 0+!
where we have used the fact that above equation we arrive at:
zo,0 = 1. Hence, taking the fourth power of the left and right hand sides in the
e4
A1(He)= vo - - + O(e5), 16
e ^ 0 + .
□
Proof of Theorem 1.5 (ii)-(iii). Let n G N and the sheet ZE C Z be fixed. Let us repeat the construction of Assumption 4.2. By Lemma 4.3 we infer that there exist exactly two analytic solutions Kn,Ej-, j = 0,1 of the implicit scalar equation det(I + eBn,E(е,к)) = 0 such that Kn,E,j(0) = 0. It can be checked that both solutions correspond to the same resonance and it suffices to analyze the solution Kn,E := Kn,E,0 only.
For all small enough e > 0 the asymptotics (4.10) yields:
arg(«n,E(e)) = 2 arg(zn,E) G ФE, if, and only if n G S(E).
Hence, if n G N \ S(E), Lemmata 4.1 and 4.3 imply that there will be no resonances in the vicinity of the point A = vn lying on Z- for sufficiently small e > 0. Thus, we have proven Theorem 1.5 (iii). While if n G S(E) we get by Lemmata 4.1 and 4.3 that there will be exactly one resonance
AE(He) = vn - («n,E(e))4,
in the vicinity of the point A = vn lying on Z- for sufficiently small e > 0 and its asymptotic expansion is a direct consequence of the asymptotic expansion (4.10) given in Lemma 4.3. Thus, the claim of Theorem 1.5 (ii) follows. □
APPENDIX
A. Krein’s formula, meromorphic continuation of resolvent, and condition on resonances
In this appendix, we use Krein’s resolvent formula for Smilansky Hamiltonian to prove Proposition 1.3 and Theorem 2.4 on meromorphic continuation of (He — A)-1 to Z The proposed continuation procedure is of an iterative nature wherein, we first extend (He — A)-1 to the sheets adjacent to the physical sheet, then to the sheets which are adjacent to the sheets being adjacent to the physical sheet and so on.
To this aim, we define for n g N0 the scalar functions C\[v0, +to) ^ yn(A) and (C\[v0, +ю))хМ ^ nn(A; x)
by:
yn(A) := rn(A)v'vn, nn(A;x) := v^4exp(—rn(A)|x|), (A.1)
where rn(-), n g N0, is as in (1.7). Next, we introduce the following operator-valued function:
T(A): f2(N0) ^ H, T(A){cn} := {cnnn(A;x)}.
For each fixed A g C \ [v0, +to) the operator T(A) is bounded and everywhere defined and the adjoint of T(A) acts as:
T(A) {un} ^ {1n(A; un)}n£No ! /n(A; u) : I nn(A; x)u(x)dx.
With these preparations, the resolvent difference of He and H0 can be expressed by [7, Thm. 6.1] (see also [4, Sec. 6]) as follows:
(He — A)-1 = (H0 — A)-1 + T(A)Y(A) [(I + eJ0(A))-1 — I]Y(A)T(A)*, A G C \ [v0, +~), (A.2)
where H0 is the Smilansky Hamiltonian with e = 0, Y(A) = diag{(2yn(A))-1/2} and J^(A) is as in (2.5). The formula (A.2) can be viewed as a particular case of abstract Krein’s formula (see e.g. [29-31]) for the resolvent difference of two self-adjoint extensions of their common densely defined symmetric restriction.
Proof of Proposition 1.3 and Theorem 2.4. Let us fix n G N0 and a sheet ZE C Z. We denote by Rn(A) the resolvent of the self-adjoint operator H2(R) э f ^ —f" + vnf in the Hilbert space L2(R). We can express the function (•; u) in (1.13) using Krein’s formula (A.2) as:
r^,e(A; u) = ((He — A)-1u <g> en,u <g> ef)
= ((Ho — A)-1u <g) en, u ® en) + ^Y(A) (I + eJ0(A))-1 — I Y(A)T(A)*u ® en, T(A)*u ® efj
= (Rn(A)u u)R + Zn(A; u)1n(A; u) ([(I + eJ0(A)) 1 — 0 Y(A)en, Y (A)* en'j
(Rn(A)u, u)r +
/n(A; u)1n ( A; u)
2yn(A)
^ + £J0(A)) en, en^ — 1
Since (Rn(A)u, u)R, yn(A), 1n(A; u), and 1n(A; u) can be easily analytically continued to Z, to extend r^,e(•; u) meromorphically to the other sheets of the component ZZ it suffices to extend:
5n,E(A) := ((I + eJ0(A))
en, en
meromorphically from Z0 to Z. The poles of the meromorphic extension of s^e(-) can be identified with the resonances of He in the sense of Definition 1.4.
To this aim, we set by definition:
5e,£(A) := ((I + eJE(A)) 1 en, en) ,
for any A G C \ [v0, +to) such that —1 G ^(eJE(A)). In what follows, we let ZE and ZF be two sheets of Z such
that ZE ~n-1 ZF with n g N04 Suppose that A ^ sE,e(G is well defined and meromorphic either on Z+ or on Z_. Next, we extend A ^ sE,e(G meromorphically from Z± to Zjjr Without loss of generality, we restrict our attention to the case that A ^ s^G) is meromorphic on Z+ and extend it meromorphically to Z_. On the open set On := C+ U C_ U (vn-1, vn), the operator-valued function:
Jef (A)
Je(A), A G C+,
JF(A), A G On \ C+,
is analytic which is essentially a consequence of analyticity on On of the entries 6y(A) (with • = E for A g C+ and • = F for A g C_) for the underlying Jacobi matrix. Thus, the operator-valued function:
On Э A ^ Aef(A) := £ (I + eJo) (Jef(A) — Jo)
is also analytic on On because of the analyticity of JEF(A). Furthermore, the values of A^ef(•) are compact operators thanks to compactness of the difference JEF(A) — J0. Taking into account that:
((I + AeEF (A))
1
e^
(I + £Jo)
1 en
^Е,£ ( A ) , A G ^,
sL(A), A G On \ C+,
we obtain from the analytic Fredholm theorem [28, Thm. VI.14] that C_ э A ^ sFe(A) is a meromorphic continuation of C+ э A ^ sE e(A) across the interval (vn-1, vn) and that the poles of C_ э A ^ sF e(A) satisfy the condition:
ker (I + £Jf(A)) = {0}, A G C_.
Starting from the physical sheet Z0, we use the above procedure iteratively to extend e(-) meromorphically to the whole of Z thus proving Proposition 1.3 and Theorem 2.4. □
Acknowledgements
This research was supported by the Czech Science Foundation (GACR) within the project 14-06818S.
4
Recall that for any sheet Ze holds Ze
i Ze .
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