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We prove that the $(k+d)$-th Neumann eigenvalue of the biharmonic operator on a bounded connected $d$-dimensional $(d\ge2)$ Lipschitz domain is not larger than its $k$-th Dirichlet eigenvalue for all $k\in\mathbb{N}$. For a special class of domains with symmetries we obtain a stronger inequality. Namely, for this class of domains, we prove that the $(k+d+1)$-th Neumann eigenvalue of the biharmonic operator does not exceed its $k$-th Dirichlet eigenvalue for all $k\in\mathbb{N}$. In particular, in two dimensions, this special class consists of domains having an axis of symmetry.

We consider the one-parametric family of self-adjoint realizations of the two-dimensional massive Dirac operator with a Lorentz scalar $\delta$-shell interaction of strength $\tau\in\mathbb{R}\setminus\{-2,0,2\}$ supported on a broken line of opening angle $2\omega$ with $\omega\in(0,\frac{\pi}{2})$. The essential spectrum of any such self-adjoint realization is symmetric with respect to the origin with a gap around zero whose size depends on the mass and, for $\tau < 0$, also on the strength of the interaction, but does not depend on $\omega$. As the main result, we prove that for any $N\in\mathbb{N}$ and strength $\tau\in(-\infty,0)\setminus\{-2\}$ the discrete spectrum of any such self-adjoint realization has at least $N$ discrete eigenvalues, with multiplicities taken into account, in the gap of the essential spectrum provided that $\omega$ is sufficiently small. Moreover, we obtain an explicit estimate on $\omega$ sufficient for this property to hold. For $\tau\in(0,\infty)\setminus\{2\}$, the discrete spectrum consists of at most one simple eigenvalue., Comment: 28 pages, 3 figures

We investigate the spectrum of the Dirac operator with infinite mass boundary conditions posed in a tubular neighborhood of a smooth compact hypersurface in $\mathbb{R}^n$ without boundary. We prove that when the tubular neighborhood shrinks to the hypersurface, the asymptotic behavior of the eigenvalues is driven by a Schr\"odinger operator involving electric and Yang-Mills potentials, both of geometric nature.

In this paper we consider the two-dimensional Schrödinger operator with an attractive potential which is a multiple of the characteristic function of an unbounded strip-shaped region, whose thickness is varying and is determined by the function $\mathbb{R}\ni x \mapsto d+\varepsilon f(x)$, where $d > 0$ is a constant, $\varepsilon > 0$ is a small parameter, and $f$ is a compactly supported continuous function. We prove that if $\int_{\mathbb{R}} f \,\mathsf{d} x > 0$, then the respective Schrödinger operator has a unique simple eigenvalue below the threshold of the essential spectrum for all sufficiently small $\varepsilon >0$ and we obtain the asymptotic expansion of this eigenvalue in the regime $\varepsilon\rightarrow 0$. An asymptotic expansion of the respective eigenfunction as $\varepsilon\rightarrow 0$ is also obtained. In the case that $\int_{\mathbb{R}} f \,\mathsf{d} x < 0$ we prove that the discrete spectrum is empty for all sufficiently small $\varepsilon > 0$. In the critical case $\int_{\mathbb{R}} f \,\mathsf{d} x = 0$, we derive a sufficient condition for the existence of a unique bound state for all sufficiently small $\varepsilon > 0$., 21pages, one figure

In the present paper we introduce the perturbed two-dimensional Robin bi-Laplacian in the exterior of a bounded simply-connected $C^2$-smooth open set. The considered perturbation is of lower order and corresponds to tension. We prove that the essential spectrum of this operator coincides with the positive semi-axis and that the negative discrete spectrum is non-empty if, and only if, the boundary parameter is negative. As the main result, we obtain an isoperimetric inequality for the lowest eigenvalue of such a perturbed Robin bi-Laplacian with a negative boundary parameter in the exterior of a bounded convex planar set under the constraint on the maximum of the curvature of the boundary with the maximizer being the exterior of the disk. The isoperimetric inequality is proved under the additional assumption that to the lowest eigenvalue for the exterior of the disk corresponds a radial eigenfunction. We provide a sufficient condition in terms of the tension parameter and the radius of the disk for this property to hold., Comment: 24 pages

