Aharonov and Bohm versus Welsh eigenvalues
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Abstract
We consider a class of twodimensional Schrödinger operator with a singular interaction of the \(\delta \) type and a fixed strength \(\beta \) supported by an infinite family of concentric, equidistantly spaced circles, and discuss what happens below the essential spectrum when the system is amended by an Aharonov–Bohm flux \(\alpha \in [0,\frac{1}{2}]\) in the center. It is shown that if \(\beta \ne 0\), there is a critical value \(\alpha _{\mathrm {crit}}\in (0,\frac{1}{2})\) such that the discrete spectrum has an accumulation point when \(\alpha <\alpha _{\mathrm {crit}}\), while for \(\alpha \ge \alpha _{\mathrm {crit}}\) the number of eigenvalues is at most finite, in particular, the discrete spectrum is empty for any fixed \(\alpha \in (0,\frac{1}{2})\) and \(\beta \) small enough.
Keywords
Singular Schrödinger operator Radial symmetry Discrete spectrum Aharonov–Bohm fluxMathematics Subject Classification
81Q10 35J101 Introduction
Schrödinger operators with radially periodic potentials attracted attention because they exhibit interesting spectral properties. It was noted early [10] that the essential spectrum threshold of such an operator coincides with that of the onedimensional Schrödinger operator describing the radial motion. More surprising appeared to be the structure of the essential spectrum which may consist of interlacing intervals of dense point and absolutely continuous nature as was first illustrated using potentials of cosine shape [11].
While this behavior can be observed in any dimension \(\ge 2\), the twodimensional case is of a particular interest because here these operators can also have a discrete spectrum below the threshold of the essential one. This fact was first observed in [5] and the national pride inspired the authors to refer to this spectrum as to Welsh eigenvalues; it was soon established that that their number is infinite if the radially symmetric potential is nonzero and belongs to \(L^1_\mathrm {loc}\,\) [17]. Moreover, the effect persists if such a regular potential is replaced by a periodic array of \(\delta \) interactions or more general singular interactions [7, 8].
The magnetic interaction we add is also chosen in the simplest possible way, namely as an Aharonov–Bohm flux \(\alpha \) at the origin of the coordinates, measured in suitable units, that gives rise to a magnetic field vanishing outside this point. The corresponding Hamiltonian will be denoted \(H_{\alpha ,\beta }\) and as we will argue, it is sufficient to consider flux values up to half of the quantum, \(\alpha \in (0,\frac{1}{2})\). Since singular interactions are involved, it may be useful to stress that we consider an Aharonov–Bohm flux alone, without any additional point interactions at origin à la [1, 6]. It is known that local magnetic fields generally, and Aharonov–Bohm fluxes in particular, can reduce the discrete spectrum, if combined with an effective potential that behaves like \(r^{2}\), on the borderline between short and long range, the effect can be dramatic [14].

there is an \(\alpha _{\mathrm {crit}}(\beta )=\alpha _{\mathrm {crit}} \in (0, \frac{1}{2})\) such that for \(\alpha \in (0,\alpha _{\mathrm {crit}})\) the discrete spectrum of \(H_{\alpha ,\beta }\) is infinite accumulating at the threshold \(E_0\), while for \(\alpha \in [\alpha _{\mathrm {crit}},\frac{1}{2})\) there is at most a finite number of eigenvalues below \(E_0\),
 the critical value \(\alpha _{\mathrm {crit}}(\beta )\) admits the following asymptotics,and$$\begin{aligned} \alpha _{\mathrm {crit}}(\beta )\rightarrow \frac{1}{2}  \quad \mathrm {for }\quad \beta \rightarrow \pm \infty \end{aligned}$$$$\begin{aligned} \alpha _{\mathrm {crit}}(\beta )\rightarrow 0 + \quad \mathrm {for }\quad \beta \rightarrow 0 , \end{aligned}$$

