By Nicolas Raymond
This publication is a synthesis of modern advances within the spectral conception of the magnetic Schrödinger operator. it may be thought of a catalog of concrete examples of magnetic spectral asymptotics.
Since the presentation contains many notions of spectral idea and semiclassical research, it starts with a concise account of thoughts and techniques utilized in the booklet and is illustrated through many uncomplicated examples.
Assuming numerous issues of view (power sequence expansions, Feshbach–Grushin discounts, WKB structures, coherent states decompositions, common types) a thought of Magnetic Harmonic Approximation is then demonstrated which permits, particularly, exact descriptions of the magnetic eigenvalues and eigenfunctions. a few elements of this conception, equivalent to these relating to spectral rate reductions or waveguides, are nonetheless obtainable to complex scholars whereas others (e.g., the dialogue of the Birkhoff general shape and its spectral outcomes, or the consequences relating to boundary magnetic wells in size 3) are meant for pro researchers.
Keywords: Magnetic Schrödinger equation, discrete spectrum, semiclassical research, magnetic harmonic approximation
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Extra info for Bound States of the Magnetic Schrödinger Operator
0; 1/. Let us prove that Zn Ä n 1. x/ D 0 elsewhere. It is clear that these functions belong to the form domain of L and that they form an orthogonal family. 0;1// : By the min-max principle, we get nC1 Ä n and this contradicts the simplicity of the eigenvalues. Let us now prove that Zn Zn 1 C 1. z0 ; z1 /. Indeed, this would imply that un vanishes at least Zn 1 C 1 times. un 1 ; un / 0 D. z0 ; z1 /, and so does un . The conclusion follows easily. 33. ˛/ its first eigenvalue. 0/ D 0 : 44 1 Elements of spectral theory The aim of this section is to establish the following lemma.
H is defined by R ˛ D m j D1 ˛j kj and RC W H ! hu; kj i/1Äj Än . Then, M W H Cm ! H Cn is bijective. H/ with kPk Ä "0 , Ã Â MCP R ; RC 0 is bijective. E0 / D n only if E0 is bijective. m and M C P is bijective if and Proof. We leave the proof of the bijectivity of M to the reader. By using a Neumann series, we can easily prove that Â Ã MCP R ; RC 0 is bijective when P is small enough. M C P/ C E0 RC D 0 ER D 0 : 36 1 Elements of spectral theory From this, we get that RC and E are surjective and that R and EC are injective.
This would be clear if there would exist a compactly supported vector potential associated with B. However, this is not the case in general (note that the naive considerations of this section are related to scattering theory in presence of magnetic fields, see the book by Yafaev ).