We consider the massless Dirac operator with the MIT bag boundary conditions on an unbounded three-dimensional circular cone. For convex cones, we prove that this operator is self-adjoint defined on four-component $H^1$--functions satisfying the MIT bag boundary conditions. The proof of this result relies on separation of variables and spectral estimates for one-dimensional fiber Dirac-type operators. Furthermore, we provide a numerical evidence for the self-adjointness on the same domain also for non-convex cones. Moreover, we prove a Hardy-type inequality for such a Dirac operator on convex cones, which, in particular, yields stability of self-adjointness under perturbations by a class of unbounded potentials. Further extensions of our results to Dirac operators with quantum dot boundary conditions are also discussed., Comment: 38 pages, 1 figure; revised version

The spectrum of the Dirichlet Laplacian on any quasi-conical open set coincides with the non-negative semi-axis. We show that there is a connected quasi-conical open set such that the respective Dirichlet Laplacian has a positive (embedded) eigenvalue. This open set is constructed as the tower of cubes of growing size connected by windows of vanishing size. Moreover, we show that the sizes of the windows in this construction can be chosen so that the absolutely continuous spectrum of the Dirichlet Laplacian is empty.

We consider the Laplace operator on a triangle, subject to attractive Robin boundary conditions. We prove that the equilateral triangle is a local maximiser of the lowest eigenvalue among all triangles of a given area provided that the negative boundary parameter is sufficiently small in absolute value, with the smallness depending on the area only. Moreover, using various trial functions, we obtain sufficient conditions for the global optimality of the equilateral triangle under fixed area constraint in the regimes of small and large couplings. We also discuss the constraint of fixed perimeter., Comment: Revised version accepted for publication in Applied Mathematics and Optimization

We consider the two-dimensional Dirac operator with Lorentz-scalar $\delta$-shell interactions on each edge of a star-graph. An orthogonal decomposition is performed which shows such an operator is unitarily equivalent to an orthogonal sum of half-line Dirac operators with off-diagonal Coulomb potentials. This decomposition reduces the computation of the deficiency indices to determining the number of eigenvalues of a one-dimensional spin-orbit operator in the interval $(-1/2,1/2)$. If the number of edges of the star graph is two or three, these deficiency indices can then be analytically determined for a range of parameters. For higher numbers of edges, it is possible to numerically calculate the deficiency indices. Among others, examples are given where the strength of the Lorentz-scalar interactions directly change the deficiency indices while other parameters are all fixed and where the deficiency indices are $(2,2)$, neither of which have been observed in the literature to the best knowledge of the authors. For those Dirac operators which are not already self-adjoint and do not have $0$ in the spectrum of the associated spin-orbit operator, the distinguished self-adjoint extension is also characterized., Comment: Revised version, 35 pages

In this work we consider the two-dimensional Dirac operator with general local singular interactions supported on a closed curve. A systematic study of the interaction is performed by decomposing it into a linear combination of four elementary interactions: electrostatic, Lorentz scalar, magnetic, and a fourth one which can be absorbed by using unitary transformations. We address the self-adjointness and the spectral description of the underlying Dirac operator, and moreover we describe its approximation by Dirac operators with regular potentials.

In this note we consider the self-adjoint Schr��dinger operator $\mathsf{A}_��$ in $L^2(\mathbb{R}^d)$, $d\geq 2$, with a $��$-potential supported on a Lipschitz hypersurface $��\subseteq\mathbb{R}^d$ of strength $��\in L^p(��)+L^\infty(��)$. We show the uniqueness of the ground state and, under some additional conditions on the coefficient $��$ and the hypersurface $��$, we determine the essential spectrum of $\mathsf{A}_��$. In the special case that $��$ is a hyperplane we obtain a Birman-Schwinger principle with a relativistic Schr��dinger operator as Birman-Schwinger operator. As an application we prove an optimization result for the bottom of the spectrum of $\mathsf{A}_��$., 23 pages, title was changed, the manuscript is submitted to the Sergey Naboko memorial volume