for any fixed \(\alpha \in (0, \frac{1}{2} )\) there exists \(\beta _0>0\) such that for any \(\beta \le \beta _0\) we have \(\sigma _\mathrm {d}(H_{\alpha ,\beta }) = \emptyset \), and moreover, \(\sigma _\mathrm {d}(H_{\frac{1}{2},\beta }) = \emptyset \) holds for any \(\beta \in \mathbb {R}\).
2 Preliminaries
Our interest here concerns the spectrum of \(H_{\alpha ;\beta }\) in the interval \((\infty ,E_0)\) which is discrete according to (2.7). Let us first collect its elementary properties.
Proposition 2.1
 (i)
\(\sharp \sigma _\mathrm {disc}(H_{0;\beta })=\infty \);
 (ii)
\(\sigma _\mathrm {disc}(H_{\frac{1}{2};\beta })=\emptyset \);
 (iii)
\(\sigma _\mathrm {disc}(H_{\alpha ;\beta })=\sigma _\mathrm {disc}(H_{1\alpha ;\beta })\);
 (iv)
if \(\sigma _\mathrm {disc}(H_{\alpha ;\beta })\ne \emptyset \), then eigenvalues of \(H_{\alpha ;\beta }\) are nondecreasing in \([0,\frac{1}{2}]\), \(\,\lambda _j(\alpha ')\ge \lambda _j(\alpha )\) holds for a fixed j if \(\alpha '\ge \alpha \).
Proof
Claim (i) follows from [8, Theorem 5.1]. Partial wave operators in the decomposition (2.3) can contribute to \(\sigma _\mathrm {disc}(H_{\alpha ;\beta })\) only if \(c_{\alpha ,l}<0\). Indeed, if \(c_{\alpha ,l}=0\) the spectrum of \(H_{\alpha ;\beta ,l}\) coincides, up to multiplicity, with that of the operator \(\mathrm {h}_\beta \) amended according to (2.2) with Dirichlet condition at \(x=0\), hence (2.6) in combination with a bracketing argument [15, Sect. XIII.15] shows that the discrete spectrum is empty and yields assertion (ii). Furthermore, in view of the minmax principle [15, Sect. XIII.1] this verifies the above claim and shows that the discrete spectrum comes from \(H_{\alpha ;\beta ,0}\) if \(\alpha \in [0,\frac{1}{2})\) and from \(H_{\alpha ;\beta ,1}\) if \(\alpha \in (\frac{1}{2},1)\). The third claim follows from the identity \(c_{\alpha ,0}=c_{1\alpha ,1}\) valid for \(\alpha \in (0,1)\), and the last one we get employing the minmax principle again. \(\square \)
It is therefore clear, as indicated in the introduction, that to describe the discrete spectrum it is sufficient to limit our attention to the values \(\alpha \in (0,\frac{1}{2})\) and to consider the operator \(H_{\alpha ;\beta ,0}\).
3 Properties of the discrete spectrum
Theorem 3.1
Proof
We note that the analogue of \(c_{\mathrm {crit}}\) for regular potentials is known in the literature as Knesser constant, cf. [16]. The obtained result allows us to prove the following claim.
Theorem 3.2
There exists an \(\alpha _{\mathrm {crit}}(\beta )=\alpha _{\mathrm {crit}}\in (0,\frac{1}{2})\) such that for \(\alpha \in (0, \alpha _{\mathrm {crit}})\) the operator \(H_{\alpha , \beta }\) has infinitely many eigenvalues accumulating at the threshold \(E_0\), the multiplicity taken into account, while for \(\alpha \in [\alpha _{\mathrm {crit}},\frac{1}{2})\) the cardinality of the discrete spectrum is finite.
Proof
4 Nonexistence of the discrete spectrum for weak \(\delta \) interactions
Theorem 4.1
Given \(\alpha \in (0, \frac{1}{2} )\) there exists a \(\beta _0 >0\) such that for any \(\beta  <\beta _0 \) the operator \(H_{\alpha ; \beta }\) has no discrete spectrum.
Proof
Lemma 4.2
Proof
Lemma 4.3
Proof
Proof of Theorem 4.1, continued
5 Oscillation theory tools
To make the paper selfcontained, we collect in this section the needed results of oscillation theory for singular potentials derived in [8]. Note that they extend the theory of Wronskian zeros for regular potentials developed in [9], related results can also be found in [19].

T is limit point in at least one endpoint \(l_\pm \)

H is defined by separated boundary conditions at the endpoints.
6 Concluding remarks
The main aim of this letter is to show that the influence of a local magnetic field on the Welsh eigenvalues depends nontrivially on the magnetic flux. In order to make the exposition easy, we focused on the simple setting with radial \(\delta \) potentials and an Aharonov–Bohm field, however, we are convinced that the conclusions extend to other potentials and other magnetic field profiles, as long as the radial symmetry and periodicity are preserved. This could be a subject of further investigation, as well as the remaining spectral properties of the present simple model such as the eigenvalue accumulation for \(\alpha \in (0,\alpha _{\mathrm {crit}})\) or (non)existence of eigenvalues for \(\alpha \in [\alpha _{\mathrm {crit}}, \frac{1}{2} )\) and an arbitrary \(\beta \ne 0\). It would be also interesting to revisit from the present point of view situations in which the radially periodic interaction is of a purely magnetic type with zero total flux [12].
Notes
Acknowledgements
The authors are obliged to the referees for careful reading of the manuscripts and making remarks that allowed to improve the presentation. The research was supported by the Project 1701706S of the Czech Science Foundation (GAČR) and the Project DEC2013/11/B/ST1/03067 of the Polish National Science Centre (NCN).
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