We consider the self-adjoint Landau Hamiltonian $H_0$ in $L^2(\mathbb{R}^2)$ whose spectrum consists of infinitely degenerate eigenvalues $��_q$, $q \in \mathbb{Z}_+$, and the perturbed operator $H_\upsilon = H_0 + \upsilon��_��$, where $��\subset \mathbb{R}^2$ is a regular Jordan $C^{1,1}$-curve, and $\upsilon \in L^p(��;\mathbb{R})$, $p>1$, has a constant sign. We investigate ${\rm Ker}(H_\upsilon -��_q)$, $q \in \mathbb{Z}_+$, and show that generically $$0 \leq {\rm dim \, Ker}(H_\upsilon -��_q) - {\rm dim \, Ker}(T_q(\upsilon ��_��)) < \infty,$$ where $T_q(\upsilon ��_��) = p_q (\upsilon ��_��)p_q$, is an operator of Berezin-Toeplitz type, acting in $p_q L^2(\mathbb{R}^2)$, and $p_q$ is the orthogonal projection on ${\rm Ker}\,(H_0 -��_q)$. If $\upsilon \neq 0$ and $q = 0$, we prove that ${\rm Ker}\,(T_0(\upsilon ��_��)) = \{0\}$. If $q \geq 1$, and $��= \mathcal{C}_r$ is a circle of radius $r$, we show that ${\rm dim \, Ker} (T_q(��_{\mathcal{C}_r})) \leq q$, and the set of $r \in (0,\infty)$ for which ${\rm dim \, Ker}(T_q(��_{\mathcal{C}_r})) \geq 1$, is infinite and discrete., 28 pages; to appear in Journal of Spectral Theory

In this paper we deal with spectral optimization for the Robin Laplacian on a family of planar domains admitting parallel coordinates, namely a fixed-width strip built over a smooth closed curve and the exterior of a convex set with a smooth boundary. We show that if the curve length is kept fixed, the first eigenvalue referring to the fixed-width strip is for any value of the Robin parameter maximized by a circular annulus. Furthermore, we prove that the second eigenvalue in the exterior of a convex domain $\Omega$ corresponding to a negative Robin parameter does not exceed the analogous quantity for a disk whose boundary has a curvature larger than or equal to the maximum of that for $\partial\Omega$., Comment: 14 pages, no figures, the title and the assumptions of the main theorem slightly modified in comparison with the first version. To appear in Mathematische Nachrichten

We obtain new Faber-Krahn-type inequalities for certain perturbations of the Dirichlet Laplacian on a bounded domain. First, we establish a two- and three-dimensional Faber-Krahn inequality for the Schr��dinger operator with point interaction: the optimiser is the ball with the point interaction supported at its centre. Next, we establish three-dimensional Faber-Krahn inequalities for one- and two-body Schr��dinger operator with attractive Coulomb interactions, the optimiser being given in terms of Coulomb attraction at the centre of the ball. The proofs of such results are based on symmetric decreasing rearrangement and Steiner rearrangement techniques; in the first model a careful analysis of certain monotonicity properties of the lowest eigenvalue is also needed., 27 pages

We consider the problem of geometric optimization of the lowest eigenvalue for the Laplacian on a compact, simply-connected two-dimensional manifold with boundary subject to an attractive Robin boundary condition. We prove that in the sub-class of manifolds with the Gauss curvature bounded from above by a constant $K_\circ \ge 0$ and under the constraint of fixed perimeter, the geodesic disk of constant curvature $K_\circ$ maximizes the lowest Robin eigenvalue. In the same geometric setting, it is proved that the spectral isoperimetric inequality holds for the lowest eigenvalue of the Dirichlet-to-Neumann operator. Finally, we adapt our methods to Robin Laplacians acting on unbounded three-dimensional cones to show that, under a constraint of fixed perimeter of the cross-section, the lowest Robin eigenvalue is maximized by the circular cone., Comment: 21 pages; references added, title changed, typos corrected; Lemma 3.8 corrected

In this article Dirac operators $A_{��, ��}$ coupled with combinations of electrostatic and Lorentz scalar $��$-shell interactions of constant strength $��$ and $��$, respectively, supported on compact surfaces $��\subset \mathbb{R}^3$ are studied. In the rigorous definition of these operators the $��$-potentials are modelled by coupling conditions at $��$. In the proof of the self-adjointness of $A_{��, ��}$ a Krein-type resolvent formula and a Birman-Schwinger principle are obtained. With their help a detailed study of the qualitative spectral properties of $A_{��, ��}$ is possible. In particular, the essential spectrum of $A_{��, ��}$ is determined, it is shown that at most finitely many discrete eigenvalues can appear, and several symmetry relations in the point spectrum are obtained. Moreover, the nonrelativistic limit of $A_{��, ��}$ is computed and it is discussed that for some special interaction strengths $A_{��, ��}$ is decoupled to two operators acting in the domains with the common boundary $��$., contribution to the proceedings of the conference "Advances in Operator Theory with Applications to Mathematical Physics" at the Chapman University, November 12-16, 2018; 20 pages

We perform quantitative spectral analysis of the self-adjoint Dirichlet Laplacian $\mathsf{H}$ on an unbounded, radially symmetric (generalized) parabolic layer $\mathcal{P}\subset\mathbb{R}^3$. It was known before that $\mathsf{H}$ has an infinite number of eigenvalues below the threshold of its essential spectrum. In the present paper, we find the discrete spectrum asymptotics for $\mathsf{H}$ by means of a consecutive reduction to the analogous asymptotic problem for an effective one-dimensional Schr\"odinger operator on the half-line with the potential the behaviour of which far away from the origin is determined by the geometry of the layer $\mathcal{P}$ at infinity., Comment: 25 pages

We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schr��dinger operator with an attractive $��'$-interaction of a fixed strength, the support of which is a $C^2$-smooth contour. Under the constraint of a fixed length of the contour, we prove that the lowest eigenvalue is maximized by the circle. The proof relies on the min-max principle and the method of parallel coordinates., contribution to the proceedings of the 3rd workshop "Mathematical Challenges of Zero-Range Physics: rigorous results and open problems"

The spectral properties of the singularly perturbed self-adjoint Landau Hamiltonian $A_��=(i \nabla + A)^2 + ����$ in $L^2(R^2)$ with a $��$-potential supported on a finite $C^{1,1}$-smooth curve $��$ are studied. Here $A = \frac{1}{2} B (-x_2, x_1)^\top$ is the vector potential, $B>0$ is the strength of the homogeneous magnetic field, and $��\in L^\infty(��)$ is a position-dependent real coefficient modeling the strength of the singular interaction on the curve $��$. After a general discussion of the qualitative spectral properties of $A_��$ and its resolvent, one of the main objectives in the present paper is a local spectral analysis of $A_��$ near the Landau levels $B(2q+1)$. Under various conditions on $��$ it is shown that the perturbation smears the Landau levels into eigenvalue clusters, and the accumulation rate of the eigenvalues within these clusters is determined in terms of the capacity of the support of $��$. Furthermore, the use of Landau Hamiltonians with $��$-perturbations as model operators for more realistic quantum systems is justified by showing that $A_��$ can be approximated in the norm resolvent sense by a family of Landau Hamiltonians with suitably scaled regular potentials., Merry Christmas

In this paper we consider the three-dimensional Schr��dinger operator with a $��$-interaction of strength $��> 0$ supported on an unbounded surface parametrized by the mapping $\mathbb{R}^2\ni x\mapsto (x,��f(x))$, where $��\in [0,\infty)$ and $f\colon \mathbb{R}^2\rightarrow\mathbb{R}$, $f\not\equiv 0$, is a $C^2$-smooth, compactly supported function. The surface supporting the interaction can be viewed as a local deformation of the plane. It is known that the essential spectrum of this Schr��dinger operator coincides with $[-\frac14��^2,+\infty)$. We prove that for all sufficiently small $��> 0$ its discrete spectrum is non-empty and consists of a unique simple eigenvalue. Moreover, we obtain an asymptotic expansion of this eigenvalue in the limit $��\rightarrow 0+$. In particular, this eigenvalue tends to $-\frac14��^2$ exponentially fast as $��\rightarrow 0+$., 21 pages, minor corrections, to appear in J. Math. Phys

The main objective of this paper is to systematically develop a spectral and scattering theory for selfadjoint Schr��dinger operators with $��$-interactions supported on closed curves in $\mathbb R^3$. We provide bounds for the number of negative eigenvalues depending on the geometry of the curve, prove an isoperimetric inequality for the principal eigenvalue, derive Schatten--von Neumann properties for the resolvent difference with the free Laplacian, and establish an explicit representation for the scattering matrix., to appear in Annales Henri Poincare

Let $��\subset\mathbb{R}^2$ be a non-closed piecewise-$C^1$ curve, which is either bounded with two free endpoints or unbounded with one free endpoint. Let $u_\pm|_��\in L^2(��)$ be the traces of a function $u$ in the Sobolev space $H^1({\mathbb R}^2\setminus ��)$ onto two faces of $��$. We prove that for a wide class of shapes of $��$ the Schr��dinger operator $\mathsf{H}_��^��$ with $��^\prime$-interaction supported on $��$ of strength $��\in L^\infty(��;\mathbb{R})$ associated with the quadratic form \[ H^1(\mathbb{R}^2\setminus��)\ni u \mapsto \int_{\mathbb{R}^2}\big|\nabla u \big|^2 \mathsf{d} x - \int_����\big| u_+|_��- u_-|_��\big|^2 \mathsf{d} s \] has no negative spectrum provided that $��$ is pointwise majorized by a strictly positive function explicitly expressed in terms of $��$. If, additionally, the domain $\mathbb{R}^2\setminus��$ is quasi-conical, we show that $��(\mathsf{H}_��^��) = [0,+\infty)$. For a bounded curve $��$ in our class and non-varying interaction strength $��\in\mathbb{R}$ we derive existence of a constant $��_* > 0$ such that $��(\mathsf{H}_��^��) = [0,+\infty)$ for all $��\in (-\infty, ��_*]$; informally speaking, bound states are absent in the weak coupling regime., 22 pages, 2 figures

We show that a Schr��dinger operator $A_{��, ��}$ with a $��$-interaction of strength $��$ supported on a bounded or unbounded $C^2$-hypersurface $��\subset \mathbb{R}^d$, $d\ge 2$, can be approximated in the norm resolvent sense by a family of Hamiltonians with suitably scaled regular potentials. The differential operator $A_{��, ��}$ with a singular interaction is regarded as a self-adjoint realization of the formal differential expression $-��- ��\langle ��_��, \cdot \rangle ��_��$, where $��\colon��\rightarrow \mathbb{R}$ is an arbitrary bounded measurable function. We discuss also some spectral consequences of this approximation result.

Let $��\subset\mathbb{R}^d$ be a $C^\infty$-smooth closed compact hypersurface, which splits the Euclidean space $\mathbb{R}^d$ into two domains $��_\pm$. In this note self-adjoint Schr��dinger operators with $��$ and $��'$-interactions supported on $��$ are studied. For large enough $m\in\mathbb{N}$ the difference of $m$th powers of resolvents of such a Schr��dinger operator and the free Laplacian is known to belong to the trace class. We prove trace formulae, in which the trace of the resolvent power difference in $L^2(\mathbb{R}^d)$ is written in terms of Neumann-to-Dirichlet maps on the boundary space $L^2(��)$.

We consider self-adjoint Schr��dinger operators in $L^2 (\mathbb{R}^d)$ with a $��$-interaction of strength $��$ and a $��'$-interaction of strength $��$, respectively, supported on a hypersurface, where $��$ and $��^{-1}$ are bounded, real-valued functions. It is known that the inequality $0 < ��\leq 4/��$ implies inequality of the eigenvalues of these two operators below the bottoms of the essential spectra. We show that this eigenvalue inequality is strict whenever $��< 4 / ��$ on a nonempty, open subset of the hypersurface. Moreover, we point out special geometries of the interaction support, such as broken lines or infinite cones, for which strict inequality of the eigenvalues even holds in the borderline case $��= 4 / ��$.

We consider a self-adjoint two-dimensional Schr��dinger operator $H_{����}$, which corresponds to the formal differential expression \[ -��- ����, \] where $��$ is a finite compactly supported positive Radon measure on ${\mathbb R}^2$ from the generalized Kato class and $��>0$ is the coupling constant. It was proven earlier that $��_{\rm ess}(H_{����}) = [0,+\infty)$. We show that for sufficiently small $��$ the condition $\sharp��_{\rm d}(H_{����}) = 1$ holds and that the corresponding unique eigenvalue has the asymptotic expansion $$ ��(��) = -(C_��+ o(1))\exp\Big(-\tfrac{4��}{����({\mathbb R}^2)}\Big), \qquad ��\rightarrow 0+, $$ with a certain constant $C_��> 0$. We obtain also the formula for the computation of $C_��$. The asymptotic expansion of the corresponding eigenfunction is provided. The statements of this paper extend Simon's results, see \cite{Si76}, to the case of potentials-measures. Also for regular potentials our results are partially new.

We investigate the spectral properties of self-adjoint Schr��dinger operators with attractive $��$-interactions of constant strength $��> 0$ supported on conical surfaces in ${\mathbb R}^3$. It is shown that the essential spectrum is given by $[-��^2/4,+\infty)$ and that the discrete spectrum is infinite and accumulates to $-��^2/4$. Furthermore, an asymptotic estimate of these eigenvalues is obtained.

We consider self-adjoint realizations of a second-order elliptic differential expression on ${\mathbb R}^n$ with singular interactions of $��$ and $��^\prime$-type supported on a compact closed smooth hypersurface in ${\mathbb R}^n$. In our main results we prove spectral asymptotics formulae with refined remainder estimates for the singular values of the resolvent difference between the standard self-adjoint realizations and the operators with a $��$ and $��^\prime$-interaction, respectively. Our technique makes use of general pseudodifferential methods, classical results on spectral asymptotics of $��$do's on closed manifolds and Krein-type resolvent formulae., to appear in J. Spectr. Theory

We investigate Schr��dinger operators with ��- and ��'-interactions supported on hypersurfaces, which separate the Euclidean space into finitely many bounded and unbounded Lipschitz domains. It turns out that the combinatorial properties of the partition and the spectral properties of the corresponding operators are related. As the main result we prove an operator inequality for the Schr��dinger operators with ��- and ��'-interactions which is based on an optimal colouring and involves the chromatic number of the partition. This inequality implies various relations for the spectra of the Schr��dinger operators and, in particular, it allows to transform known results for Schr��dinger operators with ��-interactions to Schr��dinger operators with ��'-interactions.

Self-adjoint Schr��dinger operators with $��$ and $��'$-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman--Schwinger principle and a variant of Krein's formula are shown. Furthermore, Schatten--von Neumann type estimates for the differences of the powers of the resolvents of the Schr��dinger operators with $��$ and $��'$-potentials, and the Schr��dinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schr��dinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity., to appear in Annales Henri Poincar\'e, 34 pages

The spectrum of the self-adjoint Schr\"odinger operator associated with the Kronig-Penney model on the half-line has a band-gap structure: its absolutely continuous spectrum consists of intervals (bands) separated by gaps. We show that if one changes strengths of interactions or locations of interaction centers by adding an oscillating and slowly decaying sequence which resembles the classical Wigner-von Neumann potential, then this structure of the absolutely continuous spectrum is preserved. At the same time in each spectral band precisely two critical points appear. At these points "instable" embedded eigenvalues may exist. We obtain locations of the critical points and discuss for each of them the possibility of an embedded eigenvalue to appear. We also show that the spectrum in gaps remains discrete., Comment: updated

We consider the Schr��dinger operator in the plane with delta-potential supported by a curve. For the cases of an infinite curve and a finite loop we give estimates on the lower bound of the spectrum expressed explicitly through the strength of the interaction and a parameter which characterizes geometry of the curve. Going further we cut the curve into finite number of pieces and estimate the bottom of the spectrum using the parameters for the pieces. As an application of the elaborated theory we consider a curve with a finite number of cusps and "leaky" quantum graphs.

We consider the four-component two-valley Dirac operator on a wedge in $\mathbb{R}^2$ with infinite mass boundary conditions, which enjoy a flip at the vertex. We show that it has deficiency indices $(1,1)$ and we parametrize all its self-adjoint extensions, relying on the fact that the underlying two-component Dirac operator is symmetric with deficiency indices $(0,1)$. The respective defect element is computed explicitly. We observe that there exists no self-adjoint extension, which can be decomposed into an orthogonal sum of two two-component operators. In physics, this effect is called mixing the valleys.